900:
52:
934:
928:
922:
916:
1337:
or sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type, or both. The Tits group is the
894:
in the 1860s and the other twenty-one were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list
1406:, p. 504) where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a closer examination than they have yet received." (At the time, the other sporadic groups had not been discovered.)
4890:
3437:
2641:
2279:
5033:
4653:
3891:
1992:
4510:
3749:
2132:
3288:
3074:
5179:"The sporadic simple groups may be roughly sorted as the Mathieu groups, the Leech lattice groups, Fischer's 3-transposition groups, the further Monster centralizers, and the half-dozen oddments."
2498:
2851:
2748:
2387:
4751:
4378:
4084:
5272:
5340:
4280:
3530:
5140:
4182:
3168:
2955:
5392:
3984:
3619:
497:
472:
435:
1933:
5517:"The finite simple groups are the building blocks of finite group theory. Most fall into a few infinite families of groups, but there are 26 (or 27 if the Tits group
5476:"This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group)."
5194:
from standard generators of each sporadic group. Most sporadic groups have multiple presentations & semi-presentations; the more prominent examples are listed.
799:
5943:
6349:
6584:
813:
357:
6538:
6022:
5890:
307:
4795:
3335:
2542:
2180:
4934:
4554:
3792:
6244:
792:
302:
859:
infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. The
6626:
6143:
5742:
1944:
4424:
3669:
2052:
6686:
4309:= 2·3·5·7·11·19 ≈ 2
3755:
1116:
718:
3212:
3001:
6006:
5952:
3450:
2761:
2431:
2014:
1302:
785:
6740:
3908:
2647:
1208:
852:
6358:
2791:
402:
216:
4394:
2691:
2330:
891:
6062:
4697:
4324:
4030:
1311:≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in
5716:
4919:= 2·3·5·11 ≈ 8
4780:= 2·3·5·11 ≈ 1
4682:= 2·3·5·7·11 ≈ 4
4539:= 2·3·5·7·11·23 ≈ 1
4117:= 2·3·5·17·19 ≈ 5
3443:
1154:
5160:
The groups of prime order, the alternating groups of degree at least 5, the infinite family of commutator groups F
4409:= 2·3·5·7·11·23 ≈ 2
6408:
5191:
6735:
911:. A connecting line means the lower group is subquotient of the upper – and no sporadic subquotient in between.
600:
334:
211:
99:
5206:
6530:
5277:
4230:
3480:
5090:
4132:
3118:
2905:
5877:
1439:
750:
540:
6345:"The Minimal Degrees of Faithful Representations of the Sporadic Simple Groups and their Covering Groups"
6192:
3762:
624:
4215:= 2·3·5·7 ≈ 6
3923:= 2·3·5·7·11 ≈ 4
3777:= 2·3·5·7·11 ≈ 9
3320:= 2·3·5·7·11·19·31 ≈ 5
2776:= 2·3·5·7·11·31·37·67 ≈ 5
1413:, p. 247). It does not show the numerous non-sporadic simple subquotients of the sporadic groups.
6603:
5075:= 2·3·5·13 ≈ 2
4015:= 2·3·5·7·11·23·29·31·37·43 ≈ 9
3558:= 2·3·5·7·13·29 ≈ 1
3465:= 2·3·5·7·11·13 ≈ 4
3197:= 2·3·5·7·11·23 ≈ 5
3103:= 2·3·5·7·11·23 ≈ 4
2890:= 2·3·5·7·11·19 ≈ 3
2527:= 2·3·5·7·11·13 ≈ 6
6670:
6599:
6464:
6074:
5872:
5794:
5345:
2416:= 2·3·5·7·11·13·17·23 ≈ 4
1885:
564:
552:
170:
104:
3938:
3573:
6476:
6460:
6417:
2857:
1356:
1323:
1178:
817:
139:
34:
5628:
Tabelle 2. Die
Entdeckung der sporadischen Gruppen (Table 2. The discovery of the sporadic groups)
480:
455:
418:
6648:
6496:
6437:
6380:
6321:
6222:
6108:
6044:
5974:
5912:
5860:
5838:
5778:
5764:
2971:
2504:
2393:
2285:
2138:
1513:
1319:
1298:
1290:
1230:
1091:
1082:
1073:
124:
96:
6703:
6674:
6640:
6622:
6552:
6534:
6259:
6206:
6157:
6139:
6036:
6018:
5996:
5934:
5904:
5886:
5868:
5830:
5756:
5738:
5164:(2)′ of groups of Lie type (containing the Tits group), and 15 families of groups of Lie type.
4896:
4757:
4659:
4516:
4384:
3897:
1897:
1475:
1334:
1108:
981:
972:
963:
954:
945:
864:
695:
529:
372:
266:
6656:
6614:
6560:
6504:
6480:
6445:
6421:
6403:
6388:
6362:
6329:
6303:
6275:
6230:
6196:
6165:
6116:
6090:
6082:
6010:
5982:
5956:
5920:
5846:
5820:
5802:
5728:
5720:
5706:
3904:
3305:
3174:
3080:
2961:
2661:
1286:
1282:
1061:
1052:
1043:
680:
672:
664:
656:
648:
636:
576:
516:
506:
348:
290:
165:
134:
6636:
6548:
6492:
6433:
6376:
6317:
6271:
6218:
6153:
6104:
6032:
5970:
5900:
5816:
5752:
899:
6660:
6632:
6589:
6564:
6544:
6508:
6488:
6449:
6429:
6392:
6372:
6333:
6313:
6279:
6267:
6252:
Jahresber. Deutsch. Math.-Verein. (Annual Report of the German
Mathematicians Association)
6234:
6214:
6169:
6149:
6135:
6120:
6100:
6028:
5986:
5966:
5924:
5896:
5882:
5864:
5850:
5812:
5748:
3654:= 2·3·5·7·17 ≈ 4
3543:
2986:= 2·3·5·7·11·13·23 ≈ 4
2875:
2871:
1634:
1278:
1274:
878:, is the largest of the sporadic groups, and all but six of the other sporadic groups are
844:
764:
757:
743:
700:
588:
511:
341:
255:
195:
75:
2676:= 2·3·5·7·13·19·31 ≈ 9
2315:= 2·3·5·7·11·13·17·23·29 ≈ 1
2165:= 2·3·5·7·11·13·17·19·23·31·47 ≈ 4
1746:, if regarded as a sporadic group, would belong in this generation: there is a subgroup S
6078:
5798:
6523:
4286:
4188:
4090:
4000:
3990:
3536:
3294:
1807:, and has no involvements in sporadic simple groups except the ones already mentioned.
1459:
1435:
1169:
1146:
1029:
1016:
1003:
994:
771:
707:
397:
314:
279:
200:
190:
175:
160:
114:
91:
5825:
5782:
1586:
is the group of automorphisms preserving a quaternionic structure (modulo its center).
6729:
6441:
6384:
6112:
6058:
6048:
5978:
5916:
5768:
2000:
1521:
1503:
1499:
1447:
1427:
1253:
1070:
942:
871:
690:
612:
446:
319:
185:
6652:
6500:
6325:
6226:
5961:
5938:
5842:
6706:
6484:
6180:
5878:
ATLAS of Finite Groups: Maximal
Subgroups and Ordinary Characters for Simple Groups
5783:"A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups"
5060:
1855:
1816:
1039:
836:
545:
244:
233:
180:
155:
150:
109:
80:
43:
6618:
5939:"The classification of finite simple groups. I. Simple groups and local analysis"
6720:
6574:
3639:
2754:
1614:
1565:
is the group of automorphisms preserving a complex structure (modulo its center)
1517:
1509:
1487:
1200:
991:
904:
879:
17:
6518:
6367:
6344:
6040:
6014:
5039:
3625:
1743:
1327:
1124:
860:
712:
440:
6644:
6556:
6263:
6210:
5733:
6711:
6201:
6161:
5908:
5760:
5724:
5045:
1386:
856:
533:
6308:
6291:
5834:
6425:
5807:
1629:′ has a triple cover which is the centralizer of an element of order 3 in
1301:
and semi-presentations. The degrees of minimal faithful representation or
933:
927:
921:
915:
51:
6678:
6095:
1724:
Finally, the
Monster group itself is considered to be in this generation.
