Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C27 ⋊ C3 |
| Sylow | C7 ⋊ C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C3×ΓL(1,𝔽8) |
| commutator, socle | C3×C7 | C3×C9 |
| Fitting | C7×C27 | C3 |
| prime radical | C3×ΓL(1,𝔽8) | C9 |
| cosocle | C7×C9 | C32 |
| hypercenter | C27 | ΓL(1,𝔽8) |
| hypocenter | C7 | C27⋊C3 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C9 ◁ C27 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
3^4,7|b^3=c,c^3=d,b^a=bd,e^a=e^287 conjugacy classes — view the full character table.
| maximal subgroup of | #11765578, #11765599, #11765614, … |
| maximal normal subgroup of | #11765578, #11765599, #11765614, … |
| maximal quotient of | C2×#10588029, #13133455, (C3×ΓL(1,𝔽8))⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 4);