Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C8×ΓL(1,𝔽8)2 |
| Fitting, Hall | C72×C8 ⋊ C32 |
| derived, Sylow | C72 ⋊ C32×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | ΓL(1,𝔽8)2 |
| commutator, hypocenter | C72 | C32×C8 |
| Frattini | C4 | C2×ΓL(1,𝔽8)2 |
| Fitting | C72×C8 | C32 |
| socle | C2×C72 | C32×C4 |
| prime radical | C2×ΓL(1,𝔽8)2 | C4 |
| cosocle | C4×C72 | C2×C32 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C8 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7), M16×ΓL(1,𝔽8)2, #72245290, … |
| maximal normal subgroup of | C8×ΓL(1,𝔽8)⋊AGL(1,𝔽7), M16×ΓL(1,𝔽8)2, #72245290, … |
Past the SmallGroups library coverage — no SmallGroups id.