Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3×C9 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3×D9 |
| commutator | C2×C3×C9 | C2×C8 |
| Frattini | C3×C8 | D32 |
| Fitting, cosocle | C3×C8×C9 | C22 |
| socle | C2×C32 | C2×C3⋊C8 |
| prime radical | C2×C3×D3 | C3⋊C8 |
| hypocenter | C3×C9 | M32 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C8 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|b^2=c,c^2=d,d^2=e,f^3=h,b^a=be,g^a=g^2,f^b=f^2h^2,h^b=h^2132 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#798, #13136961, #13136962, … |
| maximal normal subgroup of | D9×#798, #13136961, #13136962, … |
| maximal quotient of | #13134985, Dic3⋊(C9⋊C16), (C2×D3)⋊(C9⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 204);