Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×A4×S4 |
| unambiguous semidirect product | (C22×C4×S4)⋊C3 |
| Fitting | C24×C4 ⋊ C3×D3 |
| derived | C22×A4 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×S4 |
| commutator, hypocenter | C22×A4 | C2×C3×C4 |
| Frattini | C2 | C2×A4×S4 |
| Fitting | C24×C4 | C3×D3 |
| socle | C25 | C2×C3×D3 |
| prime radical | C2×A4×S4 | C2 |
| cosocle | C23×A4 | C22×C3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C4 | |
634 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#5912, C4×S42, #72228053, … |
| maximal normal subgroup of | A4×#5912, C4×S42, #72228053, … |
| maximal quotient of | A4×AGL(1,𝔽5)×S4, A4×(C2×S4)⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157485);