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×#19833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#1621 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
608 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)(5,10)(6,8)(11,18)(12,16)(13,17)(14,19)(20,21)(22,26)(23,30)(24,32)(25,28)(27,29)(31,34)(33,35);(4,5,9,12)(6,10,8,16)(7,14,15,18)(11,17,19,13)(20,22,25,24)(21,26,28,32)(23,29,31,33)(27,34,35,30);(4,6,9,8)(5,10,12,16)(7,13,15,17)(11,18,19,14)(20,23,25,31)(21,27,28,35)(22,29,24,33)(26,34,32,30);(4,7,9,15)(5,11,12,19)(6,13,8,17)(10,18,16,14)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12821);