Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#1635 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C42 | C23×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle | C22×C3 | C3×#120 |
| prime radical | C3×#547 | C2×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ G | |
351 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,16)(12,14,17,15)(18,19,23,27)(20,28,31,32)(21,26,22,25)(24,30,33,29);(11,14)(12,17)(15,16)(18,20,23,31)(19,24,27,33)(21,30,22,29)(25,32,26,28);(19,25)(21,22)(24,32)(26,27)(28,33)(29,30)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 367);