Finite Group DatabasebetaFinite Group Databasebeta| order | 308 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽11) (order 1320) · Inn(G) = D11 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×D11 |
| unambiguous semidirect product | C11⋊(C22×C7) |
| unambiguous semidirect product | (C2×C11)⋊(C2×C7) |
| Fitting, socle | C2×C7×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C22×C7 |
| Hall | C7×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D11 |
| commutator, cosocle, hypocenter | C11 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2×C7 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 154 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4,5,6,7,8,9)(11,12)(13,14)(15,16)(17,18)(19,20);(1,2)(10,11,13,15,17,19,20,18,16,14,12)98 conjugacy classes — view the full character table.
| maximal subgroup of | D11×Dic7, #10597505, #10597506, … |
| maximal normal subgroup of | D11×Dic7, #10597505, #10597506, … |
| maximal quotient of | C7×Dic22, C4×C7×D11, C7×D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(308, 6);