1431:
70:
1735:
and a group of order 11 is the centralizer of an element of order 11 in
6611:
The Atlas of Finite Groups - Ten Years On (LMS Lecture Note Series 249)
6404:"Smallest degrees of representations of exceptional groups of Lie type"
6086:
6005:. Mathematical Surveys and Monographs. Vol. 40. Providence, R.I.:
6000:
5710:
863:
is sometimes regarded as a sporadic group because it is not strictly a
412:
326:
1717:
and a group of order 7 is the centralizer of an element of order 7 in
1700:
and a group of order 5 is the centralizer of an element of order 5 in
1682:
and a group of order 3 is the centralizer of an element of order 3 in
2018:
1595:
Consists of subgroups which are closely related to the
Monster group
27:
Finite simple group type not classified as Lie, cyclic or alternating
6525:
Symmetry and the
Monster: One of the Greatest Quests of Mathematics
4885:{\displaystyle o{\bigl (}{\bigr )}=o{\bigl (}ababab^{2}{\bigr )}=6}
3432:{\displaystyle o{\bigl (}abab(b^{2}(b^{2})^{abab})^{5}{\bigr )}=5}
2636:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=12}
2274:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=23}
5028:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=4}
4648:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=8}
3886:{\displaystyle o{\bigl (}(ab)^{2}(abab^{2})^{2}ab^{2}{\bigr )}=7}
6613:. Cambridge, U.K: Cambridge University Press. pp. 261–273.
1375:
these finite simple groups coincide with the groups of Lie type
820:
which do not fit into any infinite family. These are called the
5527:
is counted also) which these infinite families do not include."
5176:, p. viii) organizes the 26 sporadic groups in likeness:
1269:
Various constructions for these groups were first compiled in
851:
itself. The mentioned classification theorem states that the
1987:{\displaystyle \langle \langle a,b\mid o(z)\rangle \rangle }
1534:
is the quotient of the automorphism group by its center {±1}
1368:; thus in a strict sense not sporadic, nor of Lie type. For
4505:{\displaystyle o{\bigl (}ab(abab^{2})^{2}ab^{2}{\bigr )}=4}
1591:
Third generation (8 groups): other subgroups of the
Monster
6578:
3744:{\displaystyle o{\bigl (}ab^{2}abab^{2}ab^{2}{\bigr )}=10}
2127:{\displaystyle o{\bigl (}(ab)^{4}(ab^{2})^{2}{\bigr )}=50}
1803:(2)′ is also a subquotient of the (pariah) Rudvalis group
5677:
5538:
1294:
6002:
The classification of the finite simple groups, Number 3
5499:
5419:
5173:
1270:
5541:, Sporadic groups & Exceptional groups of Lie type)
6721:
6292:"Matrix generators for exceptional groups of Lie type"
6290:
Howlett, R. B.; Rylands, L. J.; Taylor, D. E. (2001).
5443:
3283:{\displaystyle o{\bigl (}(uvv)^{3}(uv)^{6}{\bigr )}=5}
3069:{\displaystyle o{\bigl (}ab(abab^{2})^{2}{\bigr )}=42}
6604:"Chapter: An Atlas of Sporadic Group Representations"
5348:
5280:
5209:
5093:
4937:
4798:
4700:
4557:
4427:
4327:
4233:
4135:
4033:
3941:
3795:
3672:
3576:
3483:
3338:
3215:
3121:
3004:
2908:
2794:
2694:
2545:
2434:
2333:
2183:
2055:
1947:
1900:
1543:
is the stabilizer of a type 2 (i.e., length 2) vector
1426:
Of the 26 sporadic groups, 20 can be seen inside the
483:
458:
421:
5470:
2493:{\displaystyle o{\bigl (}a^{bb}(ab)^{14}{\bigr )}=5}
1862:
Table of the sporadic group orders (with Tits group)
6465:"Semi-Presentations for the Sporadic Simple Groups"
6195:Department of Mathematics and Statistics: 106−118,
867:, in which case there would be 27 sporadic groups.
6522:
5386:
5334:
5266:
5134:
5027:
4884:
4745:
4647:
4504:
4372:
4274:
4176:
4078:
3978:
3885:
3743:
3613:
3524:
3431:
3282:
3162:
3068:
2949:
2845:
2742:
2635:
2492:
2381:
2273:
2126:
1986:
1927:
491:
466:
429:
6245:"Die Sporadischen Gruppen (The Sporadic Groups)"
2846:{\displaystyle o{\bigl (}ababab^{2}{\bigr )}=67}
1774:(2)′ is also a subquotient of the Fischer group
1474:= 11, 12, 22, 23 and 24 are multiply transitive
6181:"Polytopes Derived from Sporadic Simple Groups"
6179:Hartley, Michael I.; Hulpke, Alexander (2010),
5662:
2743:{\displaystyle o{\bigl (}(ab)^{3}b{\bigr )}=21}
2382:{\displaystyle o{\bigl (}(ab)^{3}b{\bigr )}=33}
1728:(This series continues further: the product of
1494:Second generation (7 groups): the Leech lattice
1454:First generation (5 groups): the Mathieu groups
1450:, and can be organized into three generations.
890:Five of the sporadic groups were discovered by
6134:. Springer Monographs in Mathematics. Berlin:
4746:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=11}
4373:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=19}
4079:{\displaystyle o{\bigl (}abab^{2}{\bigr )}=10}
5944:Bulletin of the American Mathematical Society
5673:
5671:
5511:
5121:
5099:
5014:
4943:
4871:
4839:
4826:
4804:
4732:
4706:
4634:
4563:
4491:
4433:
4359:
4333:
4261:
4239:
4163:
4141:
4065:
4039:
3872:
3801:
3730:
3678:
3511:
3489:
3418:
3344:
3269:
3221:
3149:
3127:
3055:
3010:
2936:
2914:
2832:
2800:
2729:
2700:
2622:
2551:
2479:
2440:
2368:
2339:
2260:
2189:
2163:4,154,781,481,226,426,191,177,580,544,000,000
2113:
2061:
793:
8:
6579:"Orders of sporadic simple groups (A001228)"
2031:808,017,424,794,512,875,886,459,904,961,710,
1981:
1978:
1951:
1948:
1891:Degree of minimal faithful Brauer character
1552:is the stabilizer of a type 3 (i.e., length
5610:
5598:
5586:
5574:
5562:
5550:
5431:
6350:LMS Journal of Computation and Mathematics
5999:; Lyons, Richard; Solomon, Ronald (1998).
5487:
1865:
1662:has a double cover which is a subgroup of
1577:is the stabilizer of a type 2-3-3 triangle
1571:is the stabilizer of a type 2-2-3 triangle
800:
786:
238:
64:
29:
6585:On-Line Encyclopedia of Integer Sequences
6366:
6307:
6200:
6094:
5960:
5824:
5806:
5732:
5378:
5368:
5347:
5326:
5310:
5297:
5279:
5258:
5236:
5223:
5208:
5120:
5119:
5098:
5097:
5092:
5013:
5012:
5006:
4993:
4983:
4961:
4942:
4941:
4936:
4870:
4869:
4863:
4838:
4837:
4825:
4824:
4803:
4802:
4797:
4731:
4730:
4724:
4705:
4704:
4699:
4633:
4632:
4626:
4613:
4603:
4581:
4562:
4561:
4556:
4490:
4489:
4483:
4470:
4460:
4432:
4431:
4426:
4358:
4357:
4351:
4332:
4331:
4326:
4260:
4259:
4238:
4237:
4232:
4162:
4161:
4140:
4139:
4134:
4064:
4063:
4057:
4038:
4037:
4032:
3961:
3940:
3871:
3870:
3864:
3851:
3841:
3819:
3800:
3799:
3794:
3729:
3728:
3722:
3709:
3690:
3677:
3676:
3671:
3596:
3575:
3510:
3509:
3488:
3487:
3482:
3417:
3416:
3410:
3391:
3381:
3368:
3343:
3342:
3337:
3268:
3267:
3261:
3242:
3220:
3219:
3214:
3148:
3147:
3126:
3125:
3120:
3054:
3053:
3047:
3037:
3009:
3008:
3003:
2935:
2934:
2913:
2912:
2907:
2831:
2830:
2824:
2799:
2798:
2793:
2728:
2727:
2718:
2699:
2698:
2693:
2621:
2620:
2614:
2601:
2591:
2569:
2550:
2549:
2544:
2478:
2477:
2471:
2449:
2439:
2438:
2433:
2367:
2366:
2357:
2338:
2337:
2332:
2259:
2258:
2252:
2239:
2229:
2207:
2188:
2187:
2182:
2112:
2111:
2105:
2095:
2079:
2060:
2059:
2054:
1946:
1899:
485:
484:
482:
460:
459:
457:
423:
422:
420:
5415:
5413:
5411:
1754:(2)′ normalising a 2C subgroup of
1403:
898:
5454:
5452:
5407:
5267:{\displaystyle u=(b^{2}(b^{2})abb)^{3}}
5153:
356:
122:
32:
6677:; Nickerson, S.J.; Bray, J.N. (1999).
