Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×A4×C32 |
| unambiguous semidirect product | (C22×C32)⋊C32 |
| unambiguous semidirect product | (C22×C3×C32)⋊C3 |
| Fitting | C22×C3×C32 ⋊ C3 |
| derived | C22 ⋊ C32×C32 |
| Sylow | C22×C32 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C32 | A4 |
| commutator, hypocenter | C22 | C32×C32 |
| Frattini | C16 | C2×C3×A4 |
| Fitting | C22×C3×C32 | C3 |
| socle | C23×C3 | C3×C16 |
| prime radical | C2×C3×A4 | C16 |
| cosocle | C22×C16 | C2×C32 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C3×C32 | |
104 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
384 conjugacy classes — view the full character table.
| maximal subgroup of | C3×A4×C64, C3×S4×C32 |
| maximal normal subgroup of | C3×A4×C64, C3×S4×C32 |
| maximal quotient of | C3×SL(2,𝔽3)×C32, C3×A4×C64, C3×#13737⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153393);