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This page contains numbers appearing in group theory.
Contents
List of numbers in group theory
- The Conway group Co1 has 101 conjugacy classes.
- The constant term in the McKay–Thompson series T_{2A} is equal to 104.
- The largest order of any element in the Monster group is 119. There is also no other sporadic group with elements of larger order.
- The Frobenius kernel of the smallest non-solvable Frobenius group is the elementary abelian group of order 121.
- It is also the fourth largest undulating square number; this has been proved by David Moews.
- According to the Redmond–Sun conjecture, the numbers 121 and 125 are the seventh largest perfect powers not separated by a prime.
- Its prime factorization is 11^{2}.
- 121 is the second Friedman number.
- The exceptional Lie algebra E_{7} has dimension 133.
- The McKay-Thompson series of monstrous moonshine span a 163-dimensional vector space.
- The exceptional Lie algebra E_{7½} has dimension 190.
- It is also the 19th triangular number, and therefore the number of tiles in a double-18 domino set.
- There are 194 conjugacy classes in the Monster group.
- The exceptional Lie algebra E_{8} has dimension 248.
- It is also the number of positive commandments in Judaism.
- Furthermore, it was the number of Bundestag constituencies during 1965–1990.
- 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.
- The constant term in the Laurent series of the j-invariant is equal to 744.
- 1,440 is the order of the automorphism group of the simple group A_{6}.^{[1]}
- It is also the number of minutes in a day.^{[2]} Consequently, it is also mentioned in Antoine de Saint Exupéry's famous book The Little Prince in chapter 14, where the planet on which the lamplighter lives has 1,440 sunsets in the course of one Earth day.
- 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.
- It is also the number of known nonnegative integers which cannot be written as a sum of five nonnegative cubes; the largest of which is 1,290,740.
- The smallest faithful linear representation of the Baby monster group over any field has dimension 4,370.
- The smallest faithful linear representation of the Baby monster group over the complex numbers has dimension 4,371.
- The coefficient of the linear term in the McKay–Thompson series T_{2A} is equal to 4,372.
- 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.
- The smallest faithful linear representation of the Monster group over any field has dimension 196,882.
- The smallest faithful linear representation of the Monster group over the complex numbers has dimension 196,883.
- The Griess algebra has dimension 196,884.
- It is also the coefficient of the linear term in the Laurent series of the j-invariant, which led to the monstrous moonshine conjecture.
- There is another irreducible representation of the Monster group with degree 21,296,876.^{[3]}
- 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.
- 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. 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.
- 4,585,351,680 is the second smallest order with more than one simple group.
- 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.
- 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.
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").^{[4]} |
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 |
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 | \(f_2(11)<n<f_1^5(f_2(7))\) |
Sources
- ↑ Automorphisms of the symmetric and alternating groups
- ↑ Orders of magnitude (time)
- ↑ OEIS, Sequence A001379. Accessed 2020-05-28.
- ↑ http://mzrg.com/rubik/orders.shtml
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