Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×A4×S4 |
| unambiguous semidirect product | A4⋊(C2×C3×A4) |
| unambiguous semidirect product | (C22×S4)⋊C32 |
| unambiguous semidirect product | (C22×C3×S4)⋊C3 |
| Fitting, socle | C24×C3 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | A4×S4 |
| commutator, cosocle, hypocenter | C22×A4 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3 | C3×D3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C3 | |
244 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S42, S4×C3⋊S4, D3×A4×S4, … |
| maximal normal subgroup of | C3×S42, S4×C3⋊S4, D3×A4×S4, … |
| maximal quotient of | C3×A4⋊(#41⋊C3), C3×A4×#226, C3×SL(2,𝔽3)×S4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4674);