Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19755 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#19880 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#1624 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5,9,7)(6,12,16,19)(8,17,18,11)(10,14,13,15)(20,21,25,28)(22,26,24,32)(23,30,31,34)(27,33,35,29);(4,6,9,16)(5,10,7,13)(8,14,18,15)(11,19,17,12)(20,22,25,24)(21,26,28,32)(23,29,31,33)(27,34,35,30);(4,7,9,5)(6,13,16,10)(8,17,18,11)(12,15,19,14)(20,23,25,31)(21,27,28,35)(22,29,24,33)(26,34,32,30);(6,14)(8,18)(10,19)(11,17)(12,13)(15,16)(20,22,25,24)(21,26,28,32)(23,29,31,33)(27,34,35,30)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12868);