Finite Group DatabasebetaFinite Group Databasebeta| order | 56 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = S4×AGL(1,𝔽7) (order 1008) · Inn(G) = D7 · Out(G) = C3×S4 |
| holomorph | G ⋊ Aut(G) = #72224594 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×D7 |
| unambiguous semidirect product | C7⋊C23 |
| unambiguous semidirect product | (C2×C7)⋊C22 |
| Fitting, socle | C22×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7 |
| commutator, cosocle, hypocenter | C7 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2)(3,4)(6,7)(8,9)(10,11);(3,4)(5,6,8,10,11,9,7);(1,2)20 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D7)⋊C4, C2×C4×D7, C2×D28, … |
| maximal normal subgroup of | (C2×D7)⋊C4, C2×C4×D7, C2×D28, … |
| maximal quotient of | C2×Dic14, C2×C4×D7, C2×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(56, 12);