Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C9×#3020 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | C2×#120 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C2×C3×#120 |
| prime radical | C3×#3020 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
1047 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)(12,20)(13,17)(14,26)(15,18)(16,22)(19,29)(21,37)(23,34)(24,35)(25,32)(27,31)(28,33)(30,38)(36,39)(40,41);(10,12)(11,16)(13,21)(14,27)(15,23)(17,30)(18,31)(19,34)(20,26)(22,29)(24,40)(25,39)(28,38)(32,41)(33,36)(35,37);(10,13,15,25)(11,17,18,32)(12,21,23,39)(14,28,29,24)(16,30,31,41)(19,35,26,33)(20,36,34,37)(22,40,27,38);(11,18)(16,31)(17,32)(19,26)(20,34)(30,41)(33,35)(36,37)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1752);