Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C213 |
| direct product | C3×#63061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6384 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)(4,5,10,9)(6,13,16,11)(7,17,8,12)(14,18,15,19)(20,21,25,23)(22,28,30,26)(24,31,32,27)(29,33,35,34);(4,6,10,16)(5,11,9,13)(7,14,8,15)(12,18,17,19)(20,22,25,30)(21,26,23,28)(24,29,32,35)(27,33,31,34);(4,7,10,8)(5,12,9,17)(6,14,16,15)(11,18,13,19)(20,23,25,21)(22,26,30,28)(24,31,32,27)(29,33,35,34);(4,6)(5,11)(7,14)(8,15)(9,13)(10,16)(12,18)(17,19)(20,24)(21,27)(22,29)(23,31)(25,32)(26,33)(28,34)(30,35);(4,8,10,7)(5,12,9,17)(6,15,16,14)(11,18,13,19);(4,9,10,5)(6,13,16,11)(7,17,8,12)(14,19,15,18)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56049);