5688:
5650:
5639:
5335:{\displaystyle v=t(b^{2}(b^{2})t)^{2}}
4275:{\displaystyle o{\bigl (}{\bigr )}=12}
3525:{\displaystyle o{\bigl (}{\bigr )}=15}
1312:
814:classification of finite simple groups
358:Classification of finite simple groups
6683:ATLAS of Finite Group Representations
6185:Contributions to Discrete Mathematics
5458:
5444:Gorenstein, Lyons & Solomon (1998
5135:{\displaystyle o{\bigl (}{\bigr )}=5}
4177:{\displaystyle o{\bigl (}{\bigr )}=9}
3163:{\displaystyle o{\bigl (}{\bigr )}=4}
2950:{\displaystyle o{\bigl (}{\bigr )}=5}
1442:). These twenty have been called the
1410:
7:
5622:
1868:
5471:Howlett, Rylands & Taylor (2001
2035:= 2·3·5·7·11·13·17·19·23·29·31
1482:points. They are all subgroups of M
1333:, which is sometimes considered of
1486:, which is a permutation group on
1293:. These are also listed online at
913:The generations of Robert Griess:
25:
2313:1,255,205,709,190,661,721,292,800
1409:The diagram at right is based on
847:except for the trivial group and
6679:"ATLAS of Group Representations"
5712:Theory of groups of finite order
1613:has a double cover which is the
932:
926:
920:
914:
50:
6687:Queen Mary University of London
6296:Journal of Symbolic Computation
6130:Griess, Jr., Robert L. (1998).
5962:10.1090/S0273-0979-1979-14551-8
5387:{\displaystyle t=abab^{3}a^{2}}
1758:, giving rise to a subgroup 2·S
6485:10.1080/10586458.2005.10128927
5323:
5316:
5303:
5290:
5255:
5242:
5229:
5216:
5116:
5104:
4990:
4967:
4958:
4948:
4821:
4809:
4610:
4587:
4578:
4568:
4467:
4444:
4256:
4244:
4158:
4146:
3979:{\displaystyle o(abab^{2})=15}
3967:
3945:
3848:
3825:
3816:
3806:
3614:{\displaystyle o(abab^{2})=29}
3602:
3580:
3506:
3494:
3407:
3388:
3374:
3361:
3258:
3248:
3239:
3226:
3144:
3132:
3044:
3021:
2931:
2919:
2715:
2705:
2598:
2575:
2566:
2556:
2468:
2458:
2354:
2344:
2236:
2213:
2204:
2194:
2102:
2085:
2076:
2066:
1975:
1969:
1922:
1901:
1307:over fields of characteristic
719:Infinite dimensional Lie group
1:
6007:American Mathematical Society
5953:American Mathematical Society
1398:The earliest use of the term
6619:10.1017/CBO9780511565830.024
5787:Proc. Natl. Acad. Sci. U.S.A
5663:Nickerson & Wilson (2011
5081:
4925:
4786:
4688:
4545:
4415:
4315:
4221:
4123:
4021:
3929:
3783:
3660:
3564:
3471:
3326:
3203:
3109:
2992:
2896:
2782:
2682:
2533:
2422:
2321:
2171:
2043:
1766:(2)′ normalising a certain Q
1617:of an element of order 2 in
853:list of finite simple groups
492:{\displaystyle \mathbb {Z} }
467:{\displaystyle \mathbb {Z} }
430:{\displaystyle \mathbb {Z} }
6359:London Mathematical Society
5715:(2nd ed.). Cambridge:
5072:
4916:
4777:
4679:
4536:
4406:
4306:
4212:
4114:
4012:
3920:
3774:
3651:
3555:
3462:
3317:
3194:
3100:
2983:
2887:
2773:
2673:
2524:
2413:
2312:
2162:
2033:757,005,754,368,000,000,000
2030:
1889:
1795:′, and of the Baby Monster
1297:, updated with their group
217:List of group theory topics
6757:
6343:Jansen, Christoph (2005).
5885:. pp. xxxiii, 1–252.
5717:Cambridge University Press
5512:Hartley & Hulpke (2010
4013:86,775,571,046,077,562,880
1814:
1770:subgroup of the Monster. F
1497:
1457:
1355:of the infinite family of
6409:Communications in Algebra
6368:10.1112/S1461157000000930
6015:10.1112/S0024609398255457
2984:4,157,776,806,543,360,000
2414:4,089,470,473,293,004,800
1854:, sometimes known as the
1686:(in conjugacy class "3C")
1322:in the classification of
907:relations between the 26
6469:Experimental Mathematics
6067:Inventiones Mathematicae
6009:. pp. xiii, 1–362.
5719:. pp. xxiv, 1–512.
5613:, pp. 146, 150−152)
1928:{\displaystyle (a,b,ab)}
816:, there are a number of
335:Elementary abelian group
212:Glossary of group theory
6533:. pp. vii, 1–255.
6531:Oxford University Press
6416:(5). Philadelphia, PA:
6202:10.11575/cdm.v5i2.61945
1821:The six exceptions are
843:that does not have any
6402:Lubeck, Frank (2001).
6309:10.1006/jsco.2000.0431
6243:Hiss, Gerhard (2003).
6132:Twelve Sporadic Groups
6059:Griess, Jr., Robert L.
5388:
5336:
5268:
5136:
5029:
4886:
4747:
4649:
4506:
4374:
4276:
4178:
4080:
3980:
3887:
3745:
3615:
3526:
3433:
3284:
3164:
3070:
2951:
2847:
2774:51,765,179,004,000,000
2744:
2674:90,745,943,887,872,000
2637:
2494:
2383:
2275:
2128:
1988:
1929:
1520:dimensions called the
938:
903:The diagram shows the
826:sporadic finite groups
822:sporadic simple groups
751:Linear algebraic group
493:
468:
431:
6426:10.1081/AGB-100002175
6193:University of Calgary
5808:10.1073/pnas.61.2.398
5725:10.1112/PLMS/S2-7.1.1
5389:
5337:
5269:
5137:
5030:
4887:
4748:
4650:
4507:
4375:
4277:
4179:
4081:
3981:
3888:
3746:
3616:
3527:
3434:
3285:
3165:
3071:
2952:
2848:
2745:
2638:
2495:
2384:
2276:
2129:
2037:·41·47·59·71 ≈ 8
1989:
1930:
902:
494:
469:
432:
6477:Taylor & Francis
6418:Taylor & Francis
6063:"The Friendly Giant"
5678:Wilson et al. (1999)
5346:
5278:
5207:
5091:
4935:
4796:
4698:
4555:
4425:
4325:
4231:
4133:
4031:
3939:
3793:
3670:
3574:
3481:
3336:
3213:
3119:
3002:
2906:
2792:
2692:
2543:
2432:
2331:
2181:
2053:
1945:
1898:
1324:finite simple groups
1295:Wilson et al. (1999)
1289:and orders of their
1271:Conway et al. (1985)
812:In the mathematical
481:
456:
419:
6741:Mathematical tables
6079:1982InMat..69....1G
5799:1968PNAS...61..398C
5653:, pp. 122–123)
5589:, pp. 146−150)
5577:, pp. 104–145)
5539:Wilson et al. (1999
5500:Conway et al. (1985
5461:, pp. 244–246)
5446:, pp. 262–302)
5420:Conway et al. (1985
5174:Conway et al. (1985
2888:273,030,912,000,000
1781:, and thus also of
1387:Ree groups of type
1291:outer automorphisms
1179:Harada-Norton group
125:Group homomorphisms
35:Algebraic structure
6704:Weisstein, Eric W.
