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×#2510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| prime radical | C22×C32×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 576 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(7,8)(9,11)(10,13)(12,14)(15,16,19,21)(17,18,22,20)(23,24,26,28)(25,29,30,27);(4,5,6)(8,9)(12,13)(15,17,19,22)(16,20,21,18)(23,24)(25,27)(26,28)(29,30);(7,9,11,8)(10,13,14,12)(15,17)(16,18)(19,22)(20,21)(23,25,26,30)(24,27,28,29);(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,27)(26,28)(29,30)342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3570);