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×#62958 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6572 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)(4,5,9,12)(6,13,7,10)(8,16,17,11)(14,18,15,19)(22,28)(26,29)(30,34)(33,35);(4,6,9,7)(5,10,12,13)(8,14,17,15)(11,18,16,19)(20,21,25,23)(22,26,28,29)(24,27,32,31)(30,33,34,35);(4,5)(6,13)(7,10)(8,11)(9,12)(14,19)(15,18)(16,17)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,35)(32,34);(4,5)(6,10)(7,13)(8,11)(9,12)(14,18)(15,19)(16,17)(20,23,25,21)(22,29,28,26)(24,27,32,31)(30,33,34,35);(4,7,9,6)(5,10,12,13)(8,15,17,14)(11,18,16,19)(20,24,25,32)(21,27,23,31)(22,30,28,34)(26,33,29,35);(4,8)(5,11)(6,14)(7,15)(9,17)(10,18)(12,16)(13,19)(20,22)(21,26)(23,29)(24,30)(25,28)(27,33)(31,35)(32,34)219 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 55946);