Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C34 |
| direct product | ΓL(1,𝔽8)×H(3,𝔽3) |
| Fitting | C7×H(3,𝔽3) ⋊ C3 |
| Sylow | C7 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C33 |
| Fitting | C7×H(3,𝔽3) | C3 |
| hypercenter | H(3,𝔽3) | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×H(3,𝔽3) |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ H(3,𝔽3) | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 63 |
55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 38);