This page contains numbers appearing in group theory.
List of numbers in group theory
- The Conway group Co1 has 101 conjugacy classes.^{[citation needed]}
- The constant term in the McKay–Thompson series T_{2A} is equal to 104.^{[citation needed]}
- The largest order of any element in the Monster group is 119. There is also no other sporadic group with elements of larger order.^{[citation needed]}
- The Frobenius kernel of the smallest non-solvable Frobenius group is the elementary abelian group of order 121.^{[citation needed]}
- The exceptional Lie algebra E_{7} has dimension 133.^{[citation needed]}
- The McKay-Thompson series of monstrous moonshine span a 163-dimensional vector space.^{[citation needed]}
- The exceptional Lie algebra E_{7½} has dimension 190.^{[citation needed]}
- There are 194 conjugacy classes in the Monster group.^{[citation needed]}
- The exceptional Lie algebra E_{8} has dimension 248.^{[citation needed]}
- 720 is equal to 6!, the factorial of 6. Consequently, it is the order of the symmetric group of degree 6, which is isomorphic to B_{2}(2), and has an outer automorphism.^{[citation needed]}
- The constant term in the Laurent series of the j-invariant is equal to 744.^{[citation needed]}
- 1,440 is the order of the automorphism group of the simple group A_{6}.^{[1]}
- There are 4,060 points in the smallest faithful permutation representation of the Rudvalis group; its one-point stabilizer is the automorphism group of the Tits group.^{[citation needed]}
- The smallest faithful linear representation of the Baby monster group over any field has dimension 4,370.^{[citation needed]}
- The smallest faithful linear representation of the Baby monster group over the complex numbers has dimension 4,371.^{[citation needed]}
- The coefficient of the linear term in the McKay–Thompson series T_{2A} is equal to 4,372.^{[citation needed]}
- There are 196,560 points in the smallest faithful permutation representation of the Conway group Co0; its one-point stabilizer is the Conway group Co2.^{[citation needed]}
- The smallest faithful linear representation of the Monster group over any field has dimension 196,882.^{[citation needed]}
- The smallest faithful linear representation of the Monster group over the complex numbers has dimension 196,883.^{[citation needed]}
- The Griess algebra has dimension 196,884.^{[citation needed]}
- It is also the coefficient of the linear term in the Laurent series of the j-invariant, which led to the monstrous moonshine conjecture.^{[citation needed]}
- There is another irreducible representation of the Monster group with degree 21,296,876.^{[2]}
- 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).
- 16,776,960 is the order of the simple group PSL(2,256), which is isomorphic to PGL(2,256) and SL(2,256). It is one of the few known groups of Lie type over a finite field of characteristic 2, for which the Sylow 2-subgroup is not the largest Sylow subgroup.^{[citation needed]}
- The order of a simple group is almost never a perfect power. The emphasis is on "almost", since for NSW primes, the order of the simple group B_{2}(p) is a square number.^{[citation needed]} The simple group B_{2}(7) has order 138,297,600, which is the smallest perfect power that is also an order of a simple group.^{[citation needed]}
