Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×D3×#41 |
| Fitting | C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C22×D3 |
| commutator, cosocle | C2×C3 | C24×C32 |
| Frattini | C2 | C23×C32×D3 |
| Fitting | C33×#41 | C2 |
| socle | C2×C33 | C24 |
| hypercenter | C32×#41 | D3 |
| hypocenter | C3 | C2×C32×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#41 | |
1144 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D3×#837, C3×#10591670, D3×#63664, … |
| maximal normal subgroup of | C3×D3×#837, C3×#10591670, D3×#63664, … |
| maximal quotient of | C32×#5517, C32×#5531, C32×#5532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4620);