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×#63069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6496 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)(6,14)(7,17)(8,11)(9,13)(10,12)(15,16)(18,19)(20,21,26,25)(22,29,24,27)(23,28,33,32)(30,35,31,34);(4,6,11,10)(5,12,8,14)(7,15,13,18)(9,16,17,19)(20,22,26,24)(21,27,25,29)(23,31,33,30)(28,35,32,34);(4,7)(5,9)(6,15)(8,17)(10,18)(11,13)(12,16)(14,19)(20,23,26,33)(21,28,25,32)(22,30,24,31)(27,34,29,35);(4,8,11,5)(6,12,10,14)(7,17,13,9)(15,16,18,19)(20,24,26,22)(21,29,25,27)(23,31,33,30)(28,35,32,34);(4,9,11,17)(5,13,8,7)(6,16,10,19)(12,18,14,15)(20,21,26,25)(22,27,24,29)(23,28,33,32)(30,34,31,35);(4,10,11,6)(5,12,8,14)(7,18,13,15)(9,16,17,19)(20,25,26,21)(22,29,24,27)(23,32,33,28)(30,35,31,34)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56057);