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×#62951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6576 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)(5,9)(10,16)(11,17)(18,19)(20,21,27,25)(22,31,30,26)(23,33,35,28)(24,32,34,29);(20,22)(21,26)(23,29)(24,28)(25,31)(27,30)(32,35)(33,34);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)(20,23)(21,28)(22,32)(24,31)(25,33)(26,34)(27,35)(29,30);(4,6,8,13)(5,10,9,16)(7,14,15,12)(11,19,17,18)(20,24,27,34)(21,29,25,32)(22,28,30,33)(23,31,35,26);(4,7,8,15)(5,11,9,17)(6,12,13,14)(10,18,16,19)(20,25,27,21)(22,31,30,26)(23,33,35,28)(24,29,34,32);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)(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, 55939);