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## G = D4×A4order 96 = 25·3

### Direct product of D4 and A4

Aliases: D4×A4, C242C6, C4⋊(C2×A4), (C22×C4)⋊C6, C22⋊(C3×D4), (C22×D4)⋊C3, (C4×A4)⋊3C2, C222(C2×A4), (C22×A4)⋊1C2, C23.6(C2×C6), C2.2(C22×A4), (C2×A4).7C22, SmallGroup(96,197)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — D4×A4
 Chief series C1 — C22 — C23 — C2×A4 — C22×A4 — D4×A4
 Lower central C22 — C23 — D4×A4
 Upper central C1 — C2 — D4

Generators and relations for D4×A4
G = < a,b,c,d,e | a4=b2=c2=d2=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 208 in 66 conjugacy classes, 18 normal (12 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22, C22 [×2], C22 [×12], C6 [×3], C2×C4 [×2], D4, D4 [×5], C23, C23 [×8], C12, A4, C2×C6 [×2], C22×C4, C2×D4 [×4], C24 [×2], C3×D4, C2×A4, C2×A4 [×2], C22×D4, C4×A4, C22×A4 [×2], D4×A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4

Character table of D4×A4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A 6B 6C 6D 6E 6F 12A 12B size 1 1 2 2 3 3 6 6 4 4 2 6 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 linear of order 3 ρ6 1 1 -1 1 1 1 1 -1 ζ3 ζ32 -1 -1 ζ32 ζ3 ζ32 ζ65 ζ3 ζ6 ζ6 ζ65 linear of order 6 ρ7 1 1 -1 -1 1 1 -1 -1 ζ32 ζ3 1 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 linear of order 6 ρ8 1 1 -1 1 1 1 1 -1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ3 ζ6 ζ32 ζ65 ζ65 ζ6 linear of order 6 ρ9 1 1 -1 -1 1 1 -1 -1 ζ3 ζ32 1 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 linear of order 6 ρ10 1 1 1 1 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 linear of order 3 ρ11 1 1 1 -1 1 1 -1 1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ65 ζ32 ζ6 ζ3 ζ65 ζ6 linear of order 6 ρ12 1 1 1 -1 1 1 -1 1 ζ3 ζ32 -1 -1 ζ32 ζ3 ζ6 ζ3 ζ65 ζ32 ζ6 ζ65 linear of order 6 ρ13 2 -2 0 0 2 -2 0 0 2 2 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 2 -2 0 0 -1-√-3 -1+√-3 0 0 1-√-3 1+√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 -2 0 0 2 -2 0 0 -1+√-3 -1-√-3 0 0 1+√-3 1-√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 3 3 -3 -3 -1 -1 1 1 0 0 3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ17 3 3 -3 3 -1 -1 -1 1 0 0 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ18 3 3 3 -3 -1 -1 1 -1 0 0 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ19 3 3 3 3 -1 -1 -1 -1 0 0 3 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ20 6 -6 0 0 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D4×A4
On 12 points - transitive group 12T51
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)
(5 7)(6 8)(9 11)(10 12)
(1 3)(2 4)(9 11)(10 12)
(1 7 9)(2 8 10)(3 5 11)(4 6 12)

G:=sub<Sym(12)| (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11), (5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(9,11)(10,12), (1,7,9)(2,8,10)(3,5,11)(4,6,12)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11), (5,7)(6,8)(9,11)(10,12), (1,3)(2,4)(9,11)(10,12), (1,7,9)(2,8,10)(3,5,11)(4,6,12) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)], [(5,7),(6,8),(9,11),(10,12)], [(1,3),(2,4),(9,11),(10,12)], [(1,7,9),(2,8,10),(3,5,11),(4,6,12)])

G:=TransitiveGroup(12,51);

On 16 points - transitive group 16T179
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)
(1 6)(2 7)(3 8)(4 5)(9 13)(10 14)(11 15)(12 16)
(1 12)(2 9)(3 10)(4 11)(5 15)(6 16)(7 13)(8 14)
(5 11 15)(6 12 16)(7 9 13)(8 10 14)

