Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting | C7×3−1+2 ⋊ C3 |
| Sylow | C7 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C33 |
| Fitting | C7×3−1+2 | C3 |
| prime radical | C7⋊H(3,𝔽3) | C3 |
| hypercenter | 3−1+2 | ΓL(1,𝔽8) |
| hypocenter | C7 | #737 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
3^4,7|b^3=d,c^a=cd,e^a=e^2,c^b=cd55 conjugacy classes — view the full character table.
| maximal subgroup of | #10588065⋊C2, C2×#10588065, #13133621, … |
| maximal normal subgroup of | #10588065⋊C2, C2×#10588065, #13133621, … |
| maximal quotient of | C2×#10588065, #13133527, #13133529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 40);