Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1308 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #95 |
| commutator | C23 | C42×C9 |
| Frattini, cosocle | C2×C3×C42 | C22×C3 |
| socle | C22×C3 | C3×#95 |
| prime radical | C23×C3 | C3×C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C2×C42×C9 ◁ G | |
183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,18,15,17,14,19)(12,20,21,25,22,16,23,24)(26,27,29,32,30,33,37,40)(28,34,36,41,35,31,38,39);(10,12,14,21,15,22,13,23)(11,16,19,24,17,20,18,25)(26,28)(27,31,33,34)(29,36)(30,35)(32,39,40,41)(37,38)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 40);