The database currently contains $544{,}831$ groups from many different sources, the largest of which is $S_{47}$ of order $47!$. In addition, it contains $275{,}339{,}797$ of their subgroups and $39{,}901{,}075$ of their irreducible complex characters.

The database of finite groups consists of groups, subgroups, characters and conjugacy classes. The set of groups included contains:

  • All groups of order up to 2000, except those whose order is larger than 500 and divisible by 128,

  • All groups with a faithful transitive permutation representation of degree up to 47, except those of degree 32 and order between 512 and $4 \times 10^{10}$,

  • All groups with a faithful permutation representation of degree up to 15,

  • All groups with a rational matrix representation of dimension up to 6,

  • Finite groups of Lie type up to certain bounds,

  • Chevalley groups $E(6,2)$, $E(7,2)$, $F(4,2)$, $G(2,2)$, $G(2,3)$, $G(2,5)$, ${}^2B(2,2)$, ${}^2B(2,8)$, ${}^2B(2,32)$, ${}^3D(4,2)$, ${}^2E(6,2)$, ${}^2F(4,2)$, ${}^2F(4,2)'$, ${}^2G(2,3)$, ${}^2G(2,27)$,

  • Subgroups of $\GL(2,\Z/N)$ for $N$ up to 124, except for $N=80,96,104,112, 120$

  • Subgroups of $\GL(2,\mathbb{F}_q)$ for $q$ up to 1000 (except $\mathbb{F}_{512}$), subgroups of $\GL(3, \mathbb{F}_q)$ for $q$ up to 13, subgroups of $\GL(4, \mathbb{F}_q)$ for $q$ up to 5, subgroups of $\GL(5,2)$,

  • Perfect groups of order up to 50,000,

  • Sporadic simple groups of order up to $10^{15}$, except $ON$ and $Suz$.

For each group, the lattice of subgroups, complex and rational character tables, representatives for the conjugacy classes, automorphism and outer automorphism group, rank, Schur multiplier and various other quantities were computed, subject to limits on time, memory and errors arising in Magma. For space reasons, complex and rational character tables are never stored if larger than $511 \times 511$, and subgroups are only computed up to automorphism if there are more than 128 conjugacy classes of subgroups. For time reasons, there are some large groups where subgroups are only computed up to conjugacy and not up to automorphism. All normal subgroups are computed when possible, except that normal subgroups of size roughly the square root of the group order are omitted when there are more than 4096 normal subgroups.

