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×#62935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(5,10)(6,11)(8,16)(15,19)(22,28)(26,30)(29,34)(33,35);(4,5)(6,13)(7,11)(8,12)(9,10)(14,19)(15,18)(16,17)(20,21,25,24)(22,26,28,30)(23,27,31,32)(29,33,34,35);(4,6,9,11)(5,7,10,13)(8,14,16,18)(12,15,17,19)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,34)(32,35);(4,5)(6,7)(8,12)(9,10)(11,13)(14,15)(16,17)(18,19)(20,23)(21,27)(22,29)(24,32)(25,31)(26,33)(28,34)(30,35);(4,7)(5,11)(6,10)(8,15)(9,13)(12,18)(14,17)(16,19)(20,24,25,21)(22,30,28,26)(23,27,31,32)(29,33,34,35);(4,8)(5,12)(6,14)(7,15)(9,16)(10,17)(11,18)(13,19)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,34)(32,35)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55923);