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×#63033 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 6 generators on 35 points:
(1,2,3)(4,5,10,8)(6,12,7,11)(9,13,17,16)(14,19,15,18)(22,29)(26,30)(31,34)(32,35);(4,6,10,7)(5,11,8,12)(9,14,17,15)(13,18,16,19)(20,21,25,28)(22,30,29,26)(23,24,27,33)(31,35,34,32);(4,6,10,7)(5,11,8,12)(9,14,17,15)(13,18,16,19)(20,22,25,29)(21,26,28,30)(23,31,27,34)(24,32,33,35);(4,7,10,6)(5,12,8,11)(9,15,17,14)(13,19,16,18);(4,8,10,5)(6,12,7,11)(9,13,17,16)(14,18,15,19)(20,23)(21,24)(22,31)(25,27)(26,32)(28,33)(29,34)(30,35);(4,9)(5,13)(6,14)(7,15)(8,16)(10,17)(11,18)(12,19)(20,24)(21,27)(22,32)(23,28)(25,33)(26,34)(29,35)(30,31)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56021);