Finite Group DatabasebetaFinite Group Databasebeta| order | 7056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#72222015 |
| Hall | C2×#87641 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ C2×D4 |
| Sylow | C72 ⋊ C2×C32×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(1,𝔽7)2 |
| commutator, cosocle | C2×C72 | C23×C32 |
| Frattini | C2 | C2×AGL(1,𝔽7)2 |
| Fitting, socle | C22×C72 | C22×C32 |
| hypocenter | C72 | C2×C32×D4 |
| LCS | G ▷ C2×C72 ▷ C72 | |
| UCS | C1 ◁ C22 | |
680 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
142 conjugacy classes — view the full character table.
| maximal quotient of | #72245459 |
Past the SmallGroups library coverage — no SmallGroups id.