Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C9 · Schur multiplier (H2) = C24 |
| direct product | C9×#2009 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #392 |
| commutator | C22×C4 | C23×C9 |
| Frattini, cosocle | C22×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#392 |
| prime radical | C3×#575 | C2×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C22×C4×C9 ◁ G | |
522 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)(12,22)(13,26)(14,20)(15,19)(16,21)(17,24)(18,28)(23,31)(25,36)(27,35)(29,34)(30,32)(33,39)(37,38)(40,41);(10,12)(11,17)(13,23)(14,30)(15,24)(16,25)(18,33)(19,36)(20,22)(21,32)(26,37)(27,40)(28,38)(29,39)(31,35)(34,41);(10,13,16,29)(11,18,21,35)(12,23,25,39)(14,28,15,27)(17,31,32,33)(19,26,20,34)(22,37,36,41)(24,40,30,38)207 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 741);