Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#1905 |
| 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 | |
465 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)(10,11,15,19,16,21,14,20)(12,22,25,32,26,35,24,17)(13,27,29,18,30,33,28,34)(23,31,38,40,39,41,37,36);(10,12)(11,17,21,32)(13,23)(14,24)(15,25)(16,26)(18,31,34,41)(19,22,20,35)(27,36,33,40)(28,37)(29,38)(30,39);(10,13,16,30)(11,18,21,34)(12,23,26,39)(14,29,15,28)(17,31,32,41)(19,27,20,33)(22,36,35,40)(24,38,25,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 637);