Numbers in group theory

This page contains numbers appearing in .
== List of numbers in group theory ==
*1,440 is the order of the of the .
*There is another irreducible representation of the Monster group with degree 21,296,876.OEIS, Sequence A001379. Accessed 2020-05-28.
*There are (5)=9,999,360 5 × 5 over , therefore it is the order of a matrix group .
== Orders of non-abelian simple groups ==
This list contains with unusual properties, such as:
# Its order has at most four distinct prime factors, or is a ;
# the p-Sylow group (where p = 2 for alternating groups) is not the largest Sylow subgroup; and/or
# 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 in a .And the of kei cars is limited to 660 .
A1(13) 	1,092 	22 × 3 × 7 × 13 	It is also the number of in a double-12 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").http://mzrg.com/rubik/orders.shtml
A1(19) 	3,420 	22 × 32 × 5 × 19 	It is also the number of in a double-18 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 s in a .
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 in a .
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 in a .
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
 	7,920 	24 × 32 × 5 × 11 	8,651 	41 × 211
 	95,040 	26 × 33 × 5 × 11 	214,871 	19 × 43 × 263
 	175,560 	23 × 3 × 5 × 7 × 11 × 19 	158,485 	5 × 29 × 1093
 	443,520 	27 × 32 × 5 × 7 × 11 	941,627 	73 × 12,899
 	604,800 	27 × 33 × 52 × 7 	1,104,344 	23 × 31 × 61 × 73
 	10,200,960 	27 × 32 × 5 × 7 × 11 × 23 	17,318,406 	2 × 3 × 7 × 412,343
 	17,971,200 	211 × 33 × 52 × 13 	50,285,950 	2 × 52 × 11 × 132 × 541
 	44,352,000 	29 × 32 × 53 × 7 × 11 	149,985,646 	2 × 3,929 × 19,087
 	50,232,960 	27 × 35 × 5 × 17 × 19 	71,564,248 	23 × 7 × 239 × 5,347
 	244,823,040 	210 × 33 × 5 × 7 × 11 × 23 	1,363,957,253 	Prime
 	898,128,000 	27 × 36 × 53 × 7 × 11 	1,719,739,392 	210 × 3 × 7 × 79,973
 	4,030,387,200 	210 × 33 × 52 × 73 × 17 	22,303,017,686 	2 × 17 × 211 × 310,889
 	145,926,144,000	214 × 33 × 53 × 7 × 13 × 29 	963,226,363,401 	32 × 1,549 × 69,093,061
 	448,345,497,600	213 × 37 × 52 × 7 × 11 × 13 	4,057,939,316,149 	7 × 19 × 127 × 27,111,439
 	460,815,505,920	29 × 34 × 5 × 73 × 11 × 19 × 31	1,169,254,703,685 	3 × 5 × 1,109 × 7,681 × 9,151
 	495,766,656,000	210 × 37 × 53 × 7 × 11 × 23 	2,547,911,497,738 	2 × 1,273,955,748,869
Group	Order 	Factorization
 	42,305,421,312,000 	218 × 36 × 53 × 7 × 11 × 23
 	64,561,751,654,400 	217 × 39 × 52 × 7 × 11 × 13
 	273,030,912,000,000 	214 × 36 × 56 × 7 × 11 × 19
 	51,765,179,004,000,000 	28 × 37 × 56 × 7 × 11 × 31 × 37 × 67
 	90,745,943,887,872,000 	215 × 310 × 53 × 72 × 13 × 19 × 31
 	4,089,470,473,293,004,800 	218 × 313 × 52 × 7 × 11 × 13 × 17 × 23
 	4,157,776,806,543,360,000 	221 × 39 × 54 × 72 × 11 × 13 × 23
 	86,775,571,046,077,562,880 	221 × 33 × 5 × 7 × 113 × 23 × 29 × 31 × 37 × 43
 	1,255,205,709,190,661,721,292,800 	221 × 316 × 52 × 73 × 11 × 13 × 17 × 23 × 29
 	  4,154,781,481,226,426,191,177,580,544,000,000 	241 × 313 × 56 × 72 × 11 × 13 × 17 × 19 × 23 × 31 × 47
 	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
==Sources==
Category:Numbers
Category:Numbers in group theory
Category:Lists
Category:Class 1
Category:Class 2
Category:Numbers with 3 digits
Category:Unnamed numbers with 4 to 6 digits
Category:Unnamed numbers with 7 to 29 digits