6475:(3). Oxfordshire:
6191:(2), Alberta, CA:
6138:. pp. 1−169.
6087:10.1007/BF01389186
5384:
5332:
5264:
5192:semi-presentations
5132:
5025:
4882:
4743:
4645:
4502:
4370:
4272:
4174:
4076:
3976:
3883:
3741:
3611:
3522:
3429:
3280:
3160:
3101:42,305,421,312,000
3066:
2947:
2843:
2740:
2633:
2525:64,561,751,654,400
2490:
2379:
2271:
2124:
1995:Semi-presentation
1984:
1925:
1514:automorphism group
1476:permutation groups
1231:Baby Monster group
939:
601:Special orthogonal
489:
464:
427:
308:Lagrange's theorem
6540:978-0-19-280722-9
6459:Nickerson, S.J.;
6024:978-0-8218-0391-2
5892:978-0-19-853199-9
5863:; Curtis, R. T.;
5734:2027/uc1.b4062919
5707:Burnside, William
5611:Griess, Jr. (1998
5601:, pp. 91−96)
5599:Griess, Jr. (1982
5587:Griess, Jr. (1998
5575:Griess, Jr. (1998
5565:, pp. 54–79)
5563:Griess, Jr. (1998
5551:Griess, Jr. (1982
5432:Griess, Jr. (1998
5145:
5144:
5056:
1646:is a subgroup of
1357:commutator groups
1304:Brauer characters
1287:Schur multipliers
1279:conjugacy classes
1109:Higman-Sims group
865:group of Lie type
810:
809:
385:
384:
267:Alternating group
224:
223:
16:(Redirected from
6748:
6717:
6716:
6707:"Sporadic Group"
6690:
6664:
6608:
6593:
6575:Sloane, N. J. A.
6568:
6528:
6512:
6453:
6396:
6370:
6337:
6311:
6283:
6249:
6237:
6204:
6173:
6124:
6098:
6052:
5990:
5964:
5928:
5854:
5828:
5810:
5772:
5736:
5692:
5686:
5680:
5675:
5666:
5660:
5654:
5648:
5642:
5637:
5631:
5620:
5614:
5608:
5602:
5596:
5590:
5584:
5578:
5572:
5566:
5560:
5554:
5548:
5542:
5536:
5530:
5526:
5509:
5503:
5497:
5491:
5488:Gorenstein (1979
5485:
5479:
5468:
5462:
5456:
5447:
5441:
5435:
5429:
5423:
5417:
5395:
5393:
5391:
5390:
5385:
5383:
5382:
5373:
5372:
5341:
5339:
5338:
5333:
5331:
5330:
5315:
5314:
5302:
5301:
5273:
5271:
5270:
5265:
5263:
5262:
5241:
5240:
5228:
5227:
5201:
5195:
5190:Here listed are
5188:
5182:
5171:
5165:
5158:
5141:
5139:
5138:
5133:
5125:
5124:
5103:
5102:
5078:
5070:
5055:
5046:
5034:
5032:
5031:
5026:
5018:
5017:
5011:
5010:
4998:
4997:
4988:
4987:
4966:
4965:
4947:
4946:
4922:
4914:
4891:
4889:
4888:
4883:
4875:
4874:
4868:
4867:
4843:
4842:
4830:
4829:
4808:
4807:
4783:
4775:
4752:
4750:
4749:
4744:
4736:
4735:
4729:
4728:
4710:
4709:
4685:
4677:
4654:
4652:
4651:
4646:
4638:
4637:
4631:
4630:
4618:
4617:
4608:
4607:
4586:
4585:
4567:
4566:
4542:
4534:
4511:
4509:
4508:
4503:
4495:
4494:
4488:
4487:
4475:
4474:
4465:
4464:
4437:
4436:
4412:
4404:
4379:
4377:
4376:
4371:
4363:
4362:
4356:
4355:
4337:
4336:
4312:
4304:
4281:
4279:
4278:
4273:
4265:
4264:
4243:
4242:
4218:
4210:
4183:
4181:
4180:
4175:
4167:
4166:
4145:
4144:
4120:
4112:
4085:
4083:
4082:
4077:
4069:
4068:
4062:
4061:
4043:
4042:
4018:
4010:
3985:
3983:
3982:
3977:
3966:
3965:
3926:
3918:
3892:
3890:
3889:
3884:
3876:
3875:
3869:
3868:
3856:
3855:
3846:
3845:
3824:
3823:
3805:
3804:
3780:
3772:
3750:
3748:
3747:
3742:
3734:
3733:
3727:
3726:
3714:
3713:
3695:
3694:
3682:
3681:
3657:
3649:
3620:
3618:
3617:
3612:
3601:
3600:
3561:
3553:
3531:
3529:
3528:
3523:
3515:
3514:
3493:
3492:
3468:
3460:
3438:
3436:
3435:
3430:
3422:
3421:
3415:
3414:
3405:
3404:
3386:
3385:
3373:
3372:
3348:
3347:
3323:
3315:
3289:
3287:
3286:
3281:
3273:
3272:
3266:
3265:
3247:
3246:
3225:
3224:
3200:
3192:
3169:
3167:
3166:
3161:
3153:
3152:
3131:
3130:
3106:
3098:
3075:
3073:
3072:
3067:
3059:
3058:
3052:
3051:
3042:
3041:
3014:
3013:
2989:
2981:
2956:
2954:
2953:
2948:
2940:
2939:
2918:
2917:
2893:
2885:
2852:
2850:
2849:
2844:
2836:
2835:
2829:
2828:
2804:
2803:
2779:
2771:
2749:
2747:
2746:
2741:
2733:
2732:
2723:
2722:
2704:
2703:
2679:
2671:
2642:
2640:
2639:
2634:
2626:
2625:
2619:
2618:
2606:
2605:
2596:
2595:
2574:
2573:
2555:
2554:
2530:
2522:
2499:
2497:
2496:
2491:
2483:
2482:
2476:
2475:
2457:
2456:
2444:
2443:
2419:
2411:
2388:
2386:
2385:
2380:
2372:
2371:
2362:
2361:
2343:
2342:
2318:
2310:
2280:
2278:
2277:
2272:
2264:
2263:
2257:
2256:
2244:
2243:
2234:
2233:
2212:
2211:
2193:
2192:
2168:
2160:
2133:
2131:
2130:
2125:
2117:
2116:
2110:
2109:
2100:
2099:
2084:
2083:
2065:
2064:
2040:
2036:
2032:
2028:
1993:
1991:
1990:
1985:
1938:
1934:
1932:
1931:
1926:
1881:
1866:
1558:
1557:
1516:of a lattice in
1384:
1374:
1367:
1354:
1345:
1283:maximal subgroup
1275:character tables
1117:McLaughlin group
936:
930:
924:
918:
845:normal subgroups
802:
795:
788:
744:Algebraic groups
517:Hyperbolic group
507:Arithmetic group
498:
496:
495:
490:
488:
473:
471:
470:
465:
463:
436:
434:
433:
428:
426:
349:Schur multiplier
303:Cauchy's theorem
291:Quaternion group
239:
65:
54:
41:
30:
21:
6756:
6755:
6751:
6750:
6749:
6747:
6746:
6745:
6736:Sporadic groups
6726:
6725:
6702:
6701:
6698:
6693:
6669:
6629:
6606:
6598:
6590:OEIS Foundation
6573:
6541:
6517:
6458:
6401:
6342:
6289:
6247:
6242:
6178:
6146:
6136:Springer-Verlag
6129:
6057:
6025:
5995:
5933:
5893:
5883:Clarendon Press
5859:
5777:
5745:
5705:
5701:
5696:
5695:
5691:, p. 2151)
5687:
5683:
5676:
5669:
5661:
5657:
5649:
5645:
5638:
5634:
5621:
5617:
5609:
5605:
5597:
5593:
5585:
5581:
5573:
5569:
5561:
5557:
5549:
5545:
5537:
5533:
5524:
5518:
5510:
5506:
5498:
5494:
5486:
5482:
5469:
5465:
5457:
5450:
5442:
5438:
5430:
5426:
5418:
5409:
5404:
5399:
5398:
5374:
5364:
5344:
5343:
5322:
5306:
5293:
5276:
5275:
5254:
5232:
5219:
5205:
5204:
5202:
5198:
5189:
5185:
5172:
5168:
5163:
5159:
5155:
5150:
5089:
5088:
5076:
5074:
5068:
5053:
5047:
5002:
4989:
4979:
4957:
4933:
4932:
4920:
4918:
4912:
4902:
4859:
4794:
4793:
4781:
4779:
4773:
4763:
4720:
4696:
4695:
4683:
4681:
4675:
4665:
4622:
4609:
4599:
4577:
4553:
4552:
4540:
4538:
4532:
4522:
4479:
4466:
4456:
4423:
4422:
4410:
4408:
4402:
4390:
4347:
4323:
4322:
4310:
4308:
4302:
4292:
4229:
4228:
4216:
4214:
4208:
4194:
4131:
4130:
4118:
4116:
4110:
4096:
4053:
4029:
4028:
4016:
4014:
4008:
3996:
3957:
3937:
3936:
3924:
3922:
3916:
3860:
3847:
3837:
3815:
3791:
3790:
3778:
3776:
3770:
3718:
3705:
3686:
3668:
3667:
3655:
3653:
3647:
3636:
3592:
3572:
3571:
3559:
3557:
3556:145,926,144,000
3551:
3479:
3478:
3466:
3464:
3463:448,345,497,600
3458:
3406:
3387:
3377:
3364:
3334:
3333:
3321:
3319:
3318:460,815,505,920
3313:
3257:
3238:
3211:
3210:
3198:
3196:
3195:495,766,656,000
3190:
3180:
3117:
3116:
3104:
3102:
3096:
3086:
3043:
3033:
3000:
2999:
2987:
2985:
2979:
2967:
2904:
2903:
2891:
2889:
2883:
2868:
2820:
2790:
2789:
2777:
2775:
2769:
2714:
