Finite Group DatabasebetaFinite Group Databasebeta| order | 264 |
| isoclinism | isoclinic class of Dic22 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽11) (order 1760) · Inn(G) = C2×D11 · Out(G) = C23×C5 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D44 |
| unambiguous semidirect product | C4⋊(C3×D11) |
| Fitting | C3×C4×C11 ⋊ C2 |
| Hall | C3×C11 ⋊ D4 |
| Sylow | C11 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D11 |
| commutator, cosocle | C2×C11 | C22×C3 |
| Frattini | C2 | C2×C3×D11 |
| Fitting | C3×C4×C11 | C2 |
| socle | C2×C3×C11 | C22 |
| hypercenter | C3×C4 | D11 |
| hypocenter | C11 | C3×D4 |
| LCS | G ▷ C2×C11 ▷ C11 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
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);(5,7)(9,18)(10,17)(11,16)(12,15)(13,14)75 conjugacy classes — view the full character table.
| maximal subgroup of | #10587141, #10587142, #10587144, … |
| maximal normal subgroup of | #10587141, #10587142, #10587144, … |
| maximal quotient of | C3×#4314, C3×D88, C3×Dic44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(264, 15);