Numbers in group theory

This page contains numbers appearing in group theory.

List of numbers in group theory

• 1,440 is the order of the automorphism group of the simple group A₆.
• There is another irreducible representation of the Monster group with degree 21,296,876.
• There are OEIS A002884(5)=9,999,360 nonsingular 5 × 5 matrices over GF(2), therefore it is the order of a matrix group GL(5,2).

Orders of non-abelian simple groups

This list contains finite non-abelian simple groups with unusual properties, such as:

1. Its order has at most four distinct prime factors, or is a powerful number;
2. the p-Sylow group (where p = 2 for alternating groups) is not the largest Sylow subgroup; and/or
3. there is an exceptional isomorphism, outer automorphism group, or Schur multiplier.

Sporadic groups have their own section.

Group(s)	Order	Factorization	Remarks
A5 ≃ A1(4) ≃ A1(5)	60	22 × 3 × 5	Exceptional Schur multiplier (for A1(4)), and the 5-Sylow group is the largest Sylow subgroup.
A1(7) ≃ A2(2)	168	23 × 3 × 7	Exceptional Schur multiplier (for A2(2)), and the 2-Sylow group is the largest Sylow subgroup.
A6 ≃ A1(9) ≃ B2(2)′	360	23 × 32 × 5	Exceptional outer automorphism group (for A6) and Schur multiplier, and the 3-Sylow group is the largest Sylow subgroup.
A1(8) ≃ 2G2(3)′	504	23 × 32 × 7	The 3-Sylow group is the largest Sylow subgroup. It is also the number of possible queen moves in starchess.
A1(11)	660	22 × 3 × 5 × 11	It is also the number of feet in a furlong. And the engine displacement of kei cars is limited to 660 cm3.
A1(13)	1,092	22 × 3 × 7 × 13	It is also the number of pips in a double-12 domino set.
A1(17)	2,448	24 × 32 × 17
A7	2,520	23 × 32 × 5 × 7	Exceptional Schur multiplier, and the 2-Sylow group is not the largest Sylow subgroup. It is also the maximum possible cycle length of any given algorithm on the Rubik's cube (one such algorithm is "RL2U'F'd").
A1(19)	3,420	22 × 32 × 5 × 19	It is also the number of pips in a double-18 domino set.
A1(16)	4,080	24 × 3 × 5 × 17	The 2-Sylow group is not the largest Sylow subgroup.
A2(3)	5,616	24 × 33 × 13
G2(2)′ ≃ 2A2(32)	6,048	25 × 33 × 7	The 2-Sylow group is the largest Sylow subgroup.
A8 ≃ A3(2); A2(4)	20,160	26 × 32 × 5 × 7	Smallest order with more than one simple group. Exceptional Schur multiplier (for A3(2) and A2(4)). It is also the number of minutes in a fortnight.
B2(3) ≃ 2A3(22)	25,920	26 × 34 × 5	Exceptional Schur multiplier (for 2A3(22)), and the 3-Sylow group is the largest Sylow subgroup. It is also the number of halakim in a day.
A9	181,440	26 × 34 × 5 × 7	Largest alternating group, for which the 2-Sylow group is not the largest Sylow subgroup. It is also the number of halakim in a week.
D4(2)	174,182,400	212 × 35 × 52 × 7	Exceptional Schur multiplier.
G2(4)	251,596,800	212 × 33 × 52 × 7 × 13	Exceptional Schur multiplier.
2A5(22)	9,196,830,720	215 × 36 × 5 × 7 × 11	Exceptional Schur multiplier.
F4(2)	3,311,126, 603,366,400	224 × 36 × 52 × 72 × 13 × 17	Exceptional Schur multiplier.
2E6(22)	76,532, 479,683,774, 853,939,200	236 × 39 × 52 × 72 × 11 × 13 × 17 × 19	Exceptional Schur multiplier.

Sporadic group-related numbers

Group	Order	Factorization	Number of subgroups	Factorization
Mathieu group M11	7,920	24 × 32 × 5 × 11	8,651	41 × 211
Mathieu group M12	95,040	26 × 33 × 5 × 11	214,871	19 × 43 × 263
Janko group J1	175,560	23 × 3 × 5 × 7 × 11 × 19	158,485	5 × 29 × 1093
Mathieu group M22	443,520	27 × 32 × 5 × 7 × 11	941,627	73 × 12,899
Janko group J2	604,800	27 × 33 × 52 × 7	1,104,344	23 × 31 × 61 × 73
Mathieu group M23	10,200,960	27 × 32 × 5 × 7 × 11 × 23	17,318,406	2 × 3 × 7 × 412,343
Tits group	17,971,200	211 × 33 × 52 × 13	50,285,950	2 × 52 × 11 × 132 × 541
Higman–Sims group	44,352,000	29 × 32 × 53 × 7 × 11	149,985,646	2 × 3,929 × 19,087
Janko group J3	50,232,960	27 × 35 × 5 × 17 × 19	71,564,248	23 × 7 × 239 × 5,347
Mathieu group M24	244,823,040	210 × 33 × 5 × 7 × 11 × 23	1,363,957,253	Prime
McLaughlin group	898,128,000	27 × 36 × 53 × 7 × 11	1,719,739,392	210 × 3 × 7 × 79,973
Held group	4,030,387,200	210 × 33 × 52 × 73 × 17	22,303,017,686	2 × 17 × 211 × 310,889
Rudvalis group	145,926,144,000	214 × 33 × 53 × 7 × 13 × 29	963,226,363,401	32 × 1,549 × 69,093,061
Suzuki group	448,345,497,600	213 × 37 × 52 × 7 × 11 × 13	4,057,939,316,149	7 × 19 × 127 × 27,111,439
O'Nan group	460,815,505,920	29 × 34 × 5 × 73 × 11 × 19 × 31	1,169,254,703,685	3 × 5 × 1,109 × 7,681 × 9,151
Conway group Co3	495,766,656,000	210 × 37 × 53 × 7 × 11 × 23	2,547,911,497,738	2 × 1,273,955,748,869

Group	Order	Factorization
Conway group Co2	42,305,421,312,000	218 × 36 × 53 × 7 × 11 × 23
Fischer group Fi22	64,561,751,654,400	217 × 39 × 52 × 7 × 11 × 13
Harada–Norton group	273,030,912,000,000	214 × 36 × 56 × 7 × 11 × 19
Lyons group	51,765,179,004,000,000	28 × 37 × 56 × 7 × 11 × 31 × 37 × 67
Thompson sporadic group	90,745,943,887,872,000	215 × 310 × 53 × 72 × 13 × 19 × 31
Fischer group Fi23	4,089,470,473,293,004,800	218 × 313 × 52 × 7 × 11 × 13 × 17 × 23
Conway group Co1	4,157,776,806,543,360,000	221 × 39 × 54 × 72 × 11 × 13 × 23
Janko group J4	86,775,571,046,077,562,880	221 × 33 × 5 × 7 × 113 × 23 × 29 × 31 × 37 × 43
Fischer group Fi24	1,255,205,709,190,661,721,292,800	221 × 316 × 52 × 73 × 11 × 13 × 17 × 23 × 29
Baby monster group	4,154,781,481,226,426, 191,177,580,544,000,000	241 × 313 × 56 × 72 × 11 × 13 × 17 × 19 × 23 × 31 × 47
Monster group	808,017,424,794,512,875,886,459,904, 961,710,757,005,754,368,000,000,000	246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71