Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
942 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)(6,14)(7,11)(8,12)(9,13)(10,16)(15,18)(17,19)(20,21)(22,29)(23,27)(24,32)(25,28)(26,30)(31,34)(33,35);(4,6)(5,10)(7,15)(8,16)(9,17)(11,14)(12,19)(13,18)(20,22,25,31)(21,26,28,35)(23,30,24,33)(27,34,32,29);(4,6)(5,10)(7,16)(8,15)(9,17)(11,19)(12,14)(13,18)(23,24)(27,32)(29,34)(30,33);(4,7,9,8)(5,11,13,12)(6,15,17,16)(10,14,18,19)(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12049);