Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C32×#2399 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C32 |
| socle, prime radical | C23×C32 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
1512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1152 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(11,12,15,18)(13,17,14,16)(19,20,23,26)(21,25,22,24);(4,5,6)(7,8)(9,10)(11,13,15,14)(12,16,18,17)(19,21,23,22)(20,24,26,25);(8,10)(11,13,15,14)(12,17,18,16)(19,22,23,21)(20,25,26,24);(11,14,15,13)(12,16,18,17)396 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 3459);