Finite Group DatabasebetaFinite Group Databasebeta| order | 31218 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C121⋊(D43⋊C3) |
| unambiguous semidirect product | (C43⋊C3)⋊D121 |
| unambiguous semidirect product | D5203⋊C3 |
| Fitting, derived, Hall | C43×C121 ⋊ C2×C3 |
| Hall | C121×C43⋊C3 ⋊ C2 |
| Hall | D5203 ⋊ C3 |
| Hall | C43⋊C3 ⋊ D121 |
| Sylow | C121 ⋊ D43⋊C3 |
| Sylow | C43 ⋊ C3×D121 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C43×C121 | C2×C3 |
| Frattini | C11 | C11⋊(D43⋊C3) |
| socle | C11×C43 | C3×D11 |
| LCS | G ▷ C43×C121 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 164 |
| transitive degree | μt(G) = 5203 |
A minimal faithful permutation representation — 2 generators on 164 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(45,47)(46,50)(48,53)(49,56)(51,59)(52,58)(54,63)(55,66)(57,68)(60,72)(61,71)(62,70)(64,77)(65,80)(67,82)(69,84)(73,89)(74,88)(75,87)(76,86)(78,95)(79,98)(81,100)(83,102)(85,104)(90,110)(91,109)(92,108)(93,107)(94,106)(96,97)(99,117)(101,119)(103,121)(105,123)(111,124)(112,128)(113,127)(114,126)(115,125)(116,133)(118,135)(120,137)(122,139)(129,140)(130,143)(131,142)(132,141)(134,147)(136,149)(138,151)(144,152)(145,154)(146,153)(148,157)(150,159)(155,160)(156,161)(158,163)(162,164);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,58,87,127,154,164,156,131,93,62,47,50,71,108,143,160,158,146,114,76,53,56,59,88,128,152,150,148,132,94,63,66,68,72,109,140,138,136,134,115,77,80,82,84,89,124,122,120,118,116,95,98,100,102,104,110,105,103,101,99,97,96,117,119,121,123,90,85,83,81,79,78,133,135,137,139,111,73,69,67,65,64,125,147,149,151,129,91,60,57,55,54,106,141,157,159,144,112,74,51,49,48,86,126,153,163,155,130,92,61,46,45,70,107,142,161,162,145,113,75,52)Polycyclic presentation:
2,3,11,43,11|c^11=e^10,c^a=c^10e,d^a=d^42,e^a=e^10,d^b=d^361033 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31218, 12);