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×#63036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6608 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)(4,5,11,9)(6,7,12,8)(10,13,18,17)(14,15,19,16)(22,29)(26,30)(31,34)(32,35);(4,6)(5,8)(7,9)(10,14)(11,12)(13,16)(15,17)(18,19)(20,21,25,28)(22,30,29,26)(23,24,27,33)(31,35,34,32);(4,7,11,8)(5,6,9,12)(10,15,18,16)(13,14,17,19)(20,22,25,29)(21,26,28,30)(23,31,27,34)(24,32,33,35);(4,8,11,7)(5,12,9,6)(10,16,18,15)(13,19,17,14);(4,9,11,5)(6,7,12,8)(10,13,18,17)(14,16,19,15)(20,23)(21,24)(22,31)(25,27)(26,32)(28,33)(29,34)(30,35);(4,10,11,18)(5,13,9,17)(6,14,12,19)(7,15,8,16)(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, 56024);