Finite Group DatabasebetaFinite Group Databasebeta| order | 12642 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | (C43⋊C3)⋊(C7⋊D7) |
| Fitting, socle, derived, Hall | C72×C43 ⋊ C2×C3 |
| Hall | C72×C43⋊C3 ⋊ C2 |
| Hall | C7⋊D301 ⋊ C3 |
| Hall | C43⋊C3 ⋊ C7⋊D7 |
| Sylow | C72 ⋊ D43⋊C3 |
| Sylow | C43 ⋊ C3×C7⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C72×C43 | C2×C3 |
| LCS | G ▷ C72×C43 | |
| UCS | C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2107 |
A minimal faithful permutation representation — 3 generators on 57 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)(45,46)(47,48)(49,50)(52,53)(54,55)(56,57);(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)(51,52,54,56,57,55,53);(44,45,47,49,50,48,46)Polycyclic presentation:
2,3,7^2,43|c^a=c^6,d^a=d^6,e^a=e^42,e^b=e^36421 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12642, 107);