Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#2401 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| socle | C23×C32 | C24 |
| prime radical | C24×C32 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
1296 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)(9,12)(10,11)(13,14)(15,16,18,20)(17,19,21,22)(23,24,26,28)(25,27,29,30);(4,5,6)(8,11)(12,14)(15,17,18,21)(16,19,20,22)(23,25)(24,27)(26,29)(28,30);(16,20)(19,22)(25,29)(27,30);(7,9,10,13)(8,12,11,14)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3461);