Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62972 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,18)(8,10)(9,11)(12,15)(13,20)(16,25)(17,32)(19,27)(21,22)(23,29)(24,34)(26,35)(28,31)(30,33);(5,10)(6,15)(7,19)(11,22)(12,25)(13,27)(14,23)(16,29)(18,24)(20,34)(26,33)(28,32);(5,11)(6,16)(10,22)(12,25)(13,24)(14,23)(15,29)(17,31)(18,27)(26,32)(28,33)(30,35);(4,6)(5,12)(7,17)(8,16)(9,15)(10,23)(11,14)(13,26)(18,33)(19,30)(20,31)(21,29)(22,25)(24,28)(27,32)(34,35);(4,5)(6,12)(7,13)(8,10)(9,11)(14,15)(16,23)(17,26)(18,20)(19,24)(21,22)(25,29)(27,34)(28,30)(31,33)(32,35);(4,7)(5,13)(6,17)(8,20)(9,19)(10,18)(11,24)(12,26)(14,28)(15,30)(16,31)(21,34)(22,27)(23,33)(25,32)(29,35)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55960);