Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C22×C3×SL(2,ℤ/4) |
| derived | A4 ⋊ C22×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | S4 |
| commutator, hypocenter | A4 | C22×C3×C4 |
| Frattini | C2 | C22×C3×S4 |
| Fitting, socle | C25×C3 | D3 |
| prime radical | C23×C3×A4 | C2 |
| cosocle | C2×A4 | C23×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C23×C3 | |
586 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | #11920744, C3×A4⋊#95, C2×Dic3×SL(2,ℤ/4), … |
| maximal normal subgroup of | #11920744, C3×A4⋊#95, C2×Dic3×SL(2,ℤ/4), … |
| maximal quotient of | C22×C3×A4⋊C8, C2×C3×#5409, C2×C3×C4×SL(2,ℤ/4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8452);