Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#3081 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C22×C3×D4 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,15)(12,16,17,14)(18,19)(20,23)(21,24)(22,26)(25,29)(27,31)(28,30)(32,33);(10,12,13,17)(11,14,15,16)(18,20)(19,23)(21,28)(22,27)(24,30)(25,32)(26,31)(29,33);(19,24)(21,29)(23,30)(25,26)(28,33)(31,32);(20,27)(23,31)(28,33)(30,32)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1813);