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×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #476 |
| commutator | C2×C8 | C22×C3×C4 |
| Frattini, cosocle | C22×C8 | C23×C3 |
| socle | C2×C3 | #1987 |
| prime radical | C23×C3 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,5,8,14,10,15,24,34,11,16,23,13,9,26,7,22)(6,17,18,29,20,33,35,12,21,30,19,31,27,28,25,32);(4,6)(5,12,16,32)(7,18)(8,25)(9,20)(10,27)(11,21)(13,28,14,33)(15,29,26,31)(17,22,30,34)(19,24)(23,35);(4,7,9,23,11,24,10,8)(5,13,15,22,16,14,26,34)(6,18,20,35,21,19,27,25)(12,28,29,30,32,33,31,17)138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 6259);