Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C22 |
| direct product | C9×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #124 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#397 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
285 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,20,23,33)(13,17)(14,18)(15,19)(16,22,30,27)(21,36,24,35)(25,31)(26,32)(28,34)(29,38,40,37)(39,41);(10,12)(11,16)(13,24)(14,27)(15,23)(17,29)(18,33)(19,30)(20,34)(21,26)(22,28)(25,37)(31,36)(32,40)(35,41)(38,39);(10,13,14,25,15,26,28,39)(11,17,18,31,19,32,34,41)(12,21,22,37,23,24,27,38)(16,29,20,35,30,40,33,36)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 812);