Finite Group DatabasebetaFinite Group Databasebeta| order | 264 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C5×D3 (order 240) · Inn(G) = D3 · Out(G) = C23×C5 |
| homology | Abelianization (H1) = C8×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C8×C11) |
| derived, Sylow | C3 ⋊ C8×C11 |
| Hall | C3×C11 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C11 | D3 |
| commutator, hypocenter | C3 | C8×C11 |
| Frattini | C4 | D3×C11 |
| Fitting | C3×C4×C11 | C2 |
| socle, prime radical | C2×C3×C11 | C4 |
| cosocle | C3×C4 | C2×C11 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C11 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 264 |
A minimal faithful permutation representation — 2 generators on 22 points:
(2,3)(4,8,10,5,7,11,6,9)(12,13,14,15,16,17,18,19,20,21,22);(1,2,3)132 conjugacy classes — view the full character table.
| maximal subgroup of | D11×C3⋊C8, D33⋊C8, #10587138, … |
| maximal normal subgroup of | D11×C3⋊C8, D33⋊C8, #10587138, … |
| maximal quotient of | C11×C3⋊C16, C2×C11×C3⋊C8, C11×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(264, 1);