Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#20521 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(8,9)(10,14)(11,15)(12,13)(16,17,21,24,20,23,29,19)(18,25,28,27,26,22,30,31);(4,5,7,6)(9,13)(14,15)(16,18,20,26)(17,22,23,25)(19,27,24,31)(21,28,29,30);(4,6,7,5)(8,10,12,11)(9,14,13,15)(17,23)(18,26)(19,24)(28,30);(8,11,12,10)(9,15,13,14)(16,19,29,23,20,24,21,17)(18,27,30,25,26,31,28,22)300 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13509);