Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C32×#1653 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(8,11)(9,13)(12,14)(15,16,20,24)(17,25,28,29)(18,21,19,23)(22,26,30,27);(4,5,6)(7,8,10,12)(9,14,13,11)(16,21)(18,19)(22,29)(23,24)(25,30)(26,27);(15,17,20,28)(16,22,24,30)(18,27,19,26)(21,29,23,25)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2713);