G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,12)(2,9)(3,10)(4,11)(5,15)(6,16)(7,13)(8,14), (5,11,15)(6,12,16)(7,9,13)(8,10,14)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16), (1,6)(2,7)(3,8)(4,5)(9,13)(10,14)(11,15)(12,16), (1,12)(2,9)(3,10)(4,11)(5,15)(6,16)(7,13)(8,14), (5,11,15)(6,12,16)(7,9,13)(8,10,14) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)], [(1,6),(2,7),(3,8),(4,5),(9,13),(10,14),(11,15),(12,16)], [(1,12),(2,9),(3,10),(4,11),(5,15),(6,16),(7,13),(8,14)], [(5,11,15),(6,12,16),(7,9,13),(8,10,14)])

G:=TransitiveGroup(16,179);

On 24 points - transitive group 24T160
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 16)(6 15)(7 14)(8 13)(17 24)(18 23)(19 22)(20 21)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(1 19 13)(2 20 14)(3 17 15)(4 18 16)(5 9 23)(6 10 24)(7 11 21)(8 12 22)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,24)(18,23)(19,22)(20,21), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,19,13)(2,20,14)(3,17,15)(4,18,16)(5,9,23)(6,10,24)(7,11,21)(8,12,22)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,24)(18,23)(19,22)(20,21), (5,7)(6,8)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16), (1,19,13)(2,20,14)(3,17,15)(4,18,16)(5,9,23)(6,10,24)(7,11,21)(8,12,22) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,16),(6,15),(7,14),(8,13),(17,24),(18,23),(19,22),(20,21)], [(5,7),(6,8),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16)], [(1,19,13),(2,20,14),(3,17,15),(4,18,16),(5,9,23),(6,10,24),(7,11,21),(8,12,22)])

G:=TransitiveGroup(24,160);

On 24 points - transitive group 24T161
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 23)(18 22)(19 21)(20 24)
(1 3)(2 4)(5 7)(6 8)(9 14)(10 15)(11 16)(12 13)(17 23)(18 24)(19 21)(20 22)
(1 8)(2 5)(3 6)(4 7)(9 11)(10 12)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 15 21)(2 16 22)(3 13 23)(4 14 24)(5 9 20)(6 10 17)(7 11 18)(8 12 19)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24), (1,3)(2,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13)(17,23)(18,24)(19,21)(20,22), (1,8)(2,5)(3,6)(4,7)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24), (1,3)(2,4)(5,7)(6,8)(9,14)(10,15)(11,16)(12,13)(17,23)(18,24)(19,21)(20,22), (1,8)(2,5)(3,6)(4,7)(9,11)(10,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,16),(12,15),(17,23),(18,22),(19,21),(20,24)], [(1,3),(2,4),(5,7),(6,8),(9,14),(10,15),(11,16),(12,13),(17,23),(18,24),(19,21),(20,22)], [(1,8),(2,5),(3,6),(4,7),(9,11),(10,12),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,15,21),(2,16,22),(3,13,23),(4,14,24),(5,9,20),(6,10,17),(7,11,18),(8,12,19)])

G:=TransitiveGroup(24,161);

On 24 points - transitive group 24T162
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(6 8)(10 12)(13 15)(18 20)(22 24)
(1 3)(2 4)(5 14)(6 15)(7 16)(8 13)(9 11)(10 12)(17 23)(18 24)(19 21)(20 22)
(1 11)(2 12)(3 9)(4 10)(5 7)(6 8)(13 15)(14 16)(17 21)(18 22)(19 23)(20 24)
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 23)(10 13 24)(11 14 21)(12 15 22)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(6,8)(10,12)(13,15)(18,20)(22,24), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,23)(18,24)(19,21)(20,22), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,23)(10,13,24)(11,14,21)(12,15,22)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(6,8)(10,12)(13,15)(18,20)(22,24), (1,3)(2,4)(5,14)(6,15)(7,16)(8,13)(9,11)(10,12)(17,23)(18,24)(19,21)(20,22), (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24), (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,23)(10,13,24)(11,14,21)(12,15,22) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(6,8),(10,12),(13,15),(18,20),(22,24)], [(1,3),(2,4),(5,14),(6,15),(7,16),(8,13),(9,11),(10,12),(17,23),(18,24),(19,21),(20,22)], [(1,11),(2,12),(3,9),(4,10),(5,7),(6,8),(13,15),(14,16),(17,21),(18,22),(19,23),(20,24)], [(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,23),(10,13,24),(11,14,21),(12,15,22)])

