Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#2082 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #397 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#397 |
| prime radical | C23×C3 | C3×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,15,20,16,21,14,19)(12,22,25,17,26,33,24,34)(13,27,29,35,30,36,31,18)(23,38,39,41,28,32,40,37);(10,12,16,26)(11,17,21,34)(13,28,30,23)(14,25,15,24)(18,32,35,38)(19,33,20,22)(27,41,36,37)(29,39,31,40);(10,13,14,29,16,30,15,31)(11,18,19,27,21,35,20,36)(12,23,24,39,26,28,25,40)(17,32,22,37,34,38,33,41)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 814);