2690:
2689:
2677:
2675:
2669:
2658:
2610:
2597:
2587:
2565:
2541:
2540:
2528:
2526:
2520:
2510:
2467:
2445:
2430:
2429:
2417:
2415:
2409:
2399:
2353:
2329:
2328:
2316:
2314:
2308:
2299:
2291:
2248:
2235:
2225:
2203:
2179:
2178:
2166:
2164:
2158:
2149:
2101:
2091:
2075:
2051:
2050:
2038:
2034:
2026:
2011:
1994:
1943:
1942:
1941:
1936:
1935:
1896:
1895:
1894:
1890:
1884:
1879:
1875:
1864:
1841:
1834:
1827:
1819:
1813:
1802:
1794:
1787:
1780:
1773:
1769:
1765:
1761:
1753:
1749:
1734:
1716:
1706:The product of
1699:
1689:The product of
1681:
1671:The product of
1668:
1661:
1652:
1645:
1635:conjugacy class
1628:
1612:
1593:
1585:
1555:
1553:
1551:
1542:
1533:
1506:
1496:
1485:
1469:
1462:
1456:
1424:
1419:
1393:
1382:
1376:
1369:
1365:
1359:
1352:
1346:
1339:
1265:
1252:Fischer-Griess
1249:
1242:
1227:
1220:
1197:
1190:
1166:
1143:
1136:
1104:
1097:
1088:
1079:
1065:
1056:
1047:
1035:
1022:
1009:
1000:
987:
978:
969:
960:
951:
912:
909:sporadic groups
896:
888:
855:consists of 18
830:sporadic groups
806:
777:
776:
765:Abelian variety
758:Reductive group
746:
736:
735:
734:
733:
684:
676:
668:
660:
652:
625:Special unitary
536:
522:
521:
503:
502:
479:
478:
454:
453:
417:
416:
408:
407:
398:Discrete groups
387:
386:
342:Frobenius group
287:
274:
263:
256:Symmetric group
252:
236:
226:
225:
76:Normal subgroup
62:
42:
33:
28:
23:
22:
18:Sporadic groups
15:
12:
11:
5:
6754:
6752:
6744:
6743:
6738:
6728:
6727:
6724:
6723:
6718:
6697:
6696:External links
6694:
6692:
6691:
6666:
6665:
6627:
6595:
6594:
6577:, ed. (1996).
6570:
6569:
6539:
6514:
6513:
6455:
6454:
6398:
6397:
6339:
6338:
6302:(4): 429–445.
6286:
6285:
6258:(4): 169−193.
6239:
6238:
6175:
6174:
6144:
6126:
6125:
6054:
6053:
6023:
5997:Gorenstein, D.
5992:
5991:
5947:. New Series.
5935:Gorenstein, D.
5930:
5929:
5891:
5856:
5855:
5793:(2): 398–400.
5774:
5773:
5743:
5702:
5700:
5697:
5694:
5693:
5681:
5667:
5655:
5643:
5632:
5630:
5629:
5615:
5603:
5591:
5579:
5567:
5555:
5543:
5531:
5529:
5528:
5522:
5504:
5492:
5480:
5478:
5477:
5463:
5448:
5436:
5434:, p. 146)
5424:
5406:
5405:
5403:
5400:
5397:
5396:
5381:
5377:
5371:
5367:
5363:
5360:
5357:
5354:
5351:
5329:
5325:
5321:
5318:
5313:
5309:
5305:
5300:
5296:
5292:
5289:
5286:
5283:
5261:
5257:
5253:
5250:
5247:
5244:
5239:
5235:
5231:
5226:
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5218:
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5212:
5196:
5183:
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5180:
5166:
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5152:
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5149:
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5142:
5131:
5128:
5123:
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5112:
5109:
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5066:
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4907:
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2518:
2515:
2512:
2508:
2501:
2500:
2489:
2486:
2481:
2474:
2470:
2466:
2463:
2460:
2455:
2452:
2448:
2442:
2437:
2427:
2424:
2421:
2412:
2407:
2404:
2401:
2397:
2390:
2389:
2378:
2375:
2370:
2365:
2360:
2356:
2352:
2349:
2346:
2341:
2336:
2326:
2323:
2320:
2311:
2306:
2303:
2300:
2297:
2289:
2282:
2281:
2270:
2267:
2262:
2255:
2251:
2247:
2242:
2238:
2232:
2228:
2224:
2221:
2218:
2215:
2210:
2206:
2202:
2199:
2196:
2191:
2186:
2176:
2173:
2170:
2161:
2156:
2153:
2150:
2147:
2135:
2134:
2123:
2120:
2115:
2108:
2104:
2098:
2094:
2090:
2087:
2082:
2078:
2074:
2071:
2068:
2063:
2058:
2048:
2045:
2042:
2029:
2024:
2021:
2012:
2009:
1997:
1996:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1939:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1892:
1888:
1882:
1877:
1873:
1870:
1863:
1860:
1839:
1832:
1825:
1815:Main article:
1812:
1809:
1800:
1792:
1785:
1778:
1771:
1767:
1763:
1759:
1751:
1747:
1732:
1726:
1725:
1722:
1714:
1704:
1697:
1687:
1679:
1669:
1666:
1659:
1654:
1650:
1643:
1638:
1626:
1621:
1610:
1592:
1589:
1588:
1587:
1583:
1578:
1572:
1566:
1560:
1549:
1544:
1540:
1535:
1531:
1495:
1492:
1483:
1465:
1460:Mathieu groups
1458:Main article:
1455:
1452:
1438:of subgroups (
1423:
1420:
1418:
1415:
1404:Burnside (1911
1400:sporadic group
1391:
1385:also known as
1380:
1363:
1350:
1267:
1266:
1263:
1250:
1247:
1240:
1228:
1225:
1218:
1209:Thompson group
1206:
1198:
1195:
1188:
1176:
1167:
1164:
1152:
1147:Rudvalis group
1144:
1141:
1134:
1122:
1114:
1106:
1102:
1095:
1086:
1077:
1071:Fischer groups
1068:
1063:
1054:
1045:
1037:
1033:
1020:
1007:
998:
989:
985:
976:
967:
958:
949:
943:Mathieu groups
887:
884:
876:friendly giant
828:, or just the
808:
807:
805:
804:
797:
790:
782:
779:
778:
775:
774:
772:Elliptic curve
768:
767:
761:
760:
754:
753:
747:
742:
741:
738:
737:
732:
731:
728:
725:
721:
717:
716:
715:
710:
708:Diffeomorphism
704:
703:
698:
693:
687:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
645:
644:
633:
632:
621:
620:
609:
608:
597:
596:
585:
584:
573:
572:
565:Special linear
561:
560:
553:General linear
549:
548:
543:
537:
528:
527:
524:
523:
520:
519:
514:
509:
501:
500:
487:
475:
462:
449:
447:Modular groups
445:
444:
443:
438:
425:
409:
406:
405:
400:
394:
393:
392:
389:
388:
383:
382:
381:
380:
375:
370:
367:
361:
360:
354:
353:
352:
351:
345:
344:
338:
337:
332:
323:
322:
320:Hall's theorem
317:
315:Sylow theorems
311:
310:
305:
297:
296:
295:
294:
288:
283:
280:Dihedral group
276:
275:
270:
264:
259:
253:
248:
237:
232:
231:
228:
227:
222:
221:
220:
219:
214:
206:
205:
204:
203:
198:
193:
188:
183:
178:
173:
171:multiplicative
168:
163:
158:
153:
145:
144:
143:
142:
137:
129:
128:
120:
119:
118:
117:
115:Wreath product
112:
107:
102:
100:direct product
94:
92:Quotient group
86:
85:
84:
83:
78:
73:
63:
60:
59:
56:
55:
47:
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6753:
6742:
6739:
6737:
6734:
6733:
6731:
6722:
6719:
6714:
6713:
6708:
6705:
6700:
6699:
6695:
6688:
6684:
6680:
6676:
6672:
6668:
6667:
6662:
6658:
6654:
6650:
6646:
6642:
6638:
6634:
6630:
6628:9780511565830
6624:
6620:
6616:
6612:
6605:
6601:
6597:
6596:
6591:
6587:
6586:
6580:
6576:
6572:
6571:
6566:
6562:
6558:
6554:
6550:
6546:
6542:
6536:
6532:
6527:
6526:
6520:
6516:
6515:
6510:
6506:
6502:
6498:
6494:
6490:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6457:
6456:
6451:
6447:
6443:
6439:
6435:
6431:
6427:
6423:
6420:: 2147−2169.
