Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C22×C32×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C25×C32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ C23×C32×C4 ◁ G | |
5160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2)(13,14,15);(3,4)(16,17,18);(5,6)(7,9)(8,10)(11,12);(6,9)(7,12)(10,11);(6,10)(9,11)396 conjugacy classes — view the full character table.
| Fitting subgroup of | C22×C3⋊D3×AGL(1,ℤ/8) |
| maximal subgroup of | C22×C3⋊D3×AGL(1,ℤ/8) |
| maximal normal subgroup of | C22×C3⋊D3×AGL(1,ℤ/8) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 4638);