Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×D3×AΓL(1,𝔽8) |
| Fitting, socle | C24×C3 ⋊ C2×ΓL(1,𝔽8) |
| derived | C3×AGL(1,𝔽8) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×AΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C3×AGL(1,𝔽8) | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C2×ΓL(1,𝔽8) |
| LCS | G ▷ C3×AGL(1,𝔽8) | |
| UCS | C1 ◁ C2 | |
202 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(3,7)(4,9,6,10,5,8)(12,13);(1,2)(4,7,9,10,5,8,6)(11,12,13)48 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×AGL(3,𝔽2), C2×D3×C22⋊AΓL(1,𝔽8), C4×D3×AΓL(1,𝔽8) |
| maximal normal subgroup of | C4×D3×AΓL(1,𝔽8) |
| maximal quotient of | C2×D3×C22⋊AΓL(1,𝔽8), C4×D3×AΓL(1,𝔽8) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,7)(4,9,6,10,5,8)(12,13), (1,2)(4,7,9,10,5,8,6)(11,12,13));