Distribution of Solvability as a function of order

order factorization type
$1$ $p$ $p^2$ $p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
solvability type
cyclic 1 303 14 12 3 1315 221 156 29 10 65 602 549 268 3548
100.00% 100.00% 31.11% 1.22% 0.00% 33.53% 18.54% 5.03% 0.07% 0.01% 0.07% 5.62% 0.96% 0.39% 0.65%
abelian and metacyclic, not cyclic 31 24 31 316 163 220 113 115 501 154 362 2030
68.89% 2.44% 0.05% 26.51% 5.25% 0.54% 0.06% 0.12% 4.68% 0.27% 0.52% 0.37%
abelian, not metacyclic 37 52 167 111 108 189 183 94 941
3.76% 0.08% 5.38% 0.27% 0.06% 0.19% 0.32% 0.14% 0.17%
nilpotent and metacyclic, not abelian 52 91 309 220 205 435 391 178 1881
5.29% 0.13% 9.96% 0.54% 0.11% 0.45% 0.69% 0.26% 0.35%
nilpotent and metabelian, not abelian or metacyclic 858 66326 238 1408 2044 6516 919 159 78468
87.28% 96.95% 7.67% 3.47% 1.06% 6.72% 1.61% 0.23% 14.40%
nilpotent, not metabelian 1910 8 142 2060
2.79% 0.02% 0.15% 0.38%
metacyclic, not nilpotent 2607 567 1121 763 672 1384 7452 7467 6582 28615
66.47% 47.57% 36.13% 1.88% 0.35% 1.43% 69.54% 13.10% 9.49% 5.25%
metabelian and supersolvable, not nilpotent or metacyclic 66 868 19651 41773 33138 1284 34164 12056 143000
5.54% 27.97% 48.45% 21.72% 34.17% 11.98% 59.95% 17.38% 26.25%
metabelian and monomial, not supersolvable 22 20 1197 3385 371 570 851 2912 9328
1.85% 0.64% 2.95% 1.76% 0.38% 5.32% 1.49% 4.20% 1.71%
metabelian, not monomial 8 70 1 2 81
0.02% 0.04% 0.00% 0.00% 0.01%
supersolvable, not nilpotent or metabelian 47 1725 8146 1795 187 569 12469
1.51% 4.25% 4.24% 1.85% 0.33% 0.82% 2.29%
monomial, not supersolvable or metabelian 4 2649 7832 1716 108 902 3176 16387
0.13% 6.53% 4.07% 1.77% 1.01% 1.58% 4.58% 3.01%
solvable, not monomial or metabelian 9 1061 5728 1356 1247 1131 10532
0.29% 2.62% 2.98% 1.40% 2.19% 1.63% 1.93%
not solvable 193 8160 23728 32081
1.80% 14.32% 34.21% 5.89%
metabelian, not supersolvable, unknown if monomial 1 266 304 572 6 117 416 1682
0.03% 0.66% 0.16% 0.59% 0.06% 0.21% 0.60% 0.31%
solvable, not metabelian, unknown if monomial 11243 121893 49171 1692 17729 201728
27.72% 63.39% 50.71% 2.97% 25.56% 37.03%
Total 1 303 45 983 68413 3922 1192 3103 40559 192283 96966 10716 56983 69362 544831
0.00% 0.06% 0.01% 0.18% 12.56% 0.72% 0.22% 0.57% 7.44% 35.29% 17.80% 1.97% 10.46% 12.73%

Distribution of nilpotency class as a function of order

order factorization type
$1$ $p$ $p^2$ $p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
nilpotency class
not 2607 655 2070 38563 189803 89504 9613 54787 68301 455903
66.47% 54.95% 66.71% 95.08% 98.71% 92.30% 89.71% 96.15% 98.47% 83.68%
0 1 1
100.00% 0.00%
1 303 45 73 86 1315 537 486 360 231 369 1103 886 724 6518
100.00% 100.00% 7.43% 0.13% 33.53% 45.05% 15.66% 0.89% 0.12% 0.38% 10.29% 1.55% 1.04% 1.20%
2 338 33218 402 805 1153 2878 722 236 39752
34.38% 48.56% 12.96% 1.98% 0.60% 2.97% 1.27% 0.34% 7.30%
3 457 25343 145 654 918 3091 461 78 31147
46.49% 37.04% 4.67% 1.61% 0.48% 3.19% 0.81% 0.11% 5.72%
4 104 7107 140 162 793 98 14 8418
10.58% 10.39% 0.35% 0.08% 0.82% 0.17% 0.02% 1.55%
5 11 1900 31 16 204 19 4 2185
1.12% 2.78% 0.08% 0.01% 0.21% 0.03% 0.01% 0.40%
6 440 3 99 6 2 550
0.64% 0.01% 0.10% 0.01% 0.00% 0.10%
7 196 2 25 3 2 228
0.29% 0.00% 0.03% 0.01% 0.00% 0.04%
8 112 1 3 1 1 118
0.16% 0.00% 0.00% 0.00% 0.00% 0.02%
9 11 11
0.02% 0.00%
Total 1 303 45 983 68413 3922 1192 3103 40559 192283 96966 10716 56983 69362 544831
0.00% 0.06% 0.01% 0.18% 12.56% 0.72% 0.22% 0.57% 7.44% 35.29% 17.80% 1.97% 10.46% 12.73%

