Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C23 |
| direct product | C9×#2505 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C23 | C24×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C24×C3 | C23×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,14,15)(12,17,16,13)(18,19,22,24)(20,21,25,23)(27,30)(28,32);(11,15)(12,16)(18,20,22,25)(19,23,24,21)(26,27)(28,31)(29,30)(32,33);(10,12,14,16)(11,13,15,17)(18,20)(19,21)(22,25)(23,24)(26,28,29,32)(27,31,30,33);(10,13)(11,16)(12,15)(14,17)(18,21)(19,20)(22,23)(24,25)(26,27)(28,31)(29,30)(32,33)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1237);