Finite Group DatabasebetaFinite Group Databasebeta| order | 264 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×C5×D3×D4 (order 480) · Inn(G) = C2×D3 · Out(G) = C23×C5 |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×D12 |
| unambiguous semidirect product | C4⋊(D3×C11) |
| Fitting | C3×C4×C11 ⋊ C2 |
| Hall | C3×C11 ⋊ D4 |
| Sylow | C3 ⋊ D4×C11 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C11 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C11 |
| Frattini | C2 | C2×D3×C11 |
| Fitting | C3×C4×C11 | C2 |
| socle | C2×C3×C11 | C22 |
| hypercenter | C4×C11 | D3 |
| hypocenter | C3 | D4×C11 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C11 ◁ C4×C11 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 132 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15)(16,17,18);(13,15)(17,18)99 conjugacy classes — view the full character table.
| maximal subgroup of | #10587141, #10587143, #10587145, … |
| maximal normal subgroup of | #10587141, #10587143, #10587145, … |
| maximal quotient of | C11×#204, C11×D24, C11×Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(264, 20);