Finite Group DatabasebetaFinite Group Databasebeta| order | 308 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C5×AGL(1,𝔽7) (order 840) · Inn(G) = D7 · Out(G) = C22×C3×C5 |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C2 |
| direct product | C2×C11×D7 |
| unambiguous semidirect product | C7⋊(C22×C11) |
| unambiguous semidirect product | (C2×C7)⋊(C2×C11) |
| Fitting, socle | C2×C7×C11 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C11 |
| Hall | C7×C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C11 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C11 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C11 | |
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)(4,5)(6,7)(8,9)(10,11,12,13,14,15,16,17,18,19,20);(1,2)(3,4,6,8,9,7,5)110 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic11, #10597505, #10597507, … |
| maximal normal subgroup of | D7×Dic11, #10597505, #10597507, … |
| maximal quotient of | C11×Dic14, C4×C11×D7, C11×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(308, 7);