Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2509 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C32 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(8,11)(9,13)(15,16,19,22)(17,21,18,20)(23,24,28,26)(25,30,29,27);(4,5,6)(7,8,10,11)(9,12,13,14)(15,17,19,18)(16,20,22,21)(23,25)(24,27)(26,30)(28,29);(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,21)(19,22)(23,26,28,24)(25,30,29,27);(7,9,10,13)(8,12,11,14)(15,18,19,17)(16,21,22,20)(23,27,28,30)(24,29,26,25)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3569);