Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
| direct product | C33×ΓL(1,𝔽8) |
| unambiguous semidirect product | C7⋊C34 |
| unambiguous semidirect product | (C3×C7)⋊C33 |
| Fitting, socle | C33×C7 ⋊ C3 |
| derived, Sylow | C7 ⋊ C34 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C33 | ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7 | C34 |
| Frattini | C1 | G |
| Fitting, socle | C33×C7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C33 | |
424 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 48);