Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×D3×D9 |
| unambiguous semidirect product | (C3×D9)⋊C24 |
| unambiguous semidirect product | (D3×C9)⋊C24 |
| unambiguous semidirect product | (C3⋊D9)⋊C24 |
| Fitting | C23×C3×C9 ⋊ C22 |
| derived, Sylow | C3×C9 ⋊ C25 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×D9 |
| commutator, cosocle, hypocenter | C3×C9 | C25 |
| Frattini | C3 | C23×D32 |
| Fitting | C23×C3×C9 | C22 |
| socle | C23×C32 | C2×D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
2378 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | #13139919, D3×C22⋊C4×D9, D9×#882, … |
| maximal normal subgroup of | #13139919, D3×C22⋊C4×D9, D9×#882, … |
| maximal quotient of | C22×D9×Dic6, C22×#88641, C22×#88634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4032);