Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C27 |
| direct product | C22×C3×#341 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3×C4 | D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C24×C3 | C4⋊C4 |
| prime radical | C25×C3 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C4 ◁ C24×C3×C4 ◁ G | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(25,26,27);(3,4);(5,6)(7,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(6,8)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)336 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 13337);