Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×C3×#416 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C24×C3 | C2×D4 |
| prime radical | C23×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ C24×C3×C4 ◁ G | |
5326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(25,26,27);(3,4);(5,6,7,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 53067);