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## G = S3×D12order 144 = 24·32

### Direct product of S3 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×D12
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×D12
 Lower central C32 — C3×C6 — S3×D12
 Upper central C1 — C2 — C4

Generators and relations for S3×D12
G = < a,b,c,d | a3=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 476 in 116 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×9], S3 [×2], S3 [×8], C6 [×2], C6 [×5], C2×C4, D4 [×4], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×2], D6 [×14], C2×C6 [×3], C2×D4, C3×S3 [×2], C3×S3 [×2], C3⋊S3 [×2], C3×C6, C4×S3, D12, D12 [×5], C3⋊D4 [×2], C2×C12, C3×D4, C22×S3 [×4], C3×Dic3, C3×C12, S32 [×4], S3×C6, S3×C6 [×2], C2×C3⋊S3 [×2], C2×D12, S3×D4, C3⋊D12 [×2], S3×C12, C3×D12, C12⋊S3, C2×S32 [×2], S3×D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], S32, C2×D12, S3×D4, C2×S32, S3×D12

Character table of S3×D12

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F 12G size 1 1 3 3 6 6 18 18 2 2 4 2 6 2 2 4 6 6 12 12 2 2 4 4 4 6 6 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 2 -2 -2 0 0 0 0 2 -1 -1 -2 2 2 -1 -1 1 1 0 0 1 1 1 -2 1 -1 -1 orthogonal lifted from D6 ρ10 2 -2 -2 2 0 0 0 0 2 2 2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 0 0 0 2 -1 -1 -2 -2 2 -1 -1 -1 -1 0 0 1 1 1 -2 1 1 1 orthogonal lifted from D6 ρ12 2 -2 2 -2 0 0 0 0 2 2 2 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 0 0 2 -2 0 0 -1 2 -1 -2 0 -1 2 -1 0 0 -1 1 -2 -2 1 1 1 0 0 orthogonal lifted from D6 ρ14 2 2 2 2 0 0 0 0 2 -1 -1 2 2 2 -1 -1 -1 -1 0 0 -1 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ15 2 2 0 0 2 2 0 0 -1 2 -1 2 0 -1 2 -1 0 0 -1 -1 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ16 2 2 -2 -2 0 0 0 0 2 -1 -1 2 -2 2 -1 -1 1 1 0 0 -1 -1 -1 2 -1 1 1 orthogonal lifted from D6 ρ17 2 2 0 0 -2 -2 0 0 -1 2 -1 2 0 -1 2 -1 0 0 1 1 2 2 -1 -1 -1 0 0 orthogonal lifted from D6 ρ18 2 2 0 0 -2 2 0 0 -1 2 -1 -2 0 -1 2 -1 0 0 1 -1 -2 -2 1 1 1 0 0 orthogonal lifted from D6 ρ19 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -2 1 1 1 -1 0 0 √3 -√3 √3 0 -√3 -√3 √3 orthogonal lifted from D12 ρ20 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 1 -1 1 0 0 √3 -√3 √3 0 -√3 √3 -√3 orthogonal lifted from D12 ρ21 2 -2 -2 2 0 0 0 0 2 -1 -1 0 0 -2 1 1 1 -1 0 0 -√3 √3 -√3 0 √3 √3 -√3 orthogonal lifted from D12 ρ22 2 -2 2 -2 0 0 0 0 2 -1 -1 0 0 -2 1 1 -1 1 0 0 -√3 √3 -√3 0 √3 -√3 √3 orthogonal lifted from D12 ρ23 4 -4 0 0 0 0 0 0 -2 4 -2 0 0 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 4 0 0 0 0 0 0 -2 -2 1 4 0 -2 -2 1 0 0 0 0 -2 -2 1 -2 1 0 0 orthogonal lifted from S32 ρ25 4 4 0 0 0 0 0 0 -2 -2 1 -4 0 -2 -2 1 0 0 0 0 2 2 -1 2 -1 0 0 orthogonal lifted from C2×S32 ρ26 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 2√3 -2√3 -√3 0 √3 0 0 orthogonal faithful ρ27 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2 2 -1 0 0 0 0 -2√3 2√3 √3 0 -√3 0 0 orthogonal faithful

Permutation representations of S3×D12
On 24 points - transitive group 24T229
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)

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

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

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

G:=TransitiveGroup(24,229);

Matrix representation of S3×D12 in GL6(ℤ)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0
,
 1 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 -1 1

G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,-1,0,0,0,0,2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;

S3×D12 in GAP, Magma, Sage, TeX

S_3\times D_{12}
% in TeX

G:=Group("S3xD12");
// GroupNames label

G:=SmallGroup(144,144);
// by ID

G=gap.SmallGroup(144,144);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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