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×#62906 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6716 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)(6,14)(7,13)(8,12)(9,11)(10,15)(16,28)(17,21)(18,25)(19,26)(20,24)(22,29)(23,30)(27,32)(31,33)(34,35);(4,6,9,17)(5,10,11,23)(7,16,18,31)(8,15,20,30)(12,14,24,21)(13,22,25,34)(19,29,32,35)(26,28,27,33);(5,11)(6,15)(10,21)(12,24)(13,25)(14,23)(16,29)(17,30)(22,33)(26,27)(28,34)(31,35);(5,12)(7,18)(10,14)(11,24)(13,27)(16,31)(19,32)(21,23)(22,33)(25,26)(28,34)(29,35);(4,7,8,19)(5,13,12,26)(6,16,15,29)(9,18,20,32)(10,22,14,28)(11,25,24,27)(17,31,30,35)(21,33,23,34);(5,11)(10,23)(12,24)(13,25)(14,21)(22,34)(26,27)(28,33)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55894);