Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C2 |
| direct product | C3⋊C8×C22⋊C9 |
| unambiguous semidirect product | C3⋊(C8×C22⋊C9) |
| unambiguous semidirect product | (C22×C3⋊C8)⋊C9 |
| derived | C22×C3 ⋊ C8×C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | D3×A4 |
| commutator, hypocenter | C22×C3 | C8×C9 |
| Fitting, cosocle | C22×C32×C4 | C2×C3 |
| socle, prime radical | C23×C32 | C3×C4 |
| LCS | G ▷ C22×C3 | |
| UCS | C1 ◁ C3×C4 | |
107 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 107 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 72 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 729);