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×#62829 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6636 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)(6,12)(7,13)(8,16)(9,10)(11,17)(14,15)(18,19)(21,25)(23,30)(26,31)(29,34);(5,10)(6,13)(11,16)(14,19)(22,28)(26,31)(29,34)(33,35);(4,6)(5,7)(8,14)(9,13)(10,12)(11,15)(16,18)(17,19)(20,21,24,25)(22,26,28,31)(23,27,30,32)(29,33,34,35);(4,7)(5,6)(8,15)(9,12)(10,13)(11,14)(16,19)(17,18);(4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,17)(18,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(4,8,9,17)(5,11,10,16)(6,14,13,19)(7,15,12,18)(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, 55817);