Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#62828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6636 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)(6,8)(7,15)(9,10)(11,13)(12,17)(14,18)(16,19)(21,25)(23,30)(26,31)(29,34);(5,10)(6,13)(12,15)(14,19)(22,28)(26,31)(29,34)(33,35);(4,6,9,13)(5,11,10,8)(7,14,17,19)(12,18,15,16)(20,21,24,25)(22,26,28,31)(23,27,30,32)(29,33,34,35);(4,7)(5,12)(6,14)(8,16)(9,17)(10,15)(11,18)(13,19);(4,5)(6,11)(7,12)(8,13)(9,10)(14,18)(15,17)(16,19)(20,22)(21,26)(23,29)(24,28)(25,31)(27,33)(30,34)(32,35);(4,8,9,11)(5,13,10,6)(7,16,17,18)(12,19,15,14)(20,23)(21,27)(22,29)(24,30)(25,32)(26,33)(28,34)(31,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55816);