- 4,585,351,680 is the second smallest order with more than one simple group.^{[citation needed]}
- The order of a simple group is almost never an Achilles number. The emphasis is on "almost", since there is a simple group ^{2}A_{2}(19^{2}) of order 16,938,986,400, which is an Achilles number.^{[citation needed]}
- 281,474,976,645,120 is the order of the simple group PSL(2,65536), which is isomorphic to PGL(2,65536) and SL(2,65536). It is one of the few known groups of Lie type over a finite field of characteristic 2, for which the Sylow 2-subgroup is not the largest Sylow subgroup.^{[citation needed]}
Orders of non-abelian simple groups
This list contains finite non-abelian simple groups with unusual properties, such as:
- Its order has at most four distinct prime factors, or is a powerful number;
- 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 |
---|---|---|---|
A_{5} ≃ A_{1}(4) ≃ A_{1}(5) | 60 | 2^{2} × 3 × 5 | Exceptional Schur multiplier (for A_{1}(4)), and the 5-Sylow group is the largest Sylow subgroup. |
A_{1}(7) ≃ A_{2}(2) | 168 | 2^{3} × 3 × 7 | Exceptional Schur multiplier (for A_{2}(2)), and the 2-Sylow group is the largest Sylow subgroup. |
A_{6} ≃ A_{1}(9) ≃ B_{2}(2)′ | 360 | 2^{3} × 3^{2} × 5 | Exceptional outer automorphism group (for A_{6}) and Schur multiplier, and the 3-Sylow group is the largest Sylow subgroup. |
A_{1}(8) ≃ ^{2}G_{2}(3)′ | 504 | 2^{3} × 3^{2} × 7 | The 3-Sylow group is the largest Sylow subgroup. It is also the number of possible queen moves in starchess. |
A_{1}(11) | 660 | 2^{2} × 3 × 5 × 11 | It is also the number of feet in a furlong. And the engine displacement of kei cars is limited to 660 cm^{3}. |
A_{1}(13) | 1,092 | 2^{2} × 3 × 7 × 13 | It is also the number of pips in a double-12 domino set. |
A_{1}(17) | 2,448 | 2^{4} × 3^{2} × 17 | |
A_{7} | 2,520 | 2^{3} × 3^{2} × 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").^{[3]} |
A_{1}(19) | 3,420 | 2^{2} × 3^{2} × 5 × 19 | It is also the number of pips in a double-18 domino set. |
A_{1}(16) | 4,080 | 2^{4} × 3 × 5 × 17 | The 2-Sylow group is not the largest Sylow subgroup. |
A_{2}(3) | 5,616 | 2^{4} × 3^{3} × 13 | |
G_{2}(2)′ ≃ ^{2}A_{2}(3^{2}) | 6,048 | 2^{5} × 3^{3} × 7 | The 2-Sylow group is the largest Sylow subgroup. |
A_{8} ≃ A_{3}(2); A_{2}(4) | 20,160 | 2^{6} × 3^{2} × 5 × 7 | Smallest order with more than one simple group. Exceptional Schur multiplier (for A_{3}(2) and A_{2}(4)). It is also the number of minutes in a fortnight. |
B_{2}(3) ≃ ^{2}A_{3}(2^{2}) | 25,920 | 2^{6} × 3^{4} × 5 | Exceptional Schur multiplier (for ^{2}A_{3}(2^{2})), and the 3-Sylow group is the largest Sylow subgroup. It is also the number of halakim in a day. |
A_{9} | 181,440 | 2^{6} × 3^{4} × 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. |
D_{4}(2) | 174,182,400 | 2^{12} × 3^{5} × 5^{2} × 7 | Exceptional Schur multiplier. |
G_{2}(4) | 251,596,800 | 2^{12} × 3^{3} × 5^{2} × 7 × 13 | Exceptional Schur multiplier. |
^{2}A_{5}(2^{2}) | 9,196,830,720 | 2^{15} × 3^{6} × 5 × 7 × 11 | Exceptional Schur multiplier. |
