Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4×C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3×C4 |
| direct product | C2×A4×C42⋊C3 |
| unambiguous semidirect product | (C23×C42⋊C3)⋊C3 |
| unambiguous semidirect product | (C2×C42×A4)⋊C3 |
| unambiguous semidirect product | (C2×C22⋊(C42⋊C3))⋊C3 |
| Fitting, Sylow | C23×C42 ⋊ C32 |
| derived | C22×C42 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4×C42⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C2×C32 |
| Frattini | C22 | C2×A42 |
| Fitting | C23×C42 | C32 |
| socle | C25 | C3×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C2 | |
324 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C42⋊C32, C2×A4×#127⋊C3, C2×A4×#126⋊C3, … |
| maximal normal subgroup of | C2×A4×#127⋊C3, C2×A4×#126⋊C3 |
| maximal quotient of | C2×C42⋊C32, C42⋊C3×#126⋊C3, C42⋊C3×#127⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156564);