Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#18014 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C42 | C24×C3 |
| socle | C22×C3 | C2×C4×D4 |
| prime radical | C3×#447 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×#431 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3)(4,5)(6,10)(7,9)(8,12)(11,15)(13,17)(14,16)(18,19)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(5,9)(7,15)(10,16)(11,12)(14,19)(17,18);(4,6)(5,10)(7,14)(8,13)(9,16)(11,18)(12,17)(15,19)(20,22,23,29,24,28,30,34)(21,25,26,32,27,31,33,35);(6,13)(10,17)(14,19)(16,18)(22,28)(25,31)(29,34)(32,35)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 11002);