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×#19863 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3303 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
594 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)(5,9)(7,15)(10,16)(11,12)(13,19)(17,18)(20,21,25,28)(22,26,24,32)(23,30,31,34)(27,33,35,29);(4,5,8,12)(6,10,14,18)(7,11,15,9)(13,17,19,16)(20,22,25,24)(21,26,28,32)(23,29,31,33)(27,34,35,30);(4,6,8,14)(5,10,12,18)(7,13,15,19)(9,16,11,17)(20,23)(21,27)(22,29)(24,33)(25,31)(26,34)(28,35)(30,32);(5,11)(7,15)(9,12)(10,17)(13,19)(16,18)(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, 12851);