Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D9×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×D9×S4 |
| unambiguous semidirect product | (C9×S4)⋊C22 |
| unambiguous semidirect product | (C9⋊S4)⋊C22 |
| unambiguous semidirect product | (A4×D9)⋊C22 |
| Fitting | C23×C9 ⋊ C2×D3 |
| derived | C9×A4 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×S4 |
| commutator, cosocle, hypocenter | C9×A4 | C23 |
| Frattini | C3 | C2×D3×S4 |
| Fitting | C23×C9 | C2×D3 |
| socle | C23×C3 | D32 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C2 | |
413 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D9×S4, D9×C4⋊S4, S4×D36, … |
| maximal normal subgroup of | C4×D9×S4, D9×C4⋊S4, S4×D36, … |
| maximal quotient of | D9×A4⋊Q8, #13164100, #13164101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3998);