F_{4}(2) | 3,311,126, 603,366,400 |
2^{24} × 3^{6} × 5^{2} × 7^{2} × 13 × 17 | Exceptional Schur multiplier. |
^{2}E_{6}(2^{2}) | 76,532, 479,683,774, 853,939,200 |
2^{36} × 3^{9} × 5^{2} × 7^{2} × 11 × 13 × 17 × 19 | Exceptional Schur multiplier. |
Group | Order | Factorization | Number of subgroups | Factorization |
---|---|---|---|---|
Mathieu group M11 | 7,920 | 2^{4} × 3^{2} × 5 × 11 | 8,651 | 41 × 211 |
Mathieu group M12 | 95,040 | 2^{6} × 3^{3} × 5 × 11 | 214,871 | 19 × 43 × 263 |
Janko group J1 | 175,560 | 2^{3} × 3 × 5 × 7 × 11 × 19 | 158,485 | 5 × 29 × 1093 |
Mathieu group M22 | 443,520 | 2^{7} × 3^{2} × 5 × 7 × 11 | 941,627 | 73 × 12,899 |
Janko group J2 | 604,800 | 2^{7} × 3^{3} × 5^{2} × 7 | 1,104,344 | 2^{3} × 31 × 61 × 73 |
Mathieu group M23 | 10,200,960 | 2^{7} × 3^{2} × 5 × 7 × 11 × 23 | 17,318,406 | 2 × 3 × 7 × 412,343 |
Tits group | 17,971,200 | 2^{11} × 3^{3} × 5^{2} × 13 | 50,285,950 | 2 × 5^{2} × 11 × 13^{2} × 541 |
Higman–Sims group | 44,352,000 | 2^{9} × 3^{2} × 5^{3} × 7 × 11 | 149,985,646 | 2 × 3,929 × 19,087 |
Janko group J3 | 50,232,960 | 2^{7} × 3^{5} × 5 × 17 × 19 | 71,564,248 | 2^{3} × 7 × 239 × 5,347 |
Mathieu group M24 | 244,823,040 | 2^{10} × 3^{3} × 5 × 7 × 11 × 23 | 1,363,957,253 | Prime |
McLaughlin group | 898,128,000 | 2^{7} × 3^{6} × 5^{3} × 7 × 11 | 1,719,739,392 | 2^{10} × 3 × 7 × 79,973 |
Held group | 4,030,387,200 | 2^{10} × 3^{3} × 5^{2} × 7^{3} × 17 | 22,303,017,686 | 2 × 17 × 211 × 310,889 |
Rudvalis group | 145,926,144,000 | 2^{14} × 3^{3} × 5^{3} × 7 × 13 × 29 | 963,226,363,401 | 3^{2} × 1,549 × 69,093,061 |
Suzuki group | 448,345,497,600 | 2^{13} × 3^{7} × 5^{2} × 7 × 11 × 13 | 4,057,939,316,149 | 7 × 19 × 127 × 27,111,439 |
O'Nan group | 460,815,505,920 | 2^{9} × 3^{4} × 5 × 7^{3} × 11 × 19 × 31 | 1,169,254,703,685 | 3 × 5 × 1,109 × 7,681 × 9,151 |
Conway group Co3 | 495,766,656,000 | 2^{10} × 3^{7} × 5^{3} × 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 | 2^{18} × 3^{6} × 5^{3} × 7 × 11 × 23 |
Fischer group Fi22 | 64,561,751,654,400 | 2^{17} × 3^{9} × 5^{2} × 7 × 11 × 13 |
Harada–Norton group | 273,030,912,000,000 | 2^{14} × 3^{6} × 5^{6} × 7 × 11 × 19 |
Lyons group | 51,765,179,004,000,000 | 2^{8} × 3^{7} × 5^{6} × 7 × 11 × 31 × 37 × 67 |
Thompson sporadic group | 90,745,943,887,872,000 | 2^{15} × 3^{10} × 5^{3} × 7^{2} × 13 × 19 × 31 |
Fischer group Fi23 | 4,089,470,473,293,004,800 | 2^{18} × 3^{13} × 5^{2} × 7 × 11 × 13 × 17 × 23 |
Conway group Co1 | 4,157,776,806,543,360,000 | 2^{21} × 3^{9} × 5^{4} × 7^{2} × 11 × 13 × 23 |
Janko group J4 | 86,775,571,046,077,562,880 | 2^{21} × 3^{3} × 5 × 7 × 11^{3} × 23 × 29 × 31 × 37 × 43 |
Fischer group Fi24 | 1,255,205,709,190,661,721,292,800 | 2^{21} × 3^{16} × 5^{2} × 7^{3} × 11 × 13 × 17 × 23 × 29 |
Baby monster group | 4,154,781,481,226,426, 191,177,580,544,000,000 |
2^{41} × 3^{13} × 5^{6} × 7^{2} × 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 |
2^{46} × 3^{20} × 5^{9} × 7^{6} × 11^{2} × 13^{3} × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 |
Sources
- ↑ Automorphisms of the symmetric and alternating groups
- ↑ OEIS, Sequence A001379. Accessed 2020-05-28.
- ↑ http://mzrg.com/rubik/orders.shtml