Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2646 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C32 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| prime radical | C24×C32 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C32 ◁ G | |
1308 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)(7,8,11,12)(9,10,13,14)(16,17)(20,21)(23,24)(25,28)(26,29)(27,30);(4,5,6)(8,12)(9,13)(15,16,19,17)(18,20,22,21)(23,25)(24,27)(26,30)(28,29);(7,9)(8,10)(11,13)(12,14)(15,17,19,16)(18,20,22,21)(23,25)(24,28)(26,30)(27,29);(7,10)(8,13)(9,12)(11,14)(15,18)(16,20)(17,21)(19,22)(23,24)(25,27)(26,29)(28,30)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3706);