Finite Group DatabasebetaFinite Group Databasebeta| order | 264 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C23×AGL(1,𝔽11) (order 880) · Inn(G) = D11 · Out(G) = C23×C5 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C3×C11⋊C8 |
| unambiguous semidirect product | C11⋊(C3×C8) |
| derived, Sylow | C11 ⋊ C3×C8 |
| Hall | C3×C11 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | D11 |
| commutator, hypocenter | C11 | C3×C8 |
| Frattini | C4 | C3×D11 |
| Fitting | C3×C4×C11 | C2 |
| socle, prime radical | C2×C3×C11 | C4 |
| cosocle | C4×C11 | C2×C3 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C3×C4 | |
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)(4,5)(6,7)(8,9)(10,11)(12,16,18,13,15,19,14,17)(20,21,22);(1,2,4,6,8,10,11,9,7,5,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C11⋊C8, D33⋊C8, #10587139, … |
| maximal normal subgroup of | D3×C11⋊C8, D33⋊C8, #10587139, … |
| maximal quotient of | C3×C11⋊C16, C2×C3×C11⋊C8, C9×C11⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(264, 2);