Finite Group DatabasebetaFinite Group Databasebeta| order | 9408 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4×C7 |
| direct product | #127×#64425 |
| Fitting | C72×#127 ⋊ C2×C3 |
| Hall | #127×C7⋊D7 ⋊ C3 |
| Hall | #3862 ⋊ C2×#127 |
| Sylow | C72 ⋊ C2×C3×#127 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#64425 |
| commutator, socle, cosocle | C22×C72 | C24×C3 |
| Fitting | C72×#127 | C2×C3 |
| hypercenter | #127 | #64425 |
| hypocenter | C72 | C2×C3×#127 |
| LCS | G ▷ C22×C72 ▷ C72 | |
| UCS | C1 ◁ C22 ◁ #127 | |
2490 conjugacy classes of subgroups; only the 288 normal subgroups are shown.
This group has 2490 conjugacy classes of subgroups — too many to draw. Only its 288 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
196 conjugacy classes — view the full character table.
| maximal subgroup of | #127×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246074, #453×#64425 |
| maximal normal subgroup of | #127×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246074, #453×#64425 |
Past the SmallGroups library coverage — no SmallGroups id.