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## G = S3×S3≀C2order 432 = 24·33

### Direct product of S3 and S3≀C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C3⋊S3 — S3×S3≀C2
 Chief series C1 — C3 — C33 — C3×C3⋊S3 — S3×C3⋊S3 — S33 — S3×S3≀C2
 Lower central C33 — C3×C3⋊S3 — S3×S3≀C2
 Upper central C1

Generators and relations for S3×S3≀C2
G = < a,b,c,d,e,f | a3=b2=c3=d3=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=d, fcf=ede-1=c-1, df=fd, fef=e-1 >

Subgroups: 1720 in 192 conjugacy classes, 28 normal (18 characteristic)
C1, C2 [×7], C3, C3 [×4], C4 [×2], C22 [×9], S3, S3 [×15], C6 [×11], C2×C4, D4 [×4], C23 [×2], C32, C32 [×4], Dic3, C12, D6 [×21], C2×C6 [×4], C2×D4, C3×S3 [×16], C3⋊S3, C3⋊S3 [×7], C3×C6 [×3], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×4], C33, C32⋊C4, C32⋊C4, S32 [×2], S32 [×14], S3×C6 [×6], C2×C3⋊S3 [×3], S3×D4, S3×C32, S3×C32 [×2], C3×C3⋊S3, C3×C3⋊S3 [×2], C33⋊C2, S3≀C2, S3≀C2 [×3], C2×C32⋊C4, C2×S32 [×4], C3×C32⋊C4, C33⋊C4, C3×S32 [×2], C3×S32 [×2], S3×C3⋊S3, S3×C3⋊S3 [×2], C324D6 [×2], C2×S3≀C2, S3×C32⋊C4, C3×S3≀C2, C33⋊D4 [×2], C322D12, S33 [×2], S3×S3≀C2
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C22×S3, S3×D4, S3≀C2, C2×S3≀C2, S3×S3≀C2

Character table of S3×S3≀C2

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 12 size 1 3 6 6 9 18 18 27 2 4 4 8 8 18 54 12 12 12 12 12 12 18 24 24 36 36 36 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 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 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 -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 -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 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 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 -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 -1 1 linear of order 2 ρ9 2 0 -2 -2 2 0 0 0 -1 2 2 -1 -1 2 0 -2 1 -2 0 0 1 -1 1 1 0 0 -1 orthogonal lifted from D6 ρ10 2 -2 0 0 -2 0 0 2 2 2 2 2 2 0 0 0 0 0 -2 -2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 0 2 -2 2 0 0 0 -1 2 2 -1 -1 -2 0 -2 -1 2 0 0 1 -1 -1 1 0 0 1 orthogonal lifted from D6 ρ12 2 0 2 2 2 0 0 0 -1 2 2 -1 -1 2 0 2 -1 2 0 0 -1 -1 -1 -1 0 0 -1 orthogonal lifted from S3 ρ13 2 2 0 0 -2 0 0 -2 2 2 2 2 2 0 0 0 0 0 2 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 0 -2 2 2 0 0 0 -1 2 2 -1 -1 -2 0 2 1 -2 0 0 -1 -1 1 -1 0 0 1 orthogonal lifted from D6 ρ15 4 -4 2 0 0 0 -2 0 4 -2 1 -2 1 0 0 0 2 -1 -1 2 0 0 -1 0 0 1 0 orthogonal lifted from C2×S3≀C2 ρ16 4 4 -2 0 0 0 -2 0 4 -2 1 -2 1 0 0 0 -2 1 1 -2 0 0 1 0 0 1 0 orthogonal lifted from S3≀C2 ρ17 4 -4 0 2 0 -2 0 0 4 1 -2 1 -2 0 0 -1 0 0 2 -1 2 0 0 -1 1 0 0 orthogonal lifted from C2×S3≀C2 ρ18 4 4 0 -2 0 -2 0 0 4 1 -2 1 -2 0 0 1 0 0 -2 1 -2 0 0 1 1 0 0 orthogonal lifted from S3≀C2 ρ19 4 4 2 0 0 0 2 0 4 -2 1 -2 1 0 0 0 2 -1 1 -2 0 0 -1 0 0 -1 0 orthogonal lifted from S3≀C2 ρ20 4 0 0 0 -4 0 0 0 -2 4 4 -2 -2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 orthogonal lifted from S3×D4 ρ21 4 4 0 2 0 2 0 0 4 1 -2 1 -2 0 0 -1 0 0 -2 1 2 0 0 -1 -1 0 0 orthogonal lifted from S3≀C2 ρ22 4 -4 0 -2 0 2 0 0 4 1 -2 1 -2 0 0 1 0 0 2 -1 -2 0 0 1 -1 0 0 orthogonal lifted from C2×S3≀C2 ρ23 4 -4 -2 0 0 0 2 0 4 -2 1 -2 1 0 0 0 -2 1 -1 2 0 0 1 0 0 -1 0 orthogonal lifted from C2×S3≀C2 ρ24 8 0 -4 0 0 0 0 0 -4 -4 2 2 -1 0 0 0 2 2 0 0 0 0 -1 0 0 0 0 orthogonal faithful ρ25 8 0 0 -4 0 0 0 0 -4 2 -4 -1 2 0 0 2 0 0 0 0 2 0 0 -1 0 0 0 orthogonal faithful ρ26 8 0 4 0 0 0 0 0 -4 -4 2 2 -1 0 0 0 -2 -2 0 0 0 0 1 0 0 0 0 orthogonal faithful ρ27 8 0 0 4 0 0 0 0 -4 2 -4 -1 2 0 0 -2 0 0 0 0 -2 0 0 1 0 0 0 orthogonal faithful

