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×#62827 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6626 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)(4,5)(6,13)(7,8)(9,10)(11,16)(12,17)(14,19)(15,18)(21,25)(23,30)(26,31)(29,34);(5,10)(7,12)(11,13)(14,18)(22,28)(26,31)(29,34)(33,35);(4,5,9,10)(6,11,16,13)(7,8,12,17)(14,15,18,19)(20,21,24,25)(22,26,28,31)(23,27,30,32)(29,33,34,35);(4,6,9,16)(5,11,10,13)(7,14,12,18)(8,15,17,19);(4,7,9,12)(5,8,10,17)(6,14,16,18)(11,15,13,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(4,8)(5,12)(6,15)(7,10)(9,17)(11,18)(13,14)(16,19)(20,23)(21,27)(22,29)(24,30)(25,32)(26,33)(28,34)(31,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55815);