Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C32 · Schur multiplier (H2) = C2 |
| direct product | C3×A4⋊C32 |
| unambiguous semidirect product | A4⋊(C3×C32) |
| derived | A4 ⋊ C3×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C16 | S4 |
| commutator, hypocenter | A4 | C3×C32 |
| Frattini | C16 | C3×S4 |
| Fitting | C22×C3×C16 | D3 |
| socle | C23×C3 | C3⋊C16 |
| prime radical | C2×C3×A4 | C16 |
| cosocle | A4×C16 | C2×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C3×C16 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 288 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×C32 |
| maximal normal subgroup of | C3×S4×C32 |
| maximal quotient of | C3×SL(2,𝔽3)⋊C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 154137);