Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27 · Schur multiplier (H2) = C3 |
| direct product | C3×C7⋊C27 |
| unambiguous semidirect product | (C3×C7)⋊C27 |
| derived, Sylow | C7 ⋊ C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C9 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C3×C27 |
| Frattini | C9 | C3×ΓL(1,𝔽8) |
| Fitting | C3×C7×C9 | C3 |
| socle, prime radical | C32×C7 | C9 |
| cosocle | C7×C9 | C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C9 | |
28 conjugacy classes of subgroups.
| transitive degree | μt(G) = 567 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7⋊C27, D3×C7⋊C27, C3⋊(D7⋊C27), … |
| maximal normal subgroup of | C3×D7⋊C27, D3×C7⋊C27, C3⋊(D7⋊C27), … |
| maximal quotient of | C2×C3×C7⋊C27, C9×C7⋊C27, C32⋊(C7⋊C27), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 6);