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×#63055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6730 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)(6,13)(7,17)(8,11)(9,12)(10,16)(14,18)(15,19)(20,21)(22,27)(23,26)(24,28)(25,30)(29,34)(31,32)(33,35);(4,6,9,16)(5,10,12,13)(7,14,8,15)(11,18,17,19)(20,21)(22,27)(23,31)(24,28)(25,30)(26,32)(29,33)(34,35);(4,7,9,8)(5,11,12,17)(6,14,16,15)(10,18,13,19)(20,22)(21,25)(23,29)(24,30)(26,33)(27,28)(31,34)(32,35);(4,6)(5,10)(7,14)(8,15)(9,16)(11,18)(12,13)(17,19)(20,23)(21,26)(22,29)(24,32)(25,33)(27,34)(28,31)(30,35);(4,8,9,7)(5,11,12,17)(6,15,16,14)(10,18,13,19)(20,22)(21,27)(23,29)(24,30)(25,28)(26,34)(31,33)(32,35);(4,5)(6,10)(7,11)(8,17)(9,12)(13,16)(14,18)(15,19)(20,21)(22,25)(23,26)(24,28)(27,30)(29,33)(31,32)(34,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56043);