Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C210 |
| direct product | C3×D3×D42 |
| Fitting | C32×D42 ⋊ C2 |
| Sylow | C32 ⋊ C2×D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C24×D3 |
| commutator, cosocle | C22×C3 | C25×C3 |
| Frattini | C22 | C24×C3×D3 |
| Fitting | C32×D42 | C2 |
| socle | C22×C32 | C25 |
| hypercenter | C3×D42 | D3 |
| hypocenter | C3 | C2×C3×D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×D42 | |
5286 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
225 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D4×S4, C3×D3×#3484, C3×D3×#3498, … |
| maximal normal subgroup of | C3×D3×#3484, C3×D3×#3498, D32×D42 |
| maximal quotient of | D42×AGL(1,ℤ/9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 136320);