Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C3×#21049 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C23×C3 | C23×C4 |
| prime radical | C25×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,5)(6,7)(9,10)(12,13,15,18,16,14,17,19);(8,9,11,10)(12,14,15,19,16,13,17,18);(5,7)(8,10,11,9);(13,14)(18,19)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 14037);