Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#3289 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | D42 |
| commutator | C2×C4 | C24×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C2×C3 | C3×D42 |
| prime radical | C3×#3289 | C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ G | |
672 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)(10,11,16,22)(12,23,27,38)(13,19,29,37)(14,21,15,20)(17,30,34,26)(18,31,35,28)(24,32,41,33)(25,39,40,36);(10,12,16,27)(11,17,22,34)(13,24,29,41)(14,30,15,26)(18,32,35,33)(19,36,37,39)(20,38,21,23)(25,31,40,28);(10,13)(11,18)(12,24)(14,28)(15,31)(16,29)(17,32)(19,20)(21,37)(22,35)(23,39)(25,30)(26,40)(27,41)(33,34)(36,38);(10,13)(11,19)(12,25)(14,31)(15,28)(16,29)(17,33)(18,20)(21,35)(22,37)(23,39)(24,30)(26,41)(27,40)(32,34)(36,38)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2021);