Finite Group DatabasebetaFinite Group Databasebeta| order | 22785 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×ΓL(1,𝔽8)×ΓL(1,𝔽32) |
| unambiguous semidirect product | C7⋊(C3×C7×ΓL(1,𝔽32)) |
| unambiguous semidirect product | (ΓL(1,𝔽8)×C31)⋊(C5×C7) |
| unambiguous semidirect product | (C7×ΓL(1,𝔽8)×C31)⋊C5 |
| Fitting, socle, Hall | C72×C31 ⋊ C3×C5 |
| derived | C7×C31 ⋊ C3×C5×C7 |
| Hall | C72×ΓL(1,𝔽32) ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8)×C31 ⋊ C5 |
| Sylow | C72 ⋊ C3×ΓL(1,𝔽32) |
| Sylow | C31 ⋊ C5×C7×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | ΓL(1,𝔽8)×ΓL(1,𝔽32) |
| commutator, cosocle, hypocenter | C7×C31 | C3×C5×C7 |
| Frattini | C1 | G |
| Fitting, socle | C72×C31 | C3×C5 |
| LCS | G ▷ C7×C31 | |
| UCS | C1 ◁ C7 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 651 |
385 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22785, 15);