Permutation representations of S3×S3≀C2
On 12 points - transitive group 12T156
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)
(2 10 5)(4 7 12)
(1 9 8)(3 6 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)

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

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

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

G:=TransitiveGroup(12,156);

On 18 points - transitive group 18T150
Generators in S18
(1 6 4)(2 5 3)(7 15 11)(8 16 12)(9 17 13)(10 18 14)
(1 6)(2 5)(11 15)(12 16)(13 17)(14 18)
(1 18 16)(2 17 15)(3 9 7)(4 10 8)(5 13 11)(6 14 12)
(1 16 18)(2 17 15)(3 9 7)(4 8 10)(5 13 11)(6 12 14)
(1 2)(3 4)(5 6)(7 8 9 10)(11 12 13 14)(15 16 17 18)
(1 2)(3 4)(5 6)(7 10)(8 9)(11 14)(12 13)(15 18)(16 17)

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

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

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

G:=TransitiveGroup(18,150);

On 24 points - transitive group 24T1322
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(1 20 21)(2 22 17)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 12 16)(8 13 9)
(1 21 20)(2 22 17)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 16 12)(8 13 9)
(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 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,1322);

On 24 points - transitive group 24T1323
Generators in S24
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(2 18 13)(4 15 20)(5 23 10)(7 12 21)
(1 17 16)(3 14 19)(6 11 24)(8 22 9)
(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)(21 23)

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

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

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

G:=TransitiveGroup(24,1323);

On 24 points - transitive group 24T1324
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 8)(2 5)(3 6)(4 7)(9 21)(10 22)(11 23)(12 24)(13 20)(14 17)(15 18)(16 19)
(1 20 21)(3 23 18)(6 11 15)(8 13 9)
(2 22 17)(4 19 24)(5 10 14)(7 16 12)
(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 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,1324);

On 27 points - transitive group 27T137
Generators in S27
(1 2 3)(4 15 26)(5 12 27)(6 13 24)(7 14 25)(8 19 21)(9 16 22)(10 17 23)(11 18 20)
(2 3)(8 21)(9 22)(10 23)(11 20)(12 27)(13 24)(14 25)(15 26)
(1 4 6)(2 15 13)(3 26 24)(5 17 18)(7 16 19)(8 25 9)(10 11 27)(12 23 20)(14 22 21)
(1 7 5)(2 14 12)(3 25 27)(4 16 17)(6 19 18)(8 11 24)(9 10 26)(13 21 20)(15 22 23)
(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)
(4 6)(8 9)(10 11)(13 15)(16 19)(17 18)(20 23)(21 22)(24 26)

G:=sub<Sym(27)| (1,2,3)(4,15,26)(5,12,27)(6,13,24)(7,14,25)(8,19,21)(9,16,22)(10,17,23)(11,18,20), (2,3)(8,21)(9,22)(10,23)(11,20)(12,27)(13,24)(14,25)(15,26), (1,4,6)(2,15,13)(3,26,24)(5,17,18)(7,16,19)(8,25,9)(10,11,27)(12,23,20)(14,22,21), (1,7,5)(2,14,12)(3,25,27)(4,16,17)(6,19,18)(8,11,24)(9,10,26)(13,21,20)(15,22,23), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,6)(8,9)(10,11)(13,15)(16,19)(17,18)(20,23)(21,22)(24,26)>;

G:=Group( (1,2,3)(4,15,26)(5,12,27)(6,13,24)(7,14,25)(8,19,21)(9,16,22)(10,17,23)(11,18,20), (2,3)(8,21)(9,22)(10,23)(11,20)(12,27)(13,24)(14,25)(15,26), (1,4,6)(2,15,13)(3,26,24)(5,17,18)(7,16,19)(8,25,9)(10,11,27)(12,23,20)(14,22,21), (1,7,5)(2,14,12)(3,25,27)(4,16,17)(6,19,18)(8,11,24)(9,10,26)(13,21,20)(15,22,23), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (4,6)(8,9)(10,11)(13,15)(16,19)(17,18)(20,23)(21,22)(24,26) );

G=PermutationGroup([(1,2,3),(4,15,26),(5,12,27),(6,13,24),(7,14,25),(8,19,21),(9,16,22),(10,17,23),(11,18,20)], [(2,3),(8,21),(9,22),(10,23),(11,20),(12,27),(13,24),(14,25),(15,26)], [(1,4,6),(2,15,13),(3,26,24),(5,17,18),(7,16,19),(8,25,9),(10,11,27),(12,23,20),(14,22,21)], [(1,7,5),(2,14,12),(3,25,27),(4,16,17),(6,19,18),(8,11,24),(9,10,26),(13,21,20),(15,22,23)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(4,6),(8,9),(10,11),(13,15),(16,19),(17,18),(20,23),(21,22),(24,26)])

G:=TransitiveGroup(27,137);

Polynomial with Galois group S3×S3≀C2 over ℚ
actionf(x)Disc(f)
12T156x12-3x10-2x9-45x8-60x7-101x6-162x5+621x4+1920x3+1944x2+864x+144-244·334·56·113

Matrix representation of S3×S3≀C2 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0],[-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

S3×S3≀C2 in GAP, Magma, Sage, TeX

S_3\times S_3\wr C_2
% in TeX

G:=Group("S3xS3wrC2");
// GroupNames label

G:=SmallGroup(432,741);
// by ID

G=gap.SmallGroup(432,741);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,135,851,298,165,348,530,14118]);
// Polycyclic

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

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