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×#2514 |
| 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 | |
1116 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)(8,10)(9,13)(15,16,20,18)(17,22,21,19)(25,29)(27,30);(4,5,6)(7,8,11,10)(9,12,13,14)(15,17)(16,19)(18,22)(20,21)(23,24,26,28)(25,30,29,27);(7,9)(8,12)(10,14)(11,13)(15,18,20,16)(17,22,21,19)(23,25,26,29)(24,27,28,30);(7,10,11,8)(9,14,13,12)(15,19,20,22)(16,21,18,17)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3574);