Finite Group DatabasebetaFinite Group Databasebeta| order | 8064 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×ΓL(1,𝔽8)×#5934 |
| Fitting | C2×C7×H(5,𝔽2) ⋊ C3×D3 |
| derived | C7×#997 ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×#5934 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ΓL(1,𝔽8)×#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C22×C3 |
| Frattini | C2 | C2×ΓL(1,𝔽8)×#1020 |
| Fitting | C2×C7×H(5,𝔽2) | C3×D3 |
| socle | C22×C7 | C3×#1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C22 | |
1808 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2)(4,5,6)(7,9,8)(11,16,14)(12,17,15);(3,4,6,8,5,7,9)(10,11,14,12)(13,15,17,16)130 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×#5934 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×#5934 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,5,6)(7,9,8)(11,16,14)(12,17,15), (3,4,6,8,5,7,9)(10,11,14,12)(13,15,17,16));