Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
| direct product | C2×#89039 |
| Fitting | C23×C33 ⋊ C22 |
| Sylow | C33 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×C3⋊D3 |
| commutator, cosocle | C2×C33 | C24 |
| Frattini | C2 | C22×D3×C3⋊D3 |
| Fitting | C23×C33 | C22 |
| socle | C22×C33 | C23 |
| hypercenter | C23 | D3×C3⋊D3 |
| hypocenter | C33 | C22×D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1952 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 1952 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
102 conjugacy classes — view the full character table.
| maximal subgroup of | #13156978, #13159065, #13166608, … |
| maximal normal subgroup of | #13156978, #13159065, #13166608, … |
| maximal quotient of | C2×#11723470, C2×#11723525, C2×#11723497, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4577);