Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C9 · Schur multiplier (H2) = C4 |
| direct product | C9×#1454 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C9 | C23 |
| commutator | C22 | C2×C42×C9 |
| Frattini, cosocle | C3×C42 | C23×C3 |
| socle | C22×C3 | C2×C3×C42 |
| prime radical | C23×C3 | C3×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C9 ◁ G | |
267 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9)(10,11,13,17,14,16,20,23)(12,15,19,21,18,22,24,25)(26,27)(28,31)(29,33)(30,32)(34,38)(35,37)(36,39)(40,41);(10,12)(11,15)(13,19)(14,18)(16,22)(17,21)(20,24)(23,25)(26,28,29,34,30,35,36,40)(27,31,33,38,32,37,39,41);(11,16)(12,18)(17,23)(19,24)(27,32)(31,37)(33,39)(38,41)504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 186);