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×#62974 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6546 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)(7,11)(8,9)(10,13)(14,18)(15,16)(17,19)(20,21,25,26)(22,27,28,33)(23,24,31,32)(29,30,34,35);(6,13)(7,15)(10,12)(11,16)(21,26)(23,31)(27,33)(29,34);(5,9)(10,12)(11,16)(17,18)(22,28)(27,33)(29,34)(30,35);(4,6)(5,10)(7,14)(8,13)(9,12)(11,17)(15,19)(16,18)(20,22,25,28)(21,27,26,33)(23,29,31,34)(24,30,32,35);(4,5)(6,10)(7,11)(8,9)(12,13)(14,17)(15,16)(18,19)(20,23)(21,24)(22,29)(25,31)(26,32)(27,30)(28,34)(33,35);(4,7)(5,11)(6,14)(8,15)(9,16)(10,17)(12,18)(13,19)(20,24)(21,23)(22,30)(25,32)(26,31)(27,29)(28,35)(33,34)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55962);