Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6356 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)(5,9)(10,16)(11,17)(18,19)(20,21)(22,28)(23,27)(24,25)(26,30)(29,34)(31,32)(33,35);(21,25)(26,28)(27,32)(33,34);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)(20,21)(22,26)(23,27)(24,25)(28,30)(29,33)(31,32)(34,35);(4,6,8,13)(5,10,9,16)(7,14,15,12)(11,19,17,18)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,35)(32,34);(4,7,8,15)(5,11,9,17)(6,12,13,14)(10,18,16,19);(4,5)(6,10)(7,11)(8,9)(12,18)(13,16)(14,19)(15,17)(20,23,24,31)(21,27,25,32)(22,29,30,35)(26,33,28,34)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55938);