Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#64306 |
| Fitting | C2×C33×C4 ⋊ C22 |
| Sylow | C33 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D32 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini | C2 | C22×C3×D32 |
| Fitting | C2×C33×C4 | C22 |
| socle | C2×C33 | C24 |
| hypercenter | C2×C3×C4 | D32 |
| hypocenter | C32 | C2×C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 | |
806 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | #13155421, #13155422, #13155431, … |
| maximal normal subgroup of | #13155421, #13155422, #13155431, … |
| maximal quotient of | C3×C4×#3796, C3×C4×#3797, C3×#10593901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4501);