Finite Group DatabasebetaFinite Group Databasebeta| order | 1764 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C3 |
| direct product | ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C22×ΓL(1,𝔽8))⋊ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7×ΓL(1,𝔽8))⋊A4 |
| Fitting, socle, derived, Hall | C22×C72 ⋊ C32 |
| Sylow | C72 ⋊ C3×A4 |
| Sylow | C22 ⋊ ΓL(1,𝔽8)2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C22×C72 | C32 |
| LCS | G ▷ C22×C72 | |
| UCS | C1 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1764, 168);