6419:
6415:
6411:
6410:
6405:
6400:
6399:
6394:
6390:
6386:
6382:
6378:
6374:
6369:
6364:
6360:
6356:
6352:
6351:
6346:
6341:
6340:
6335:
6331:
6327:
6323:
6319:
6315:
6310:
6305:
6301:
6297:
6293:
6288:
6287:
6281:
6277:
6273:
6269:
6265:
6261:
6257:
6253:
6246:
6241:
6240:
6236:
6232:
6228:
6224:
6220:
6216:
6212:
6208:
6203:
6198:
6194:
6190:
6186:
6182:
6177:
6176:
6171:
6167:
6163:
6159:
6155:
6151:
6147:
6145:9783540627784
6141:
6137:
6133:
6128:
6127:
6122:
6118:
6114:
6110:
6106:
6102:
6097:
6096:2027.42/46608
6092:
6088:
6084:
6080:
6076:
6072:
6068:
6064:
6060:
6056:
6055:
6050:
6046:
6042:
6038:
6034:
6030:
6026:
6020:
6016:
6012:
6008:
6004:
6003:
5998:
5994:
5993:
5988:
5984:
5980:
5976:
5972:
5968:
5963:
5958:
5954:
5950:
5946:
5945:
5940:
5936:
5932:
5931:
5926:
5922:
5918:
5914:
5910:
5906:
5902:
5898:
5894:
5888:
5884:
5880:
5879:
5874:
5873:Wilson, R. A.
5870:
5869:Parker, R. A.
5866:
5865:Norton, S. P.
5862:
5861:Conway, J. H.
5858:
5857:
5852:
5848:
5844:
5840:
5836:
5832:
5827:
5822:
5818:
5814:
5809:
5804:
5800:
5796:
5792:
5788:
5784:
5780:
5779:Conway, J. H.
5776:
5775:
5770:
5766:
5762:
5758:
5754:
5750:
5746:
5744:0-486-49575-2
5740:
5735:
5730:
5726:
5722:
5718:
5714:
5713:
5708:
5704:
5703:
5698:
5690:
5685:
5682:
5679:
5674:
5672:
5668:
5664:
5659:
5656:
5652:
5647:
5644:
5641:
5640:Sloane (1996)
5636:
5633:
5627:
5626:
5624:
5619:
5616:
5612:
5607:
5604:
5600:
5595:
5592:
5588:
5583:
5580:
5576:
5571:
5568:
5564:
5559:
5556:
5553:, p. 91)
5552:
5547:
5544:
5540:
5535:
5532:
5521:
5516:
5515:
5513:
5508:
5505:
5501:
5496:
5493:
5489:
5484:
5481:
5475:
5474:
5472:
5467:
5464:
5460:
5455:
5453:
5449:
5445:
5440:
5437:
5433:
5428:
5425:
5421:
5416:
5414:
5412:
5408:
5401:
5379:
5375:
5369:
5365:
5361:
5358:
5355:
5352:
5349:
5327:
5319:
5311:
5307:
5298:
5294:
5287:
5284:
5281:
5259:
5251:
5248:
5245:
5237:
5233:
5224:
5220:
5213:
5210:
5200:
5197:
5193:
5187:
5184:
5178:
5177:
5175:
5170:
5167:
5157:
5154:
5147:
5129:
5126:
5113:
5110:
5107:
5094:
5087:
5084:
5067:
5064:
5062:
5059:
5057:
5050:
5043:
5042:
5038:
5037:
5022:
5019:
5007:
5003:
4999:
4994:
4984:
4980:
4976:
4973:
4970:
4962:
4954:
4951:
4938:
4931:
4928:
4911:
4908:
4905:
4903:
4899:
4895:
4894:
4879:
4876:
4864:
4860:
4856:
4853:
4850:
4847:
4844:
4834:
4831:
4818:
4815:
4812:
4799:
4792:
4789:
4772:
4769:
4766:
4764:
4760:
4756:
4755:
4740:
4737:
4725:
4721:
4717:
4714:
4711:
4701:
4694:
4691:
4674:
4671:
4668:
4666:
4662:
4658:
4657:
4642:
4639:
4627:
4623:
4619:
4614:
4604:
4600:
4596:
4593:
4590:
4582:
4574:
4571:
4558:
4551:
4548:
4531:
4528:
4525:
4523:
4519:
4515:
4514:
4499:
4496:
4484:
4480:
4476:
4471:
4461:
4457:
4453:
4450:
4447:
4441:
4438:
4428:
4421:
4418:
4401:
4398:
4396:
4393:
4391:
4387:
4383:
4382:
4367:
4364:
4352:
4348:
4344:
4341:
4338:
4328:
4321:
4318:
4301:
4298:
4295:
4293:
4289:
4285:
4284:
4269:
4266:
4253:
4250:
4247:
4234:
4227:
4224:
4207:
4204:
4201:
4199:
4195:
4191:
4187:
4186:
4171:
4168:
4155:
4152:
4149:
4136:
4129:
4126:
4109:
4106:
4103:
4101:
4097:
4093:
4089:
4088:
4073:
4070:
4058:
4054:
4050:
4047:
4044:
4034:
4027:
4024:
4007:
4004:
4002:
3999:
3997:
3993:
3989:
3988:
3973:
3970:
3962:
3958:
3954:
3951:
3948:
3942:
3935:
3932:
3915:
3912:
3910:
3906:
3903:
3901:
3900:
3896:
3895:
3880:
3877:
3865:
3861:
3857:
3852:
3842:
3838:
3834:
3831:
3828:
3820:
3812:
3809:
3796:
3789:
3786:
3769:
3766:
3764:
3761:
3759:
3758:
3754:
3753:
3738:
3735:
3723:
3719:
3715:
3710:
3706:
3702:
3699:
3696:
3691:
3687:
3683:
3673:
3666:
3663:
3652:4,030,387,200
3646:
3643:
3641:
3638:
3633:
3629:
3628:
3624:
3623:
3608:
3605:
3597:
3593:
3589:
3586:
3583:
3577:
3570:
3567:
3550:
3547:
3545:
3542:
3540:
3539:
3535:
3534:
3519:
3516:
3503:
3500:
3497:
3484:
3477:
3474:
3457:
3454:
3452:
3449:
3447:
3446:
3442:
3441:
3426:
3423:
3411:
3401:
3398:
3395:
3392:
3382:
3378:
3369:
3365:
3358:
3355:
3352:
3349:
3339:
3332:
3329:
3312:
3309:
3307:
3304:
3302:
3298:
3297:
3293:
3292:
3277:
3274:
3262:
3254:
3251:
3243:
3235:
3232:
3229:
3216:
3209:
3206:
3189:
3186:
3183:
3181:
3177:
3173:
3172:
3157:
3154:
3141:
3138:
3135:
3122:
3115:
3112:
3095:
3092:
3089:
3087:
3083:
3079:
3078:
3063:
3060:
3048:
3038:
3034:
