Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C22×C4 |
| direct product | C9×#2532 |
| 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 | |
609 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)(10,11,14,12)(13,16,17,15)(19,20)(21,25)(28,32)(31,33);(11,12)(15,16)(18,19,22,20)(21,23,25,24)(26,27,30,29)(28,31,32,33);(10,12,14,11)(13,16,17,15)(18,20,22,19)(21,24,25,23)(26,28)(27,31)(29,33)(30,32);(10,13,14,17)(11,15,12,16)(18,21)(19,23)(20,24)(22,25)(26,29,30,27)(28,33,32,31)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1264);