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×#63070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C26 |
| commutator, Frattini, cosocle | C22 | C26×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
6588 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)(4,5,11,10)(6,12,15,8)(7,16,9,13)(14,18,17,19)(20,21)(22,30)(23,33)(24,27)(25,29)(26,28)(31,32)(34,35);(4,6)(5,12)(7,17)(8,10)(9,14)(11,15)(13,18)(16,19)(20,22,27,26)(21,28,24,30)(23,31,29,34)(25,32,33,35);(4,7,11,9)(5,13,10,16)(6,14,15,17)(8,18,12,19)(20,23)(21,25)(22,31)(24,33)(26,34)(27,29)(28,32)(30,35);(4,8)(5,6)(7,18)(9,19)(10,15)(11,12)(13,14)(16,17)(20,24,27,21)(22,28,26,30)(23,33,29,25)(31,32,34,35);(4,9,11,7)(5,13,10,16)(6,14,15,17)(8,19,12,18)(20,25,27,33)(21,29,24,23)(22,32,26,35)(28,34,30,31);(4,10,11,5)(6,8,15,12)(7,16,9,13)(14,18,17,19)(20,26,27,22)(21,28,24,30)(23,34,29,31)(25,32,33,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56058);