11,053 Pages

## 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
A5A1(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.
A6A1(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").[4]
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.
A8A3(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.

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

## Approximations of these numbers

For 20,160:

Notation Approximation
Scientific notation $$2.016 \times 10^4$$ (exact)
Arrow notation $$142↑2$$
Chained arrow notation $$142→2$$
Hyperfactorial array notation $$7!$$
Fast-growing hierarchy $$f_2(11)$$
Hardy hierarchy $$H_{\omega^2}(11)$$
Slow-growing hierarchy $$g_{\omega^2}(142)$$

For 25,920:

Notation Approximation
Scientific notation $$2.5920*10^4$$ (exact)
Fast-growing hierarchy $$f_2(11)<n<f_1^5(f_2(7))$$

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