Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C3×D4×C22⋊A4 |
| unambiguous semidirect product | C4⋊(C2×C3×C22⋊A4) |
| Fitting | C24×C3×D4 ⋊ C3 |
| Sylow | C24×D4 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C22×C22⋊A4 |
| commutator, cosocle | C25 | C22×C32 |
| Frattini | C2 | C22×C3×C22⋊A4 |
| Fitting | C24×C3×D4 | C3 |
| socle | C25×C3 | C22×C3 |
| hypercenter | C3×D4 | C22⋊A4 |
| hypocenter | C24 | C32×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2×C3 ◁ C3×D4 | |
3816 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 3816 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D4×C22⋊A4, D4×(C22⋊A4)⋊C9, D4×#88881, … |
| maximal normal subgroup of | C32×D4×C22⋊A4, D4×(C22⋊A4)⋊C9, D4×#88881, … |
| maximal quotient of | C32×D4×C22⋊A4, D4×#88907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 153899);