Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C9 · Schur multiplier (H2) = C29 |
| direct product | C2×C9×#546 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C25×C9 |
| Frattini, cosocle | C22×C3 | C25×C3 |
| socle | C23×C3 | C24×C3 |
| prime radical | C2×C3×#546 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
2850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(15,16,17,18,19,20,21,22,23);(4,6)(7,8,11,12)(9,13,14,10);(8,12)(10,13);(3,4)(5,6)(7,9)(8,10)(11,14)(12,13);(7,10)(8,9)(11,13)(12,14)450 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2195);