Finite Group DatabasebetaFinite Group Databasebeta| order | 22764 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C271 · Schur multiplier (H2) = C2 |
| direct product | C22⋊ΓL(1,𝔽8)×C271 |
| unambiguous semidirect product | C22⋊(ΓL(1,𝔽8)×C271) |
| unambiguous semidirect product | C7⋊(A4×C271) |
| Fitting, socle, Hall | C22×C7×C271 ⋊ C3 |
| derived, Hall | C22×C7 ⋊ C3×C271 |
| Hall | C7×C271 ⋊ A4 |
| Hall | C22×C271 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ A4×C271 |
| Sylow | C22 ⋊ ΓL(1,𝔽8)×C271 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C271 | C22⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7 | C3×C271 |
| Frattini | C1 | G |
| Fitting, socle | C22×C7×C271 | C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C271 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 7588 |
3252 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22764, 51);