Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of A4×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×SL(2,ℤ/4) |
| unambiguous semidirect product | A4⋊(C4×A4) |
| unambiguous semidirect product | (C22×SL(2,ℤ/4))⋊C3 |
| derived | C22×A4 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×S4 |
| commutator, hypocenter | C22×A4 | C3×C4 |
| Fitting, socle | C25 | C3×D3 |
| prime radical | C2×A42 | C2 |
| cosocle | C23×A4 | C2×C3 |
| LCS | G ▷ C22×A4 | |
| UCS | C1 ◁ C2 | |
167 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | S4×SL(2,ℤ/4), #11923458, #11923460, … |
| maximal normal subgroup of | S4×SL(2,ℤ/4), #11923458, #11923460, … |
| maximal quotient of | A4×A4⋊C8, SL(2,𝔽3)×SL(2,ℤ/4), A4×#859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8275);