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G = D4×S4order 192 = 26·3

Direct product of D4 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×A4 — D4×S4
 Chief series C1 — C22 — A4 — C2×A4 — C2×S4 — C22×S4 — D4×S4
 Lower central A4 — C2×A4 — D4×S4
 Upper central C1 — C2 — D4

Generators and relations for D4×S4
G = < a,b,c,d,e,f | a4=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 1026 in 234 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2 [×10], C3, C4, C4 [×7], C22, C22 [×2], C22 [×30], S3 [×4], C6 [×3], C2×C4 [×13], D4, D4 [×28], C23, C23 [×21], Dic3, C12, A4, D6 [×7], C2×C6 [×2], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×3], C2×D4 [×28], C24 [×2], C24 [×2], C4×S3, D12, C3⋊D4 [×2], C3×D4, S4 [×2], S4 [×2], C2×A4, C2×A4 [×2], C22×S3 [×2], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4, C22×D4 [×3], A4⋊C4, C4×A4, S3×D4, C2×S4, C2×S4 [×2], C2×S4 [×4], C22×A4 [×2], D42, C4×S4, C4⋊S4, A4⋊D4 [×2], D4×A4, C22×S4 [×2], D4×S4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, S4, C22×S3, S3×D4, C2×S4 [×3], C22×S4, D4×S4

Character table of D4×S4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 12 size 1 1 2 2 3 3 6 6 6 6 12 12 8 2 6 6 6 12 12 12 12 8 16 16 16 ρ1 1 1 1 1 1 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 -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 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 -1 1 -1 -1 1 linear of order 2 ρ5 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ6 1 1 -1 1 1 1 1 -1 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 -2 0 0 2 -2 -2 0 0 2 0 0 2 0 0 2 -2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 2 2 0 -2 2 0 0 0 -1 -2 -2 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from D6 ρ11 2 -2 0 0 2 -2 2 0 0 -2 0 0 2 0 0 -2 2 0 0 0 0 -2 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 2 2 0 2 2 0 0 0 -1 2 2 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 2 -2 -2 2 2 0 -2 -2 0 0 0 -1 2 2 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from D6 ρ14 2 2 2 -2 2 2 0 2 -2 0 0 0 -1 -2 -2 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from D6 ρ15 3 3 3 -3 -1 -1 -1 -1 1 -1 -1 1 0 -3 1 1 1 -1 -1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ16 3 3 -3 -3 -1 -1 1 1 1 1 -1 -1 0 3 -1 -1 -1 -1 1 1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 3 -3 3 -1 -1 -1 1 -1 -1 1 -1 0 -3 1 1 1 -1 1 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ18 3 3 3 3 -1 -1 1 -1 -1 1 1 1 0 3 -1 -1 -1 -1 -1 -1 1 0 0 0 0 orthogonal lifted from S4 ρ19 3 3 3 3 -1 -1 -1 -1 -1 -1 -1 -1 0 3 -1 1 1 1 1 1 -1 0 0 0 0 orthogonal lifted from S4 ρ20 3 3 -3 3 -1 -1 1 1 -1 1 -1 1 0 -3 1 -1 -1 1 -1 1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ21 3 3 -3 -3 -1 -1 -1 1 1 -1 1 1 0 3 -1 1 1 1 -1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ22 3 3 3 -3 -1 -1 1 -1 1 1 1 -1 0 -3 1 -1 -1 1 1 -1 -1 0 0 0 0 orthogonal lifted from C2×S4 ρ23 4 -4 0 0 4 -4 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 0 0 0 orthogonal lifted from S3×D4 ρ24 6 -6 0 0 -2 2 -2 0 0 2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 0 orthogonal faithful ρ25 6 -6 0 0 -2 2 2 0 0 -2 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D4×S4
On 12 points - transitive group 12T86
Generators in S12
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)
(1 3)(2 4)(9 11)(10 12)
(5 7)(6 8)(9 11)(10 12)
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(5 10)(6 11)(7 12)(8 9)

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

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

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

G:=TransitiveGroup(12,86);

On 16 points - transitive group 16T421
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 4)(5 7)(9 11)(13 15)
(1 14)(2 15)(3 16)(4 13)(5 11)(6 12)(7 9)(8 10)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)
(5 15 11)(6 16 12)(7 13 9)(8 14 10)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 11)(10 12)

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

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

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

G:=TransitiveGroup(16,421);

On 24 points - transitive group 24T358
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)(9 11)(13 15)(18 20)(22 24)
(1 17)(2 18)(3 19)(4 20)(5 7)(6 8)(9 22)(10 23)(11 24)(12 21)(13 15)(14 16)
(1 3)(2 4)(5 13)(6 14)(7 15)(8 16)(9 24)(10 21)(11 22)(12 23)(17 19)(18 20)
(1 21 14)(2 22 15)(3 23 16)(4 24 13)(5 18 11)(6 19 12)(7 20 9)(8 17 10)
(1 3)(2 4)(5 11)(6 12)(7 9)(8 10)(13 22)(14 23)(15 24)(16 21)

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

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

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

G:=TransitiveGroup(24,358);

