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×#62934 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6454 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)(6,12)(10,14)(13,18)(17,19)(20,21)(22,23)(24,28)(25,26)(27,31)(29,30)(32,33)(34,35);(4,5,9,8)(6,10,12,14)(7,11,15,16)(13,17,18,19)(21,26)(22,27)(28,33)(29,34);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,18)(16,19)(20,22,25,27)(21,23,26,31)(24,29,32,34)(28,30,33,35);(4,7)(5,11)(6,13)(8,16)(9,15)(10,17)(12,18)(14,19)(20,23)(21,27)(22,26)(24,30)(25,31)(28,34)(29,33)(32,35);(4,8,9,5)(6,14,12,10)(7,11,15,16)(13,17,18,19)(20,24,25,32)(21,28,26,33)(22,29,27,34)(23,30,31,35);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,18)(16,19)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55922);