Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63054 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6624 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)(6,11)(7,10)(8,12)(9,14)(13,18)(15,16)(17,19)(20,21)(22,29)(23,33)(24,27)(25,28)(26,32)(30,31)(34,35);(4,5)(6,11)(7,15)(8,12)(9,14)(10,16)(13,17)(18,19)(20,22,25,32)(21,26,28,29)(23,30,27,34)(24,31,33,35);(4,6)(5,9)(7,13)(8,14)(10,17)(11,12)(15,18)(16,19)(20,23,25,27)(21,24,28,33)(22,30,32,34)(26,31,29,35);(4,7)(5,10)(6,13)(8,16)(9,17)(11,18)(12,15)(14,19)(20,22)(21,26)(23,30)(24,31)(25,32)(27,34)(28,29)(33,35);(4,6)(5,11)(7,13)(8,14)(9,12)(10,18)(15,17)(16,19)(20,21)(22,26)(23,24)(25,28)(27,33)(29,32)(30,31)(34,35);(4,5)(6,9)(7,10)(8,12)(11,14)(13,17)(15,16)(18,19)(20,24)(21,27)(22,31)(23,28)(25,33)(26,34)(29,30)(32,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56042);