Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#13273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #476 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C23×C3 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3)(5,7)(8,9,13,17,12,29,28,39,15,20,27,37,14,19,11,18)(10,21,25,16,31,34,30,33,26,32,22,36,24,38,23,35);(4,5,6,7)(8,10,12,24,15,26,14,31)(9,16,19,33,20,36,29,35)(11,22,13,30,28,25,27,23)(17,32,18,38,37,21,39,34);(8,11,14,27,15,28,12,13)(9,17,29,39,20,37,19,18)(10,22,31,23,26,25,24,30)(16,32,35,34,36,21,33,38)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6261);