Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2402 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C9 | C23 |
| commutator | C23 | C24×C9 |
| Frattini, socle, prime radical, cosocle | C23×C3 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C9 ◁ G | |
600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19)(20,23)(21,25)(22,24)(26,27,29,31)(28,30,32,33);(11,14)(15,17)(18,20,22,21)(19,23,24,25)(26,28)(27,30)(29,32)(31,33);(18,21,22,20)(19,23,24,25)(28,32)(30,33);(10,12,13,16)(11,15,14,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1134);