Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1912 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #386 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#386 |
| prime radical | C2×C3×#41 | C22×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
471 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)(10,11,15,19,16,20,14,30)(12,21,25,32,24,36,23,17)(13,26,28,34,29,18,27,33)(22,35,39,31,38,41,37,40);(10,12)(11,17,20,32)(13,22)(14,23)(15,25)(16,24)(18,31,26,40)(19,21,30,36)(27,37)(28,39)(29,38)(33,41,34,35);(10,13,16,29)(11,18,20,26)(12,22,24,38)(14,28,15,27)(17,31,32,40)(19,34,30,33)(21,35,36,41)(23,39,25,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 644);