Finite Group DatabasebetaFinite Group Databasebeta| order | 56 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = D4×AGL(1,𝔽7) (order 336) · Inn(G) = D7 · Out(G) = C3×D4 |
| holomorph | G ⋊ Aut(G) = #72215186 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic7 |
| unambiguous semidirect product | (C2×C7)⋊C4 |
| derived, Sylow | C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D7 |
| commutator, hypocenter | C7 | C2×C4 |
| Frattini | C2 | C2×D7 |
| Fitting, socle, prime radical | C22×C7 | C2 |
| cosocle | C2×C7 | C22 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4,5,6)(8,9)(10,11)(12,13);(1,2)(7,8,10,12,13,11,9)20 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic7, Dic7⋊C4, C4⋊Dic7, … |
| maximal normal subgroup of | C4×Dic7, Dic7⋊C4, C4⋊Dic7, … |
| maximal quotient of | C2×C7⋊C8, #1136, C4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(56, 6);