Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#19079 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C3 |
| prime radical | C3×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,14)(7,15)(8,13)(9,12)(10,11)(16,19)(17,18)(20,21,24,26)(22,28,30,35)(23,31,33,27)(25,32,34,29);(4,6)(5,7)(8,18)(9,15)(10,16)(11,17)(12,19)(13,14)(20,22)(21,25)(23,32)(24,30)(26,34)(27,35)(28,31)(29,33);(4,7)(5,6)(8,14)(9,19)(10,17)(11,16)(12,15)(13,18)(20,21)(22,25)(23,31)(24,26)(27,33)(28,32)(29,35)(30,34);(5,11)(7,17)(12,13)(14,19)(21,26)(25,34)(27,31)(28,35)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 12067);