Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×D3×#611 |
| Fitting | C23×C3×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C22×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle | C22×C32 | C22×D3 |
| hypercenter | C23 | D3×D9 |
| hypocenter | C3×C9 | C22×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
762 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
126 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63365, D3×#63447, #13148201, … |
| maximal normal subgroup of | D3×#63365, D3×#63447, #13148201, … |
| maximal quotient of | C2×#11720955, C2×#11720903, C2×#11720984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2499);