Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×SD16⋊C8 |
| unambiguous semidirect product | SD16⋊(C8×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C2×D4 |
| commutator | C4 | C22×C8×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle | C22×C3 | C3×C4×D4 |
| prime radical | C2×C3×D4 | C2×C3×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ C4×C8×C9 ◁ G | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,14)(13,15)(16,17)(18,19,21,24,22,20,23,25);(11,14)(12,17)(15,16);(11,15)(14,16)(18,20,21,25,22,19,23,24)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 310);