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×#62938 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6672 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)(6,12)(8,16)(15,18)(22,28)(26,30)(29,34)(33,35);(4,5)(6,11)(7,12)(8,13)(9,10)(14,18)(15,19)(16,17)(20,21,25,24)(22,26,28,30)(23,27,31,32)(29,33,34,35);(4,6)(5,7)(8,14)(9,12)(10,11)(13,15)(16,19)(17,18)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,34)(32,35);(4,6)(5,11)(7,10)(8,14)(9,12)(13,18)(15,17)(16,19)(20,23)(21,27)(22,29)(24,32)(25,31)(26,33)(28,34)(30,35);(4,7,9,11)(5,12,10,6)(8,15,16,18)(13,19,17,14)(20,24,25,21)(22,30,28,26)(23,27,31,32)(29,33,34,35);(4,8,9,16)(5,13,10,17)(6,14,12,19)(7,15,11,18)(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, 55926);