Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C22×C33 |
| direct product | C32×#120⋊C3 |
| unambiguous semidirect product | #120⋊C33 |
| unambiguous semidirect product | (C3×#120)⋊C32 |
| Fitting | C32×#120 ⋊ C3 |
| derived | C24 ⋊ C2×C33 |
| Sylow | #120 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #120⋊C3 |
| commutator, cosocle, hypocenter | C24 | C2×C33 |
| Frattini | C22 | C2×C32×A4 |
| Fitting | C32×#120 | C3 |
| socle | C22×C32 | C2×A4 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C32 | |
338 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C32×#4625, C3×C3⋊#4625, C3⋊(C3⋊#4625), … |
| maximal normal subgroup of | C32×#4625, C3×C3⋊#4625, C3⋊(C3⋊#4625), … |
| maximal quotient of | C32×#379⋊C3, C32×#393⋊C3, C32×#395⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3699);