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×#2647 |
| 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 | |
1080 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)(8,9)(12,13)(15,16)(17,20)(18,21)(19,22)(23,24,27,25)(26,29,30,28);(4,5,6)(7,8,11,9)(10,12,14,13)(15,17)(16,18)(19,21)(20,22)(23,25,27,24)(26,28,30,29);(7,9,11,8)(10,12,14,13)(15,17)(16,20)(18,22)(19,21)(23,26,27,30)(24,28,25,29);(7,10,11,14)(8,12,9,13)(15,18,19,20)(16,21,22,17)288 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3707);