Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C8 · Schur multiplier (H2) = C2×C3 |
| direct product | C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| Fitting | C72×C8 ⋊ C2×C32 |
| derived, Sylow | C72 ⋊ C2×C32×C8 |
| Hall | C8×C7⋊D7 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ C2×C8 |
308 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
208 conjugacy classes — view the full character table.
| maximal subgroup of | M16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), SD16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), D8×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
| maximal normal subgroup of | M16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), SD16×ΓL(1,𝔽8)⋊AGL(1,𝔽7), D8×ΓL(1,𝔽8)⋊AGL(1,𝔽7) |
Past the SmallGroups library coverage — no SmallGroups id.