Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#3022 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C2×#120 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C2×C3×#120 |
| prime radical | C3×#585 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
1056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,16,21)(12,22,25,34)(13,18,27,33)(14,20,15,19)(17,28,31,24)(23,37,40,36)(26,35,29,32)(30,38,41,39);(10,12)(11,17)(13,23)(14,24)(15,28)(16,25)(18,30)(19,22)(20,34)(21,31)(26,38)(27,40)(29,39)(32,36)(33,41)(35,37);(10,13)(11,18)(12,23)(14,26)(15,29)(16,27)(17,30)(19,32)(20,35)(21,33)(22,36)(24,38)(25,40)(28,39)(31,41)(34,37);(10,14,16,15)(11,19,21,20)(12,24,25,28)(13,26,27,29)(17,22,31,34)(18,32,33,35)(23,38,40,39)(30,36,41,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1754);