Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3×C4 |
| direct product | C32×#2678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
2406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2,3)(7,8,9,10)(15,16)(17,21)(18,19)(20,22);(4,5,6)(12,14)(16,19)(17,22);(7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(21,22);(7,8)(9,10)(15,17)(16,20)(18,22)(19,21)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3738);