Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C9 · Schur multiplier (H2) = C2×C4 |
| direct product | C9×#1833 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #117 |
| commutator | C2×C4 | C22×C4×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#382 |
| prime radical | C23×C3 | C3×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C9×#117 ◁ G | |
258 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)(12,20,23,16)(13,24,26,39)(14,19)(15,18)(17,25,30,40)(21,34)(22,33,37,29)(27,31)(28,35)(32,41)(36,38);(10,12,14,22,15,23,27,37)(11,16,19,29,18,20,31,33)(13,21,25,35,26,36,40,41)(17,28,24,38,30,32,39,34);(10,13)(11,17,18,30)(12,21)(14,25)(15,26)(16,28,20,32)(19,24,31,39)(22,35)(23,36)(27,40)(29,38,33,34)(37,41)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 565);