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×#13272 |
| 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 | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(8,9,11,17,13,21,12,16,15,18,22,23,14,10,19,20);(4,5,6,7)(8,10,14,9,15,21,13,18)(11,17,19,16,22,23,12,20);(8,11,13,12,15,22,14,19)(9,16,10,17,18,20,21,23)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6260);