Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2049 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #394 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#394 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
381 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)(10,11,16,21)(12,22,25,37)(13,26,30,35)(14,20,15,19)(17,31,33,24)(18,28,36,29)(23,34,27,32)(38,41,40,39);(10,12)(11,17)(13,27)(14,31)(15,24)(16,25)(18,34)(19,37)(20,22)(21,33)(23,30)(26,40)(28,39)(29,41)(32,36)(35,38);(10,13,14,28,16,30,15,29)(11,18,19,35,21,36,20,26)(12,23,24,39,25,27,31,41)(17,32,22,38,33,34,37,40)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 781);