Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25×C4 |
| direct product | C9×#2310 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C9 | C24 |
| commutator | C22 | C23×C4×C9 |
| Frattini, socle, cosocle | C23×C3 | C24×C3 |
| prime radical | C24×C3 | C23×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C9 ◁ G | |
1176 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)(11,13)(14,15,16,17)(19,23)(20,26)(21,29)(31,33);(18,19,22,23)(20,27,26,24)(21,25,29,30)(28,33,32,31);(10,11)(12,13)(14,15)(16,17)(18,20,22,26)(19,24,23,27)(21,28,29,32)(25,31,30,33);(18,21)(19,25)(20,28)(22,29)(23,30)(24,31)(26,32)(27,33)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1042);