Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#63715 |
| Fitting | C3×C9×#41 ⋊ C2 |
| Sylow | C3×C9 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C23×D9 |
| commutator, cosocle | C2×C9 | C24×C3 |
| Frattini | C2×C3 | C23×C3×D3 |
| Fitting | C3×C9×#41 | C2 |
| socle | C2×C32 | C23×D3 |
| hypercenter | C3×#41 | D9 |
| hypocenter | C9 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×#41 | |
538 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 538 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
171 conjugacy classes — view the full character table.
| maximal subgroup of | #13141008, #13141026, #13141033, … |
| maximal normal subgroup of | #13141008, #13141026, #13141033, … |
| maximal quotient of | C3×#10589634, C3×#10589621, C3×#10589623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2599);