Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×ΓL(1,𝔽8)×GL(3,𝔽2) |
| Fitting | C2×C7 ⋊ C3×GL(3,𝔽2) |
| socle | C2×C7×GL(3,𝔽2) ⋊ C3 |
| derived | C7×GL(3,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×GL(3,𝔽2) |
| commutator, hypocenter | C7×GL(3,𝔽2) | C2×C3 |
| Frattini | C1 | G |
| Fitting | C2×C7 | C3×GL(3,𝔽2) |
| socle | C2×C7×GL(3,𝔽2) | C3 |
| cosocle | C7 | C2×C3×GL(3,𝔽2) |
| LCS | G ▷ C7×GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
188 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 98 |
A minimal faithful permutation representation — 2 generators on 16 points:
(3,4,6,7,8,9,5)(10,11,13)(12,14,16);(1,2)(3,6,7)(4,5,9)(11,12)(14,15)60 conjugacy classes — view the full character table.
| maximal subgroup of | GL(3,𝔽2)×SL(2,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,6,7,8,9,5)(10,11,13)(12,14,16), (1,2)(3,6,7)(4,5,9)(11,12)(14,15));