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×#62969 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6664 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)(10,13)(14,18)(17,19)(20,21,27,25)(22,31,30,26)(23,33,35,28)(24,32,34,29);(4,5,9,7)(6,10,12,13)(8,11,16,15)(14,17,18,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,30)(32,35)(33,34);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,23)(21,28)(22,32)(24,31)(25,33)(26,34)(27,35)(29,30);(4,7,9,5)(6,13,12,10)(8,11,16,15)(14,17,18,19)(20,24,27,34)(21,29,25,32)(22,28,30,33)(23,31,35,26);(4,8,9,16)(5,11,7,15)(6,14,12,18)(10,17,13,19)(20,25,27,21)(22,31,30,26)(23,33,35,28)(24,29,34,32);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,26,27,31)(21,30,25,22)(23,34,35,24)(28,29,33,32)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55957);