Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#1459 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | C22⋊C4 |
| commutator | C22 | C22×C32×C8 |
| Frattini, cosocle | C22×C4 | C23×C32 |
| socle | C22×C32 | C2×C22⋊C4 |
| prime radical | C22×C32×D4 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C23×C32×C4 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(7,8)(9,13,14,11)(10,12)(15,16,18,21,19,22,17,20);(4,5,6)(7,9)(8,11)(10,14)(12,13);(15,17,19,18)(16,20,22,21)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2519);