Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C4 · Schur multiplier (H2) = C25×C3×C4 |
| direct product | C32×#2311 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator | C22 | C23×C32×C4 |
| Frattini, cosocle | C23 | C24×C32 |
| prime radical | C22×C32×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(8,10)(11,12,13,14)(15,16,18,19)(17,21,22,20);(4,5,6)(16,19)(20,21);(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(15,17,18,22)(16,20,19,21)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3371);