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×#1645 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | #120 |
| commutator | C42 | C23×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle, prime radical | C22×C3 | C3×#120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C42×C9 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(11,14)(12,16)(15,17)(18,19,22,25)(20,26,27,33)(21,29,30,24)(23,31,32,28);(10,11,13,15)(12,17,16,14)(21,30)(24,29)(26,33)(28,31);(18,20,22,27)(19,23,25,32)(21,31,30,28)(24,33,29,26)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 377);