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×#1626 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| prime radical | C32×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(8,12)(9,10)(13,14)(16,20)(17,21)(18,22);(4,5,6)(7,8)(9,12)(10,14)(11,13)(15,16,19,17)(18,21,22,20);(7,9,11,10)(8,12,13,14)(15,17,19,16)(18,20,22,21)234 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 2686);