Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×#121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
990 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)(4,5,7,6)(8,9)(10,19)(11,23)(12,17)(13,28)(14,18)(15,21)(16,26)(20,34)(22,30)(24,38)(25,32)(27,31)(29,39)(33,36)(35,37);(4,6,7,5)(8,10,13,21,14,22,12,27)(9,15,17,19,18,31,28,30)(11,20,25,35,26,36,24,39)(16,29,32,33,23,37,38,34);(8,11,13,25,14,26,12,24)(9,16,17,32,18,23,28,38)(10,20,21,35,22,36,27,39)(15,29,19,33,31,37,30,34);(8,12,14,13)(9,17,18,28)(10,21,22,27)(11,24,26,25)(15,30,31,19)(16,32,23,38)(20,35,36,39)(29,34,37,33)174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14065);