Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#30487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3)(4,5)(6,7)(8,9,12,18,13,19,14,15)(10,20,21,29,23,16,28,32)(11,24,25,17,26,30,27,33)(22,36,37,34,38,39,35,31);(9,15)(10,21)(11,25)(12,14)(17,33)(18,19)(23,28)(26,27)(29,32)(31,36)(34,39)(35,37);(5,7)(8,10,13,23)(9,16,19,20)(11,22,26,38)(12,21,14,28)(15,29,18,32)(17,31,33,34)(24,39,30,36)(25,37,27,35);(5,7)(8,11)(9,17)(10,22)(12,27)(13,26)(14,25)(15,30)(16,31)(18,24)(19,33)(20,34)(21,35)(23,38)(28,37)(29,36)(32,39)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 23475);