Finite Group DatabasebetaFinite Group Databasebeta| order | 264 |
| isoclinism | isoclinic class of D11 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×D11 |
| unambiguous semidirect product | C11⋊(C23×C3) |
| unambiguous semidirect product | (C2×C11)⋊(C22×C3) |
| Fitting, socle | C22×C3×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C23×C3 |
| Hall | C3×C11 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3 | D11 |
| commutator, cosocle, hypocenter | C11 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C22×C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4)(5,6,7)(9,10)(11,12)(13,14)(15,16)(17,18);(3,4)(8,9,11,13,15,17,18,16,14,12,10);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D11)⋊Dic3, C3×(C2×D11)⋊C4, C2×Dic3×D11, … |
| maximal normal subgroup of | (C2×D11)⋊Dic3, C3×(C2×D11)⋊C4, C2×Dic3×D11, … |
| maximal quotient of | C2×C3×Dic22, C2×C3×C4×D11, C2×C3×D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(264, 36);