Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6626 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)(5,10)(6,12)(13,18)(14,19)(20,21,25,28)(22,29,30,26)(23,24,27,33)(31,35,34,32);(4,5)(6,11)(7,12)(8,13)(9,10)(14,16)(15,19)(17,18)(22,30)(26,29)(31,34)(32,35);(4,5)(6,11)(7,12)(8,13)(9,10)(14,16)(15,19)(17,18)(20,22)(21,26)(23,31)(24,32)(25,30)(27,34)(28,29)(33,35);(4,6,9,12)(5,11,10,7)(8,14,17,19)(13,16,18,15);(4,7,9,11)(5,12,10,6)(8,16,17,15)(13,14,18,19)(20,23)(21,24)(22,31)(25,27)(26,32)(28,33)(29,35)(30,34);(4,8)(5,13)(6,14)(7,15)(9,17)(10,18)(11,16)(12,19)(20,24)(21,27)(22,32)(23,28)(25,33)(26,34)(29,31)(30,35)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56017);