Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C9 | #121 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C2×C3×#121 |
| prime radical | C3×#3067 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C9 ◁ C22×C4×C9 ◁ G | |
678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,22)(13,27)(14,19)(15,20)(16,21)(17,24)(18,28)(23,37)(25,33)(26,32)(29,34)(30,35)(31,39)(36,38)(40,41);(10,12,14,24,16,26,15,25)(11,17,20,22,21,33,19,32)(13,23,28,38,29,39,30,40)(18,31,27,36,35,37,34,41);(10,13)(11,18)(12,23)(14,28)(15,30)(16,29)(17,31)(19,34)(20,27)(21,35)(22,36)(24,38)(25,40)(26,39)(32,41)(33,37);(10,14,16,15)(11,19,21,20)(12,24,26,25)(13,28,29,30)(17,32,33,22)(18,34,35,27)(23,38,39,40)(31,41,37,36)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1799);