Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×A4⋊S4 |
| unambiguous semidirect product | A4⋊(C3×S4) |
| Fitting, socle | C24×C3 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4⋊S4 |
| commutator, cosocle, hypocenter | A42 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C3 | |
228 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4678);