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×#62967 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6444 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)(20,21,25,26)(22,23,28,31)(24,27,32,33)(29,30,34,35);(4,5,9,7)(6,10,12,13)(8,11,16,15)(14,17,18,19)(21,26)(22,28)(27,33)(29,34);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,22)(21,23)(24,29)(25,28)(26,31)(27,30)(32,34)(33,35);(4,7,9,5)(6,13,12,10)(8,11,16,15)(14,17,18,19)(20,23)(21,22)(24,30)(25,31)(26,28)(27,29)(32,35)(33,34);(4,8,9,16)(5,11,7,15)(6,14,12,18)(10,17,13,19);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,24,25,32)(21,27,26,33)(22,29,28,34)(23,30,31,35)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55955);