Finite Group DatabasebetaFinite Group Databasebeta| order | 56 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C3×GL(3,𝔽2) (order 1008) · Inn(G) = C1 · Out(G) = C2×C3×GL(3,𝔽2) |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽7)×AGL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C7 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(7,8,9,10,11,12,13);(3,4);(5,6)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(56, 13);