Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C42 · Schur multiplier (H2) = C3×C4 |
| direct product | C42×ΓL(1,𝔽8)2 |
| Fitting, Hall | C42×C72 ⋊ C32 |
| derived, Sylow | C72 ⋊ C32×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | ΓL(1,𝔽8)2 |
| commutator, hypocenter | C72 | C32×C42 |
| Frattini | C22 | C22×ΓL(1,𝔽8)2 |
| Fitting | C42×C72 | C32 |
| socle, cosocle | C22×C72 | C22×C32 |
| prime radical | C22×ΓL(1,𝔽8)2 | C22 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C42 | |
420 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
400 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C42×ΓL(1,𝔽8)⋊AGL(1,𝔽7), ΓL(1,𝔽8)2×#127, #72245701, … |
| maximal normal subgroup of | C42×ΓL(1,𝔽8)⋊AGL(1,𝔽7), ΓL(1,𝔽8)2×#127, #72245701, … |
Past the SmallGroups library coverage — no SmallGroups id.