Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C27 |
| direct product | C3×#42470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| prime radical | C24×C3 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 35 points:
(1,2,3)(12,13,16,14)(15,18,19,17)(20,21)(22,27)(23,31)(24,29)(25,26)(28,34)(30,33)(32,35);(4,5)(6,10)(7,8)(9,11)(12,14,16,13)(15,18,19,17)(21,26)(24,27)(28,31)(33,35);(5,8)(9,10)(12,15)(13,17)(14,18)(16,19)(20,22,25,29)(21,27,26,24)(23,32,34,30)(28,33,31,35);(4,5)(6,9)(7,8)(10,11)(20,23,25,34)(21,28,26,31)(22,30,29,32)(24,33,27,35);(4,6,7,11)(5,9,8,10)(20,24,25,27)(21,29,26,22)(23,33,34,35)(28,32,31,30)156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 35458);