3030:
3027:
3024:
3018:
3015:
3005:
2998:
2995:
2978:
2975:
2973:
2970:
2968:
2964:
2960:
2959:
2944:
2941:
2928:
2925:
2922:
2909:
2902:
2899:
2882:
2879:
2877:
2873:
2870:
2865:
2861:
2860:
2856:
2855:
2840:
2837:
2825:
2821:
2817:
2814:
2811:
2808:
2805:
2795:
2788:
2785:
2768:
2765:
2763:
2760:
2758:
2757:
2753:
2752:
2737:
2734:
2724:
2719:
2711:
2708:
2695:
2688:
2685:
2668:
2665:
2663:
2660:
2655:
2651:
2650:
2646:
2645:
2630:
2627:
2615:
2611:
2607:
2602:
2592:
2588:
2584:
2581:
2578:
2570:
2562:
2559:
2546:
2539:
2536:
2519:
2516:
2513:
2511:
2507:
2503:
2502:
2487:
2484:
2472:
2464:
2461:
2453:
2450:
2446:
2435:
2428:
2425:
2408:
2405:
2402:
2400:
2396:
2392:
2391:
2376:
2373:
2363:
2358:
2350:
2347:
2334:
2327:
2324:
2307:
2304:
2301:
2296:
2292:
2288:
2284:
2283:
2268:
2265:
2253:
2249:
2245:
2240:
2230:
2226:
2222:
2219:
2216:
2208:
2200:
2197:
2184:
2177:
2174:
2157:
2154:
2151:
2146:
2142:
2141:
2137:
2136:
2121:
2118:
2106:
2096:
2092:
2088:
2080:
2072:
2069:
2056:
2049:
2046:
2025:
2022:
2020:
2016:
2013:
2008:
2004:
2003:
1999:
1998:
1972:
1966:
1963:
1960:
1957:
1954:
1940:
1919:
1916:
1913:
1910:
1907:
1904:
1893:
1887:
1883:
1878:
1874:
1871:
1867:
1861:
1859:
1857:
1853:
1849:
1845:
1838:
1831:
1824:
1818:
1810:
1808:
1806:
1798:
1791:
1784:
1777:
1757:
1745:
1740:
1738:
1731:
1723:
1720:
1713:
1709:
1705:
1703:
1696:
1692:
1688:
1685:
1678:
1674:
1670:
1665:
1658:
1655:
1649:
1642:
1639:
1636:
1632:
1625:
1622:
1620:
1616:
1609:
1605:
1602:
1601:
1600:
1598:
1590:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1548:
1545:
1539:
1536:
1530:
1527:
1526:
1525:
1523:
1522:Leech lattice
1519:
1515:
1511:
1505:
1504:Conway groups
1501:
1500:Leech lattice
1493:
1491:
1489:
1481:
1477:
1473:
1468:
1461:
1453:
1451:
1449:
1448:Robert Griess
1445:
1441:
1437:
1433:
1429:
1428:monster group
1421:
1416:
1414:
1412:
1407:
1405:
1401:
1396:
1394:
1390:
1379:
1372:
1362:
1358:
1349:
1343:
1336:
1332:
1329:
1325:
1321:
1316:
1314:
1313:Jansen (2005)
1310:
1306:
1305:
1300:
1299:presentations
1296:
1292:
1288:
1285:, as well as
1284:
1281:and lists of
1280:
1277:, individual
1276:
1272:
1262:
1258:
1255:
1254:Monster group
1251:
1246:
1239:
1235:
1232:
1229:
1224:
1217:
1213:
1210:
1207:
1205:
1202:
1199:
1194:
1187:
1183:
1180:
1177:
1174:
1171:
1168:
1163:
1159:
1156:
1153:
1151:
1148:
1145:
1140:
1133:
1129:
1126:
1123:
1121:
1118:
1115:
1113:
1110:
1107:
1105:
1101:
1094:
1089:
1085:
1080:
1076:
1072:
1069:
1067:
1066:
1058:
1057:
1049:
1048:
1041:
1040:Conway groups
1038:
1036:
1032:
1027:
1026:
1019:
1014:
1013:
1006:
1001:
997:
993:
990:
988:
984:
979:
975:
970:
966:
961:
957:
952:
948:
944:
941:
940:
935:
929:
923:
917:
910:
906:
901:
897:
893:
892:Émile Mathieu
885:
883:
881:
877:
873:
872:monster group
868:
866:
862:
858:
854:
850:
846:
842:
838:
833:
831:
827:
823:
819:
815:
803:
798:
796:
791:
789:
784:
783:
781:
780:
773:
770:
769:
766:
763:
762:
759:
756:
755:
752:
749:
748:
745:
740:
739:
729:
726:
723:
722:
720:
714:
711:
709:
706:
705:
702:
699:
697:
694:
692:
689:
688:
685:
679:
677:
671:
669:
663:
661:
655:
653:
647:
646:
642:
638:
635:
634:
630:
626:
623:
622:
618:
614:
611:
610:
606:
602:
599:
598:
594:
590:
587:
586:
582:
578:
575:
574:
570:
566:
563:
562:
558:
554:
551:
550:
547:
544:
542:
539:
538:
535:
531:
526:
525:
518:
515:
513:
510:
508:
505:
504:
476:
451:
450:
448:
442:
439:
414:
411:
410:
404:
401:
399:
396:
395:
391:
390:
379:
376:
374:
371:
368:
365:
364:
363:
362:
359:
355:
350:
347:
346:
343:
340:
339:
336:
333:
331:
329:
325:
324:
321:
318:
316:
313:
312:
309:
306:
304:
301:
300:
299:
298:
292:
289:
286:
281:
278:
277:
273:
268:
265:
262:
257:
254:
251:
246:
243:
242:
241:
240:
235:
234:Finite groups
230:
229:
218:
215:
213:
210:
209:
208:
207:
202:
199:
197:
194:
192:
189:
187:
184:
182:
179:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
148:
147:
146:
141:
138:
136:
133:
132:
131:
130:
127:
126:
121:
116:
113:
111:
108:
106:
103:
101:
98:
95:
93:
90:
89:
88:
87:
82:
79:
77:
74:
72:
69:
68:
67:
66:
61:Basic notions
58:
57:
53:
49:
48:
45:
40:
36:
31:
19:
6710:
6682:
6675:Parker, R.A.
6671:Wilson, R.A.
6610:
6582:
6529:. New York:
6524:
6472:
6468:
6461:Wilson, R.A.
