Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#1672 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
450 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)(15,16,19,22)(17,23,25,30)(18,26,28,21)(20,27,29,24);(4,5,6)(7,8,9,11,10,12,13,14);(15,17,19,25)(16,20,22,29)(18,27,28,24)(21,30,26,23)261 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2732);