Finite Group DatabasebetaFinite Group Databasebeta| order | 9408 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | H(5,𝔽2)×ΓL(1,𝔽8)⋊D7 |
| Fitting | C72×H(5,𝔽2) ⋊ C2×C3 |
| Hall | H(5,𝔽2)×C7⋊D7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×H(5,𝔽2) |
| Sylow | C72 ⋊ C2×C3×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×ΓL(1,𝔽8)⋊D7 |
| commutator, socle, cosocle | C2×C72 | C25×C3 |
| Fitting | C72×H(5,𝔽2) | C2×C3 |
| hypercenter | H(5,𝔽2) | ΓL(1,𝔽8)⋊D7 |
| hypocenter | C72 | C2×C3×H(5,𝔽2) |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
4490 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
374 conjugacy classes. The character table is too large to display.
| maximal subgroup of | B(4,𝔽2)×ΓL(1,𝔽8)⋊D7, H(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #453×ΓL(1,𝔽8)⋊D7 |
| maximal normal subgroup of | B(4,𝔽2)×ΓL(1,𝔽8)⋊D7, H(5,𝔽2)×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #453×ΓL(1,𝔽8)⋊D7 |
Past the SmallGroups library coverage — no SmallGroups id.