Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| prime radical | C3×#537 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(5,9)(10,16)(11,17)(18,19)(20,21,25,23)(22,28,30,26)(24,31,32,27)(29,33,35,34);(20,22,25,30)(21,26,23,28)(24,29,32,35)(27,33,31,34);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)(20,23,25,21)(22,26,30,28)(24,31,32,27)(29,33,35,34);(4,6,8,13)(5,10,9,16)(7,14,15,12)(11,19,17,18)(20,24,25,32)(21,27,23,31)(22,29,30,35)(26,33,28,34);(4,7,8,15)(5,11,9,17)(6,12,13,14)(10,18,16,19);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55935);