Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4×#126⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | A4×#126⋊C3 |
| unambiguous semidirect product | (C22×#126⋊C3)⋊C3 |
| unambiguous semidirect product | (A4×#126)⋊C3 |
| unambiguous semidirect product | #85646⋊C3 |
| Fitting, Sylow | C22×#126 ⋊ C32 |
| derived | C22×C42 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C42 | C2×C32 |
| Frattini | C22 | C2×A42 |
| Fitting | C22×#126 | C32 |
| socle | C24 | C2×C3×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 | |
204 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | #72213872, A4×#560⋊C3, C2×A4×#126⋊C3, … |
| maximal normal subgroup of | A4×#560⋊C3, C2×A4×#126⋊C3 |
| maximal quotient of | #72213872, A4×#396⋊C3, #72222714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156568);