Distribution of rank as a function of order

order factorization type
$1$ $p$ $p^2$ $p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
rank
0 1 1
100.00% 0.00%
1 303 14 12 3 1315 221 156 29 10 65 602 549 268 3547
100.00% 31.11% 1.22% 0.00% 33.53% 18.54% 5.03% 0.07% 0.01% 0.07% 5.62% 0.96% 0.39% 0.65%
2 31 256 1427 2607 932 2155 11200 40871 22289 9501 23313 47982 162564
68.89% 26.04% 2.09% 66.47% 78.19% 69.45% 27.61% 21.26% 22.99% 88.66% 40.91% 69.18% 29.84%
3 529 9871 39 702 20185 84769 49915 613 22040 18629 207292
53.81% 14.43% 3.27% 22.62% 49.77% 44.09% 51.48% 5.72% 38.68% 26.86% 38.05%
4 168 25380 88 8545 55010 21755 10588 2309 123843
17.09% 37.10% 2.84% 21.07% 28.61% 22.44% 18.58% 3.33% 22.73%
5 16 30576 2 565 10116 2720 460 160 44615
1.63% 44.69% 0.06% 1.39% 5.26% 2.81% 0.81% 0.23% 8.19%
6 2 901 33 1378 178 24 14 2530
0.20% 1.32% 0.08% 0.72% 0.18% 0.04% 0.02% 0.46%
7 193 2 119 30 5 349
0.28% 0.00% 0.06% 0.03% 0.01% 0.06%
8 43 9 10 4 66
0.06% 0.00% 0.01% 0.01% 0.01%
9 15 1 3 19
0.02% 0.00% 0.00% 0.00%
10 3 1 4
0.00% 0.00% 0.00%
12 1 1
0.00% 0.00%
Total 1 303 45 983 68413 3922 1192 3103 40559 192283 96966 10716 56983 69362 544831
0.00% 0.06% 0.01% 0.18% 12.56% 0.72% 0.22% 0.57% 7.44% 35.29% 17.80% 1.97% 10.46% 12.73%

Distribution of derived length among solvable groups as a function of order

order factorization type
$1$ $p$ $p^2$ $p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
derived length
0 1 1
100.00% 0.00%
1 303 45 73 86 1315 537 486 360 231 369 1103 886 724 6518
100.00% 100.00% 7.43% 0.13% 33.53% 45.05% 15.66% 0.89% 0.12% 0.38% 10.48% 1.81% 1.59% 1.27%
2 910 66417 2607 655 2557 23513 48453 42417 9312 43909 22305 263055
92.57% 97.08% 66.47% 54.95% 82.40% 57.97% 25.20% 43.74% 88.49% 89.94% 48.88% 51.30%
3 1902 58 9427 46801 34597 108 2584 7332 102809
2.78% 1.87% 23.24% 24.34% 35.68% 1.03% 5.29% 16.07% 20.05%
4 8 2 6563 77776 19182 1444 8591 113566
0.01% 0.06% 16.18% 40.45% 19.78% 2.96% 18.83% 22.15%
5 696 16811 401 5409 23317
1.72% 8.74% 0.41% 11.85% 4.55%
6 1982 1257 3239
1.03% 2.75% 0.63%
7 221 16 237
0.11% 0.04% 0.05%
8 8 8
0.00% 0.00%
Total 1 303 45 983 68413 3922 1192 3103 40559 192283 96966 10523 48823 45634 512750
0.00% 0.06% 0.01% 0.19% 13.34% 0.76% 0.23% 0.61% 7.91% 37.50% 18.91% 2.05% 9.52% 8.90%

Distribution of automorphism group order as a function of order for nonabelian groups

