Finite Group DatabasebetaFinite Group Databasebeta| order | 26862 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C37⋊C3)⋊D121 |
| unambiguous semidirect product | C121⋊(D37⋊C3) |
| unambiguous semidirect product | D4477⋊C3 |
| Fitting, derived, Hall | C37×C121 ⋊ C2×C3 |
| Hall | C37⋊C3×C121 ⋊ C2 |
| Hall | D4477 ⋊ C3 |
| Sylow | C121 ⋊ D37⋊C3 |
| Hall | C37⋊C3 ⋊ D121 |
| Sylow | C37 ⋊ C3×D121 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C37×C121 | C2×C3 |
| Frattini | C11 | C11⋊(D37⋊C3) |
| socle | C11×C37 | C3×D11 |
| LCS | G ▷ C37×C121 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 158 |
| transitive degree | μt(G) = 4477 |
A minimal faithful permutation representation — 2 generators on 158 points:
(2,6,7,3,4,8)(5,14,15,9,10,23)(11,26,27,17,18,34)(12,25,21,19,16,13)(20,36,37,29,28,32)(22,24,35,31,30,33)(39,41)(40,44)(42,47)(43,50)(45,53)(46,52)(48,57)(49,60)(51,62)(54,66)(55,65)(56,64)(58,71)(59,74)(61,76)(63,78)(67,83)(68,82)(69,81)(70,80)(72,89)(73,92)(75,94)(77,96)(79,98)(84,104)(85,103)(86,102)(87,101)(88,100)(90,91)(93,111)(95,113)(97,115)(99,117)(105,118)(106,122)(107,121)(108,120)(109,119)(110,127)(112,129)(114,131)(116,133)(123,134)(124,137)(125,136)(126,135)(128,141)(130,143)(132,145)(138,146)(139,148)(140,147)(142,151)(144,153)(149,154)(150,155)(152,157)(156,158);(1,2,5,11,18,28,33,34,31,21,7,6,12,20,30,10,16,23,32,37,15,25,14,24,29,19,4,8,13,22,27,35,36,26,17,9,3)(38,52,81,121,148,158,150,125,87,56,41,44,65,102,137,154,152,140,108,70,47,50,53,82,122,146,144,142,126,88,57,60,62,66,103,134,132,130,128,109,71,74,76,78,83,118,116,114,112,110,89,92,94,96,98,104,99,97,95,93,91,90,111,113,115,117,84,79,77,75,73,72,127,129,131,133,105,67,63,61,59,58,119,141,143,145,123,85,54,51,49,48,100,135,151,153,138,106,68,45,43,42,80,120,147,157,149,124,86,55,40,39,64,101,136,155,156,139,107,69,46)Polycyclic presentation:
2,3,11,37,11|c^11=e^10,c^a=c^10e,d^a=d^36,e^a=e^10,d^b=d^26912 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(26862, 12);