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×#2497 |
| 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 | |
642 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)(10,11,13,14)(12,16,17,15)(19,22)(20,24)(26,27)(28,32)(29,31)(30,33);(11,14)(15,16)(18,19)(20,23)(21,22)(24,25)(26,27,29,31)(28,30,33,32);(10,11)(12,15)(13,14)(16,17)(18,19)(20,23)(21,22)(24,25)(26,27)(28,30)(29,31)(32,33);(10,12,13,17)(11,15,14,16)(18,20,21,24)(19,23,22,25)(26,28,29,33)(27,30,31,32)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1229);