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## G = S3×D11order 132 = 22·3·11

### Direct product of S3 and D11

Aliases: S3×D11, D33⋊C2, C31D22, C111D6, C33⋊C22, (S3×C11)⋊C2, (C3×D11)⋊C2, SmallGroup(132,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — S3×D11
 Chief series C1 — C11 — C33 — C3×D11 — S3×D11
 Lower central C33 — S3×D11
 Upper central C1

Generators and relations for S3×D11
G = < a,b,c,d | a3=b2=c11=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
11C2
33C2
33C22
11C6
11S3
3C22
3D11
11D6
3D22

Character table of S3×D11

 class 1 2A 2B 2C 3 6 11A 11B 11C 11D 11E 22A 22B 22C 22D 22E 33A 33B 33C 33D 33E size 1 3 11 33 2 22 2 2 2 2 2 6 6 6 6 6 4 4 4 4 4 ρ1 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 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 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 linear of order 2 ρ5 2 0 -2 0 -1 1 2 2 2 2 2 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 0 2 0 -1 -1 2 2 2 2 2 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 2 0 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 orthogonal lifted from D22 ρ8 2 2 0 0 2 0 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 orthogonal lifted from D11 ρ9 2 2 0 0 2 0 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 orthogonal lifted from D11 ρ10 2 -2 0 0 2 0 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 orthogonal lifted from D22 ρ11 2 2 0 0 2 0 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 orthogonal lifted from D11 ρ12 2 -2 0 0 2 0 ζ119+ζ112 ζ116+ζ115 ζ1110+ζ11 ζ118+ζ113 ζ117+ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ117+ζ114 ζ119+ζ112 orthogonal lifted from D22 ρ13 2 -2 0 0 2 0 ζ118+ζ113 ζ119+ζ112 ζ117+ζ114 ζ1110+ζ11 ζ116+ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 ζ1110+ζ11 ζ119+ζ112 ζ117+ζ114 ζ116+ζ115 ζ118+ζ113 orthogonal lifted from D22 ρ14 2 2 0 0 2 0 ζ117+ζ114 ζ1110+ζ11 ζ119+ζ112 ζ116+ζ115 ζ118+ζ113 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 ζ116+ζ115 ζ1110+ζ11 ζ119+ζ112 ζ118+ζ113 ζ117+ζ114 orthogonal lifted from D11 ρ15 2 -2 0 0 2 0 ζ116+ζ115 ζ117+ζ114 ζ118+ζ113 ζ119+ζ112 ζ1110+ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ1110+ζ11 ζ116+ζ115 orthogonal lifted from D22 ρ16 2 2 0 0 2 0 ζ1110+ζ11 ζ118+ζ113 ζ116+ζ115 ζ117+ζ114 ζ119+ζ112 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 ζ117+ζ114 ζ118+ζ113 ζ116+ζ115 ζ119+ζ112 ζ1110+ζ11 orthogonal lifted from D11 ρ17 4 0 0 0 -2 0 2ζ119+2ζ112 2ζ116+2ζ115 2ζ1110+2ζ11 2ζ118+2ζ113 2ζ117+2ζ114 0 0 0 0 0 -ζ118-ζ113 -ζ116-ζ115 -ζ1110-ζ11 -ζ117-ζ114 -ζ119-ζ112 orthogonal faithful ρ18 4 0 0 0 -2 0 2ζ116+2ζ115 2ζ117+2ζ114 2ζ118+2ζ113 2ζ119+2ζ112 2ζ1110+2ζ11 0 0 0 0 0 -ζ119-ζ112 -ζ117-ζ114 -ζ118-ζ113 -ζ1110-ζ11 -ζ116-ζ115 orthogonal faithful ρ19 4 0 0 0 -2 0 2ζ118+2ζ113 2ζ119+2ζ112 2ζ117+2ζ114 2ζ1110+2ζ11 2ζ116+2ζ115 0 0 0 0 0 -ζ1110-ζ11 -ζ119-ζ112 -ζ117-ζ114 -ζ116-ζ115 -ζ118-ζ113 orthogonal faithful ρ20 4 0 0 0 -2 0 2ζ1110+2ζ11 2ζ118+2ζ113 2ζ116+2ζ115 2ζ117+2ζ114 2ζ119+2ζ112 0 0 0 0 0 -ζ117-ζ114 -ζ118-ζ113 -ζ116-ζ115 -ζ119-ζ112 -ζ1110-ζ11 orthogonal faithful ρ21 4 0 0 0 -2 0 2ζ117+2ζ114 2ζ1110+2ζ11 2ζ119+2ζ112 2ζ116+2ζ115 2ζ118+2ζ113 0 0 0 0 0 -ζ116-ζ115 -ζ1110-ζ11 -ζ119-ζ112 -ζ118-ζ113 -ζ117-ζ114 orthogonal faithful

Smallest permutation representation of S3×D11
On 33 points
Generators in S33
(1 21 32)(2 22 33)(3 12 23)(4 13 24)(5 14 25)(6 15 26)(7 16 27)(8 17 28)(9 18 29)(10 19 30)(11 20 31)
(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 31)(21 32)(22 33)
(1 2 3 4 5 6 7 8 9 10 11)(12 13 14 15 16 17 18 19 20 21 22)(23 24 25 26 27 28 29 30 31 32 33)
(1 11)(2 10)(3 9)(4 8)(5 7)(12 18)(13 17)(14 16)(19 22)(20 21)(23 29)(24 28)(25 27)(30 33)(31 32)

G:=sub<Sym(33)| (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32)>;

G:=Group( (1,21,32)(2,22,33)(3,12,23)(4,13,24)(5,14,25)(6,15,26)(7,16,27)(8,17,28)(9,18,29)(10,19,30)(11,20,31), (12,23)(13,24)(14,25)(15,26)(16,27)(17,28)(18,29)(19,30)(20,31)(21,32)(22,33), (1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33), (1,11)(2,10)(3,9)(4,8)(5,7)(12,18)(13,17)(14,16)(19,22)(20,21)(23,29)(24,28)(25,27)(30,33)(31,32) );

G=PermutationGroup([[(1,21,32),(2,22,33),(3,12,23),(4,13,24),(5,14,25),(6,15,26),(7,16,27),(8,17,28),(9,18,29),(10,19,30),(11,20,31)], [(12,23),(13,24),(14,25),(15,26),(16,27),(17,28),(18,29),(19,30),(20,31),(21,32),(22,33)], [(1,2,3,4,5,6,7,8,9,10,11),(12,13,14,15,16,17,18,19,20,21,22),(23,24,25,26,27,28,29,30,31,32,33)], [(1,11),(2,10),(3,9),(4,8),(5,7),(12,18),(13,17),(14,16),(19,22),(20,21),(23,29),(24,28),(25,27),(30,33),(31,32)]])

S3×D11 is a maximal subgroup of   D33⋊S3
S3×D11 is a maximal quotient of   D33⋊C4  C33⋊D4  C3⋊D44  C11⋊D12  C33⋊Q8  D33⋊S3

Matrix representation of S3×D11 in GL4(𝔽67) generated by

 1 0 0 0 0 1 0 0 0 0 0 66 0 0 1 66
,
 66 0 0 0 0 66 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 66 24 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,66,66],[66,0,0,0,0,66,0,0,0,0,0,1,0,0,1,0],[0,66,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3×D11 in GAP, Magma, Sage, TeX

S_3\times D_{11}
% in TeX

G:=Group("S3xD11");
// GroupNames label

G:=SmallGroup(132,5);
// by ID

G=gap.SmallGroup(132,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-11,54,1923]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^11=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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