order factorization type
$p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
aut order
1-7 1 1
0.04% 0.00%
8-32 10 2 7 10 29
1.10% 0.08% 1.07% 0.38% 0.01%
33-128 47 2 17 12 57 24 29 27 16 231
5.16% 0.00% 0.65% 1.83% 2.18% 0.06% 0.03% 0.28% 0.03% 0.04%
129-512 139 173 63 21 135 178 82 399 163 269 77 1699
15.27% 0.25% 2.42% 3.21% 5.16% 0.44% 0.04% 0.43% 1.70% 0.48% 0.11% 0.32%
513-2048 89 2262 246 31 287 1095 788 2622 609 2038 1069 11136
9.78% 3.31% 9.44% 4.73% 10.97% 2.73% 0.41% 2.80% 6.34% 3.64% 1.56% 2.09%
2049-8192 84 9115 343 48 392 3815 5485 9518 1067 6996 3909 40772
9.23% 13.34% 13.16% 7.33% 14.98% 9.50% 2.88% 10.15% 11.11% 12.48% 5.71% 7.64%
8193-65536 222 42181 672 139 634 11730 28961 20878 2127 23916 10631 142091
24.40% 61.73% 25.78% 21.22% 24.23% 29.22% 15.21% 22.27% 22.14% 42.67% 15.53% 26.63%
65537- 319 14594 1263 397 1102 23308 155058 60284 5614 22811 52782 337532
35.05% 21.36% 48.45% 60.61% 42.11% 58.05% 81.45% 64.32% 58.44% 40.70% 77.09% 63.27%
Total 910 68327 2607 655 2617 40150 190374 93730 9607 56046 68468 533491
0.17% 12.81% 0.49% 0.12% 0.49% 7.53% 35.68% 17.57% 1.80% 10.51% 12.83%

Distribution of outer aut. group order as a function of order for nonabelian groups

order factorization type
$p^{3-6}$ $p^{7+}$ $pq,pqr,\ldots$ $p^2q,p^2q^2$ $p^3q,p^4q$ $p^{3+}q^2$ $p^{3+}q^{3+}$ $p^{5+}q$ $p^{1,2}q^{1,2}r^{1,2}\cdots$ $p^{3+}qr\cdots$ other Total
outer order
1 39 1 3 43 672 17 33 377 1784 2969
1.50% 0.15% 0.11% 0.11% 0.35% 0.02% 0.34% 0.67% 2.61% 0.56%
2-7 18 22 262 28 77 781 27160 463 586 1935 20241 51573
1.98% 0.03% 10.05% 4.27% 2.94% 1.95% 14.27% 0.49% 6.10% 3.45% 29.56% 9.67%
8-32 161 2084 421 66 268 8401 79513 12275 1251 11314 20400 136154
17.69% 3.05% 16.15% 10.08% 10.24% 20.92% 41.77% 13.10% 13.02% 20.19% 29.79% 25.52%
33-128 170 7544 654 140 618 11330 36236 22221 2028 15306 9910 106157
18.68% 11.04% 25.09% 21.37% 23.61% 28.22% 19.03% 23.71% 21.11% 27.31% 14.47% 19.90%
129-512 178 12076 928 216 932 9281 22677 37843 2215 17280 6559 110185
19.56% 17.67% 35.60% 32.98% 35.61% 23.12% 11.91% 40.37% 23.06% 30.83% 9.58% 20.65%
513-2048 137 30283 173 79 504 5040 12195 13212 1519 6246 4569 73957
15.05% 44.32% 6.64% 12.06% 19.26% 12.55% 6.41% 14.10% 15.81% 11.14% 6.67% 13.86%
2049-8192 98 8476 91 57 124 2561 6586 4141 1062 2159 2734 28089
10.77% 12.41% 3.49% 8.70% 4.74% 6.38% 3.46% 4.42% 11.05% 3.85% 3.99% 5.27%
8193-65536 88 4411 39 56 57 1906 2681 2100 770 1044 1609 14761
9.67% 6.46% 1.50% 8.55% 2.18% 4.75% 1.41% 2.24% 8.01% 1.86% 2.35% 2.77%
65537- 60 3431 12 34 807 2654 1458 143 385 662 9646
6.59% 5.02% 1.83% 1.30% 2.01% 1.39% 1.56% 1.49% 0.69% 0.97% 1.81%
Total 910 68327 2607 655 2617 40150 190374 93730 9607 56046 68468 533491
0.17% 12.81% 0.49% 0.12% 0.49% 7.53% 35.68% 17.57% 1.80% 10.51% 12.83%