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×#62973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6798 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)(4,5,8,13)(6,14,17,10)(7,18,19,26)(9,12,21,28)(11,25,27,20)(15,30,33,35)(16,23,34,24)(22,32,29,31);(4,6)(5,10)(7,15)(8,17)(9,16)(11,22)(12,24)(13,14)(18,35)(19,33)(20,32)(21,34)(23,28)(25,31)(26,30)(27,29);(4,7)(5,11)(6,15)(8,19)(9,20)(10,22)(12,18)(13,27)(14,29)(16,32)(17,33)(21,25)(23,30)(24,35)(26,28)(31,34);(5,12)(6,16)(7,19)(11,26)(13,28)(15,31)(17,34)(18,27)(20,25)(22,29)(30,35)(32,33);(6,17)(7,20)(10,14)(11,18)(15,31)(16,34)(19,25)(22,30)(23,24)(26,27)(29,35)(32,33);(5,13)(6,16)(10,23)(11,27)(12,28)(14,24)(15,32)(17,34)(18,26)(22,30)(29,35)(31,33)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55961);