Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#7180 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | #106 |
| commutator | C8 | C3×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| socle | C2×C3 | #1279 |
| prime radical | C23×C3 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ C22×C3×C8 ◁ C2×C3×C4×C8 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3)(4,5,8,14,10,16,26,31,11,17,25,13,9,15,7,24)(6,18,20,29,22,34,19,30,23,32,27,28,21,33,35,12);(4,6)(5,12,15,28,17,30,16,29)(7,19)(8,27)(9,21)(10,22)(11,23)(13,18,31,33,14,32,24,34)(20,25)(26,35)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 168);