Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2950 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×D4 |
| commutator | C4 | C23×C4×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C22×C3×D4 |
| prime radical | C23×C3 | C2×C3×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C9×#117 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,12)(14,15)(16,24)(17,21)(18,30)(19,22)(20,26)(23,32)(25,38)(27,37)(28,36)(29,42)(31,34)(33,40)(35,41)(39,45)(43,44);(14,16,19,27)(15,20,22,34)(17,25,28,39)(18,31,32,26)(21,33,36,44)(23,37,30,24)(29,43,41,40)(35,45,42,38);(10,12,13,11)(14,17,19,28)(15,21,22,36)(16,25,27,39)(18,29,32,41)(20,33,34,44)(23,35,30,42)(24,38,37,45)(26,40,31,43);(15,22)(17,28)(20,34)(23,30)(24,37)(25,39)(29,41)(40,43)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1682);