Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×SD16⋊C8 |
| unambiguous semidirect product | SD16⋊(C32×C8) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C2×D4 |
| commutator | C4 | C22×C32×C8 |
| Frattini, cosocle | C42 | C23×C32 |
| socle | C22×C32 | C4×D4 |
| prime radical | C2×C32×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C32×C4×C8 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8)(9,11)(10,12)(13,14)(15,16,18,21,19,17,20,22);(4,5,6)(8,11)(9,14)(12,13);(8,12)(11,13)(15,17,18,22,19,16,20,21)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2638);