6413:
6407:
6354:
6348:
6299:
6295:
6255:
6251:
6188:
6184:
6131:
6070:
6066:
6001:
5948:
5942:
5876:
5790:
5786:
5711:
5689:Lubeck (2001
5684:
5658:
5651:Jansen (2005
5646:
5635:
5618:
5606:
5594:
5582:
5570:
5558:
5546:
5534:
5519:
5507:
5495:
5483:
5466:
5439:
5427:
5199:
5186:
5169:
5156:
5048:
5040:
4897:
4758:
4660:
4517:
4385:
4287:
4197:
4189:
4099:
4091:
3991:
3898:
3756:
3631:
3626:
3537:
3444:
3300:
3295:
3175:
3081:
2962:
2863:
2858:
2755:
2653:
2648:
2505:
2394:
2294:
2286:
2144:
2139:
2006:
2001:
1851:
1847:
1843:
1836:
1829:
1822:
1820:
1817:Pariah group
1804:
1796:
1789:
1782:
1775:
1755:
1741:
1736:
1729:
1727:
1718:
1711:
1707:
1701:
1694:
1690:
1683:
1676:
1672:
1663:
1656:
1647:
1640:
1630:
1623:
1618:
1607:
1603:
1596:
1594:
1580:
1574:
1568:
1562:
1546:
1537:
1528:
1510:subquotients
1507:
1479:
1471:
1466:
1463:
1444:happy family
1443:
1425:
1422:Happy Family
1417:Organization
1408:
1399:
1397:
1388:
1377:
1370:
1360:
1347:
1341:
1330:
1317:
1308:
1303:
1273:, including
1268:
1260:
1256:
1244:
1237:
1233:
1222:
1215:
1211:
1203:
1192:
1185:
1181:
1172:
1161:
1157:
1155:Suzuki group
1149:
1138:
1131:
1127:
1119:
1111:
1099:
1092:
1083:
1074:
1060:
1051:
1042:
1030:
1024:
1017:
1011:
1004:
995:
992:Janko groups
982:
973:
964:
955:
946:
908:
889:
880:subquotients
875:
869:
848:
840:
837:simple group
834:
829:
825:
821:
811:
640:
628:
616:
604:
592:
580:
568:
556:
377:
327:
284:
271:
260:
249:
245:Cyclic group
123:
110:Free product
81:Group action
44:Group theory
39:Group theory
38:
6600:Wilson, R.A
6519:Ronan, Mark
6479:: 359−371.
6361:: 122−144.
5699:Works cited
5459:Ronan (2006
4407:244,823,040
3775:898,128,000
1872:Discoverer
1615:centralizer
1411:Ronan (2006
1344:= 0)-member
1201:Lyons group
1170:O'Nan group
1098:′ or
905:subquotient
839:is a group
530:Topological
369:alternating
6730:Categories
6661:0914.20016
6565:1113.00002
6509:1087.20025
6450:1004.20003
6393:1089.20006
6334:0987.20003
6280:1042.20007
6235:1320.51021
6170:0908.20007
6121:0498.20013
6041:6907721813
5987:0414.20009
5955:: 43–199.
5925:0568.20001
5881:. Oxford:
5851:0186.32401
5625:, p. 172)
5623:Hiss (2003
5422:, p. viii)
5402:References
5073:17,971,200
4790:2B, 3B, 11
4692:2A, 4A, 11
4537:10,200,960
4419:2B, 3A, 23
4127:2A, 3A, 19
4115:50,232,960
4025:2A, 4A, 37
3933:2A, 5A, 11
3921:44,352,000
3787:2A, 5A, 11
3763:McLaughlin
3664:2A, 7C, 17
3568:2B, 4A, 13
3475:2B, 3B, 13
3330:2A, 4A, 11
3207:2A, 7C, 17
3113:2A, 5A, 28
2996:2B, 3C, 40
2900:2A, 3B, 22
2537:2A, 13, 11
2426:2B, 3D, 28
2325:2A, 3E, 29
2175:2C, 3A, 55
2047:2A, 3B, 29
1937:Generators
1880:Generation
1744:Tits group
1498:See also:
1328:Tits group
1318:A further
1125:Held group
861:Tits group
637:Symplectic
577:Orthogonal
534:Lie groups
441:Free group
166:continuous
105:Direct sum
6712:MathWorld
6645:726827806
6557:180766312
6442:122060727
6385:121362819
6264:0012-0456
6211:1715-0868
6113:123597150
6073:: 1−102.
6049:209854856
5979:121953006
5917:117473588
5769:117347785
5665:, p. 365)
5514:, p.106)
5502:, p.viii)
5473:, p.429)
5085:2A, 3, 13
4225:2B, 3B, 7
2786:2, 5A, 14
2686:2, 3A, 19
1982:⟩
1979:⟩
1964:∣
1952:⟨
1949:⟨
1436:quotients
1432:subgroups
1320:exception
857:countably
824:, or the
701:Conformal
589:Euclidean
196:nilpotent
6653:59394831
6602:(1998).
6521:(2006).
6501:13100616
6463:(2011).
6326:14682147
6284:(German)
6227:40845205
6162:38910263
6061:(1982).
5937:(1979).
5909:12106933
5875:(1985).
5843:29358882
5835:16591697
5781:(1968).
5761:54407807
5709:(1911).
5490:, p.111)
4929:2, 4, 11
4549:2, 4, 23
3544:Rudvalis
2662:Thompson
1559:) vector
1508:All the
1490:points.
1440:sections
1335:Lie type
1165:3−
696:Poincaré
541:Solenoid
413:Integers
403:Lattices
378:sporadic
373:Lie type
201:solvable
191:dihedral
176:additive
161:infinite
71:Subgroup
6637:1647427
6549:2215662
6493:2172713
6434:1837968
6377:2153793
6318:1823074
6272:2033760
6219:2791293
6154:1707296
6105:0671653
6075:Bibcode
6033:1490581
5971:0513750
5901:0827219
5817:0237634
5795:Bibcode
5753:0069818
4906:Mathieu
4767:Mathieu
4680:443,520
4669:Mathieu
4526:Mathieu
4395:Mathieu
4319:2, 3, 7
4307:175,560
4213:604,800
2514:Fischer
2403:Fischer
2302:Fischer
2152:Fischer
2015:Fischer
1856:pariahs
1811:Pariahs
1762:×F
1750:×F
1554:√
1512:of the
1402:may be
1326:is the
882:of it.
691:Lorentz
613:Unitary
512:Lattice
452:PSL(2,
186:abelian
97:(Semi-)
6659:
6651:
6643:
6635:
6625:
6563:
6555:
6547:
6537:
6507:
6499:
6491:
6448:
6440:
6432:
6391:
6383:
6375:
6332:
6324:
6316:
6278:
6270:
6262:
6233:
6225:
6217:
6209:
6168:
6160:
6152:
6142:
6119:
6111:
6103:
6047:
6039:
6031:
6021:
5985:
5977:
5969:
5923:
5915:
5907:
5899:
5889:
5849:
5841:
5833:
5826:225171
5823:
5815:
5767:
5759:
5751:
5741:
5203:Where
4778:95,040
4303:Pariah
4111:Pariah
4009:Pariah
3905:Higman
3552:Pariah
3451:Suzuki
3314:Pariah
3184:Conway
3090:Conway
2972:Conway
2876:Norton
2872:Harada
2770:Pariah
2044:196883
2019:Griess
1869:Group
1850:, and
1373:> 0
937:Pariah
818:groups
546:Circle
477:SL(2,
366:cyclic
330:-group
181:cyclic
156:finite
151:simple
135:kernel
6649:S2CID
6607:(PDF)
6497:S2CID
6438:S2CID
6381:S2CID
6322:S2CID
6248:(PDF)
6223:S2CID
6109:S2CID
6045:S2CID
5975:S2CID
5951:(1).
5913:S2CID
5839:S2CID
5765:S2CID
5342:with
5148:Notes
5065:1964
4917:7,920
4909:1861
4770:1861
4672:1861
4529:1861
4399:1861
4299:1965
4296:Janko
4205:1968
4202:Janko
4107:1968
4104:Janko
4005:1976
4001:Janko
3913:1967
3767:1969
3644:1969
3548:1972
3455:1969
3327:10944
3310:1976
3306:O'Nan
3187:1969
3093:1969
2976:1969
2880:1976
2766:1972
2762:Lyons
2666:1976
2517:1971
2406:1971
2305:1971
2155:1973
2023:1973
1886:Order
1876:Year
1637:"3A")
931:3rd,
925:2nd,
919:1st,
886:Names
874:, or
730:Sp(∞)
727:SU(∞)
140:image
6641:OCLC
6623:ISBN
6583:The
6553:OCLC
6535:ISBN
6260:ISSN
6207:ISSN
6158:OCLC
6140:ISBN
6037:OCLC
6019:ISBN
5905:OCLC
5887:ISBN
5831:PMID
5757:OCLC
5739:ISBN
5525:(2)′
5274:and
5061:Tits
5054:(2)′
4022:1333
3909:Sims
3640:Held
2783:2480
2322:8671
2172:4371
1788:and
1742:The
1633:(in
1502:and
1470:for
1383:(2),
1366:(2)′
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