Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#1010 |
| Fitting | C33×#41 ⋊ C2 |
| Sylow | C33 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C23×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 | C32×2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×#41 | |
1036 conjugacy classes of subgroups; only the 510 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 510 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
243 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#10591684, C3×#10591685, C3×#10591687, … |
| maximal normal subgroup of | C3×#10591684, C3×#10591685, C3×#10591687, … |
| maximal quotient of | C32×#5516, C32×#5531, C32×#5533, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4622);