Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C9×#1358 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #331 |
| commutator | C42 | C2×C4×C9 |
| Frattini, cosocle | C3×C4×C8 | C22×C3 |
| socle | C2×C3 | C3×#331 |
| prime radical | C22×C3 | C3×#100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ C4×C8×C9 ◁ G | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,18,14,17,26,33,15,20,27,36,29,19,37,35)(12,21,22,38,23,32,39,41,24,31,40,34,28,16,25,30);(10,12,15,24)(11,16,20,32)(13,25,27,39)(14,28,29,23)(17,31,19,21)(18,34,36,38)(22,26,40,37)(30,33,41,35)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 90);