Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | #41×GO+(4,𝔽2) |
| Fitting | C32×#41 ⋊ D4 |
| Sylow | C32 ⋊ D4×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3⋊D3 | C25 |
| Frattini | C2 | C23×GO+(4,𝔽2) |
| Fitting | C32×#41 | D4 |
| socle | C2×C32 | C23×D4 |
| hypercenter | #41 | GO+(4,𝔽2) |
| hypocenter | C32 | D4×#41 |
| LCS | G ▷ C2×C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ #41 | |
2538 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | #41×AΓL(1,𝔽9), AGL(1,ℤ/8)×GO+(4,𝔽2), #72220239, … |
| maximal normal subgroup of | #41×AΓL(1,𝔽9), AGL(1,ℤ/8)×GO+(4,𝔽2), #72220239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 136870);