G:=TransitiveGroup(24,162);

On 24 points - transitive group 24T163
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)(17 24)(18 23)(19 22)(20 21)
(9 14)(10 15)(11 16)(12 13)(17 21)(18 22)(19 23)(20 24)
(1 6)(2 7)(3 8)(4 5)(17 21)(18 22)(19 23)(20 24)
(1 15 23)(2 16 24)(3 13 21)(4 14 22)(5 9 18)(6 10 19)(7 11 20)(8 12 17)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)(17,24)(18,23)(19,22)(20,21), (9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,6)(2,7)(3,8)(4,5)(17,21)(18,22)(19,23)(20,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)(17,24)(18,23)(19,22)(20,21), (9,14)(10,15)(11,16)(12,13)(17,21)(18,22)(19,23)(20,24), (1,6)(2,7)(3,8)(4,5)(17,21)(18,22)(19,23)(20,24), (1,15,23)(2,16,24)(3,13,21)(4,14,22)(5,9,18)(6,10,19)(7,11,20)(8,12,17) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16),(17,24),(18,23),(19,22),(20,21)], [(9,14),(10,15),(11,16),(12,13),(17,21),(18,22),(19,23),(20,24)], [(1,6),(2,7),(3,8),(4,5),(17,21),(18,22),(19,23),(20,24)], [(1,15,23),(2,16,24),(3,13,21),(4,14,22),(5,9,18),(6,10,19),(7,11,20),(8,12,17)])

G:=TransitiveGroup(24,163);

On 24 points - transitive group 24T164
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(22 24)
(5 22)(6 23)(7 24)(8 21)(13 20)(14 17)(15 18)(16 19)
(1 11)(2 12)(3 9)(4 10)(5 22)(6 23)(7 24)(8 21)
(1 17 8)(2 18 5)(3 19 6)(4 20 7)(9 16 23)(10 13 24)(11 14 21)(12 15 22)

G:=sub<Sym(24)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (5,22)(6,23)(7,24)(8,21)(13,20)(14,17)(15,18)(16,19), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21), (1,17,8)(2,18,5)(3,19,6)(4,20,7)(9,16,23)(10,13,24)(11,14,21)(12,15,22)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(22,24), (5,22)(6,23)(7,24)(8,21)(13,20)(14,17)(15,18)(16,19), (1,11)(2,12)(3,9)(4,10)(5,22)(6,23)(7,24)(8,21), (1,17,8)(2,18,5)(3,19,6)(4,20,7)(9,16,23)(10,13,24)(11,14,21)(12,15,22) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(5,7),(10,12),(13,15),(18,20),(22,24)], [(5,22),(6,23),(7,24),(8,21),(13,20),(14,17),(15,18),(16,19)], [(1,11),(2,12),(3,9),(4,10),(5,22),(6,23),(7,24),(8,21)], [(1,17,8),(2,18,5),(3,19,6),(4,20,7),(9,16,23),(10,13,24),(11,14,21),(12,15,22)])

G:=TransitiveGroup(24,164);

D4×A4 is a maximal subgroup of   A4⋊SD16  D4⋊S4  D42S4
D4×A4 is a maximal quotient of   SL2(𝔽3)⋊5D4  SL2(𝔽3)⋊6D4  Q16.A4  SD16.A4  D8.A4

Polynomial with Galois group D4×A4 over ℚ
actionf(x)Disc(f)
12T51x12-19x10+133x8-422x6+588x4-279x2+27212·315·78·136·294

Matrix representation of D4×A4 in GL5(ℤ)

 0 1 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 -1 -1 -1
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 -1 -1 -1 0 0 1 0 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 0 0 0 1 0

G:=sub<GL(5,Integers())| [0,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,-1,0,0,1,0,-1,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,-1,0,0,0,1,-1,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,-1,0,0,0,0,-1,1,0,0,0,-1,0] >;

D4×A4 in GAP, Magma, Sage, TeX

D_4\times A_4
% in TeX

G:=Group("D4xA4");
// GroupNames label

G:=SmallGroup(96,197);
// by ID

G=gap.SmallGroup(96,197);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,2,169,376,665]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^2=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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