Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,12)(8,10)(9,13)(11,18)(15,27)(16,23)(17,28)(19,26)(20,22)(21,30)(24,33)(25,31)(29,35)(32,34);(5,10)(6,15)(7,19)(12,25)(13,23)(17,32)(18,30)(20,31)(22,26)(24,28)(29,33)(34,35);(4,6,16,30)(5,11,23,27)(7,17,31,32)(8,15,9,18)(10,21,13,14)(12,24,25,35)(19,33,20,29)(22,34,26,28);(5,10)(11,21)(12,22)(13,23)(14,27)(24,34)(25,26)(28,35);(4,7,8,20)(5,12,10,22)(6,17,15,29)(9,19,16,31)(11,24,21,34)(13,26,23,25)(14,28,27,35)(18,33,30,32);(5,13)(10,23)(11,14)(12,26)(21,27)(22,25)(24,28)(34,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55814);