Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #6226 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C3 |
| direct product | C22×#6233 |
| Fitting | C24×C3×C9 ⋊ C2 |
| Sylow | C3×C9 ⋊ C22×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×C3⋊D9 |
| commutator, cosocle | C2×C3×C9 | C24 |
| Frattini | C2×C3 | C23×C3⋊D3 |
| Fitting | C24×C3×C9 | C2 |
| socle | C23×C32 | C2×D3 |
| hypercenter | C24 | C3⋊D9 |
| hypocenter | C3×C9 | C22×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C23 ◁ C24 | |
1180 conjugacy classes of subgroups; only the 321 normal subgroups are shown.
This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 321 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
228 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720330, #13141125, #13141126, … |
| maximal normal subgroup of | C2×#11720330, #13141125, #13141126, … |
| maximal quotient of | C22×#88534, C2×#11721813, C22×#88536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2645);