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×#62968 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6444 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)(6,12)(10,13)(14,18)(17,19)(20,21)(22,28)(23,26)(24,27)(25,30)(29,34)(31,33)(32,35);(4,5,9,7)(6,10,12,13)(8,11,16,15)(14,17,18,19)(20,21,24,27)(22,25,30,28)(23,26,31,33)(29,32,35,34);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,21)(22,25)(23,26)(24,27)(28,30)(29,32)(31,33)(34,35);(4,7,9,5)(6,13,12,10)(8,11,16,15)(14,17,18,19)(20,22)(21,25)(23,29)(24,30)(26,32)(27,28)(31,35)(33,34);(4,8,9,16)(5,11,7,15)(6,14,12,18)(10,17,13,19);(4,6)(5,10)(7,13)(8,14)(9,12)(11,17)(15,19)(16,18)(20,23,24,31)(21,26,27,33)(22,29,30,35)(25,32,28,34)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55956);