On 24 points - transitive group 24T393
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 21)(6 24)(7 23)(8 22)(13 17)(14 20)(15 19)(16 18)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 15)(14 16)(17 19)(18 20)
(1 3)(2 4)(5 24)(6 21)(7 22)(8 23)(9 11)(10 12)(13 20)(14 17)(15 18)(16 19)
(1 8 18)(2 5 19)(3 6 20)(4 7 17)(9 23 13)(10 24 14)(11 21 15)(12 22 16)
(1 10)(2 11)(3 12)(4 9)(5 15)(6 16)(7 13)(8 14)(17 23)(18 24)(19 21)(20 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)(5,21)(6,24)(7,23)(8,22)(13,17)(14,20)(15,19)(16,18), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,15)(14,16)(17,19)(18,20), (1,3)(2,4)(5,24)(6,21)(7,22)(8,23)(9,11)(10,12)(13,20)(14,17)(15,18)(16,19), (1,8,18)(2,5,19)(3,6,20)(4,7,17)(9,23,13)(10,24,14)(11,21,15)(12,22,16), (1,10)(2,11)(3,12)(4,9)(5,15)(6,16)(7,13)(8,14)(17,23)(18,24)(19,21)(20,22)>;

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

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

G:=TransitiveGroup(24,393);

On 24 points - transitive group 24T434
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)
(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(1 11 14)(2 12 15)(3 9 16)(4 10 13)(5 20 22)(6 17 23)(7 18 24)(8 19 21)
(1 23)(2 24)(3 21)(4 22)(5 13)(6 14)(7 15)(8 16)(9 19)(10 20)(11 17)(12 18)

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

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

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

G:=TransitiveGroup(24,434);

On 24 points - transitive group 24T435
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 9)(2 12)(3 11)(4 10)(5 14)(6 13)(7 16)(8 15)(17 22)(18 21)(19 24)(20 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 15 19)(2 16 20)(3 13 17)(4 14 18)(5 21 10)(6 22 11)(7 23 12)(8 24 9)
(1 3)(2 4)(5 23)(6 24)(7 21)(8 22)(9 11)(10 12)(13 19)(14 20)(15 17)(16 18)

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

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

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

G:=TransitiveGroup(24,435);

On 24 points - transitive group 24T436
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)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)
(5 7)(6 8)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)
(1 13 19)(2 14 20)(3 15 17)(4 16 18)(5 23 9)(6 24 10)(7 21 11)(8 22 12)
(1 11)(2 12)(3 9)(4 10)(5 17)(6 18)(7 19)(8 20)(13 21)(14 22)(15 23)(16 24)

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

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

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

G:=TransitiveGroup(24,436);

On 24 points - transitive group 24T437
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 23)(6 22)(7 21)(8 24)(13 18)(14 17)(15 20)(16 19)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)
(5 22)(6 23)(7 24)(8 21)(13 19)(14 20)(15 17)(16 18)
(1 8 19)(2 5 20)(3 6 17)(4 7 18)(9 21 13)(10 22 14)(11 23 15)(12 24 16)
(5 20)(6 17)(7 18)(8 19)(13 21)(14 22)(15 23)(16 24)

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

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

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

G:=TransitiveGroup(24,437);

On 24 points - transitive group 24T438
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 19)(6 18)(7 17)(8 20)(13 22)(14 21)(15 24)(16 23)
(1 9)(2 10)(3 11)(4 12)(13 23)(14 24)(15 21)(16 22)
(5 18)(6 19)(7 20)(8 17)(13 23)(14 24)(15 21)(16 22)
(1 15 19)(2 16 20)(3 13 17)(4 14 18)(5 12 24)(6 9 21)(7 10 22)(8 11 23)
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)(17 23)(18 24)(19 21)(20 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)(5,19)(6,18)(7,17)(8,20)(13,22)(14,21)(15,24)(16,23), (1,9)(2,10)(3,11)(4,12)(13,23)(14,24)(15,21)(16,22), (5,18)(6,19)(7,20)(8,17)(13,23)(14,24)(15,21)(16,22), (1,15,19)(2,16,20)(3,13,17)(4,14,18)(5,12,24)(6,9,21)(7,10,22)(8,11,23), (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,23)(18,24)(19,21)(20,22)>;

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

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

G:=TransitiveGroup(24,438);

On 24 points - transitive group 24T439
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)(17 19)(22 24)
(1 8)(2 5)(3 6)(4 7)(9 18)(10 19)(11 20)(12 17)
(9 18)(10 19)(11 20)(12 17)(13 22)(14 23)(15 24)(16 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 19 15)(6 20 16)(7 17 13)(8 18 14)
(9 23)(10 24)(11 21)(12 22)(13 17)(14 18)(15 19)(16 20)

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

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

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

G:=TransitiveGroup(24,439);

On 24 points - transitive group 24T440
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)(17 19)(22 24)
(1 8)(2 5)(3 6)(4 7)(9 18)(10 19)(11 20)(12 17)
(9 18)(10 19)(11 20)(12 17)(13 22)(14 23)(15 24)(16 21)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 19 15)(6 20 16)(7 17 13)(8 18 14)
(1 8)(2 5)(3 6)(4 7)(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)

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

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

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

G:=TransitiveGroup(24,440);

Polynomial with Galois group D4×S4 over ℚ
actionf(x)Disc(f)
12T86x12-24x10+225x8-1044x6+2473x4-2691x2+891212·312·56·115·134·434

Matrix representation of D4×S4 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 0 1 0 0 -1 -1 -1 0 0 1 0 0
,
 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 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

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,-1,1,0,0,0,-1,0,0,0,1,-1,0],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

D4×S4 in GAP, Magma, Sage, TeX

D_4\times S_4
% in TeX

G:=Group("D4xS4");
// GroupNames label

G:=SmallGroup(192,1472);
// by ID

G=gap.SmallGroup(192,1472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,135,1124,4037,285,2358,475]);
// Polycyclic

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

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