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) = C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×Dic7 |
| unambiguous semidirect product | C7⋊(C4×C11) |
| derived, Sylow | C7 ⋊ C4×C11 |
| Hall | C7×C11 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C11 | D7 |
| commutator, hypocenter | C7 | C4×C11 |
| Frattini | C2 | C11×D7 |
| Fitting, socle, prime radical | C2×C7×C11 | C2 |
| cosocle | C2×C7 | C2×C11 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C11 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 308 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,18,19,20,21,22);(12,14,13,15)(17,22)(18,21)(19,20)110 conjugacy classes — view the full character table.
| maximal subgroup of | D11×Dic7, D77⋊C4, #10597506, … |
| maximal normal subgroup of | D11×Dic7, D77⋊C4, #10597506, … |
| maximal quotient of | C11×C7⋊C8, C2×C11×Dic7, C3×C11×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(308, 1);