Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×C4×SL(2,ℤ/4) |
| unambiguous semidirect product | (C2×A4)⋊(C3×C42) |
| derived | A4 ⋊ C2×C3×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C4 | S4 |
| commutator, hypocenter | A4 | C2×C3×C42 |
| Frattini | C22 | C22×C3×S4 |
| Fitting | C24×C3×C4 | D3 |
| socle | C25×C3 | C2×D3 |
| prime radical | C23×C3×A4 | C22 |
| cosocle | C22×A4 | C23×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C3×C4 | |
1212 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C4×GL(2,ℤ/4) |
| maximal normal subgroup of | C2×C3×C4×GL(2,ℤ/4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156773);