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## G = S3×D13order 156 = 22·3·13

### Direct product of S3 and D13

Aliases: S3×D13, D39⋊C2, C31D26, C131D6, C39⋊C22, (S3×C13)⋊C2, (C3×D13)⋊C2, SmallGroup(156,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — S3×D13
 Chief series C1 — C13 — C39 — C3×D13 — S3×D13
 Lower central C39 — S3×D13
 Upper central C1

Generators and relations for S3×D13
G = < a,b,c,d | a3=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

3C2
13C2
39C2
39C22
13C6
13S3
3C26
3D13
13D6
3D26

Character table of S3×D13

 class 1 2A 2B 2C 3 6 13A 13B 13C 13D 13E 13F 26A 26B 26C 26D 26E 26F 39A 39B 39C 39D 39E 39F size 1 3 13 39 2 26 2 2 2 2 2 2 6 6 6 6 6 6 4 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 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 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 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 linear of order 2 ρ5 2 0 -2 0 -1 1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 0 2 0 -1 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 0 0 2 0 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 ζ1312+ζ13 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 orthogonal lifted from D13 ρ8 2 -2 0 0 2 0 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1310+ζ133 ζ139+ζ134 ζ1311+ζ132 -ζ139-ζ134 -ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 -ζ137-ζ136 -ζ1312-ζ13 ζ1312+ζ13 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ137+ζ136 orthogonal lifted from D26 ρ9 2 2 0 0 2 0 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 ζ137+ζ136 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 orthogonal lifted from D13 ρ10 2 2 0 0 2 0 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 ζ1310+ζ133 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 orthogonal lifted from D13 ρ11 2 -2 0 0 2 0 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 -ζ137-ζ136 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 -ζ139-ζ134 -ζ138-ζ135 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 orthogonal lifted from D26 ρ12 2 -2 0 0 2 0 ζ138+ζ135 ζ1310+ζ133 ζ1311+ζ132 ζ139+ζ134 ζ1312+ζ13 ζ137+ζ136 -ζ1312-ζ13 -ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 -ζ138-ζ135 -ζ1310-ζ133 ζ1310+ζ133 ζ1312+ζ13 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ138+ζ135 orthogonal lifted from D26 ρ13 2 2 0 0 2 0 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 ζ1311+ζ132 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 orthogonal lifted from D13 ρ14 2 -2 0 0 2 0 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 -ζ1310-ζ133 -ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 -ζ1311-ζ132 -ζ139-ζ134 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 orthogonal lifted from D26 ρ15 2 2 0 0 2 0 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1311+ζ132 ζ137+ζ136 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 ζ138+ζ135 ζ138+ζ135 ζ137+ζ136 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ139+ζ134 orthogonal lifted from D13 ρ16 2 2 0 0 2 0 ζ1311+ζ132 ζ139+ζ134 ζ137+ζ136 ζ1312+ζ13 ζ1310+ζ133 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 ζ139+ζ134 ζ139+ζ134 ζ1310+ζ133 ζ137+ζ136 ζ1312+ζ13 ζ138+ζ135 ζ1311+ζ132 orthogonal lifted from D13 ρ17 2 -2 0 0 2 0 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ138+ζ135 ζ1311+ζ132 ζ1312+ζ13 -ζ1311-ζ132 -ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 -ζ1310-ζ133 -ζ137-ζ136 ζ137+ζ136 ζ1311+ζ132 ζ139+ζ134 ζ138+ζ135 ζ1312+ζ13 ζ1310+ζ133 orthogonal lifted from D26 ρ18 2 -2 0 0 2 0 ζ1312+ζ13 ζ1311+ζ132 ζ1310+ζ133 ζ137+ζ136 ζ138+ζ135 ζ139+ζ134 -ζ138-ζ135 -ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 -ζ1312-ζ13 -ζ1311-ζ132 ζ1311+ζ132 ζ138+ζ135 ζ1310+ζ133 ζ137+ζ136 ζ139+ζ134 ζ1312+ζ13 orthogonal lifted from D26 ρ19 4 0 0 0 -2 0 2ζ137+2ζ136 2ζ1312+2ζ13 2ζ138+2ζ135 2ζ1310+2ζ133 2ζ139+2ζ134 2ζ1311+2ζ132 0 0 0 0 0 0 -ζ1312-ζ13 -ζ139-ζ134 -ζ138-ζ135 -ζ1310-ζ133 -ζ1311-ζ132 -ζ137-ζ136 orthogonal faithful ρ20 4 0 0 0 -2 0 2ζ1311+2ζ132 2ζ139+2ζ134 2ζ137+2ζ136 2ζ1312+2ζ13 2ζ1310+2ζ133 2ζ138+2ζ135 0 0 0 0 0 0 -ζ139-ζ134 -ζ1310-ζ133 -ζ137-ζ136 -ζ1312-ζ13 -ζ138-ζ135 -ζ1311-ζ132 orthogonal faithful ρ21 4 0 0 0 -2 0 2ζ1310+2ζ133 2ζ137+2ζ136 2ζ139+2ζ134 2ζ138+2ζ135 2ζ1311+2ζ132 2ζ1312+2ζ13 0 0 0 0 0 0 -ζ137-ζ136 -ζ1311-ζ132 -ζ139-ζ134 -ζ138-ζ135 -ζ1312-ζ13 -ζ1310-ζ133 orthogonal faithful ρ22 4 0 0 0 -2 0 2ζ139+2ζ134 2ζ138+2ζ135 2ζ1312+2ζ13 2ζ1311+2ζ132 2ζ137+2ζ136 2ζ1310+2ζ133 0 0 0 0 0 0 -ζ138-ζ135 -ζ137-ζ136 -ζ1312-ζ13 -ζ1311-ζ132 -ζ1310-ζ133 -ζ139-ζ134 orthogonal faithful ρ23 4 0 0 0 -2 0 2ζ138+2ζ135 2ζ1310+2ζ133 2ζ1311+2ζ132 2ζ139+2ζ134 2ζ1312+2ζ13 2ζ137+2ζ136 0 0 0 0 0 0 -ζ1310-ζ133 -ζ1312-ζ13 -ζ1311-ζ132 -ζ139-ζ134 -ζ137-ζ136 -ζ138-ζ135 orthogonal faithful ρ24 4 0 0 0 -2 0 2ζ1312+2ζ13 2ζ1311+2ζ132 2ζ1310+2ζ133 2ζ137+2ζ136 2ζ138+2ζ135 2ζ139+2ζ134 0 0 0 0 0 0 -ζ1311-ζ132 -ζ138-ζ135 -ζ1310-ζ133 -ζ137-ζ136 -ζ139-ζ134 -ζ1312-ζ13 orthogonal faithful

Smallest permutation representation of S3×D13
On 39 points
Generators in S39
(1 21 27)(2 22 28)(3 23 29)(4 24 30)(5 25 31)(6 26 32)(7 14 33)(8 15 34)(9 16 35)(10 17 36)(11 18 37)(12 19 38)(13 20 39)
(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 27)(22 28)(23 29)(24 30)(25 31)(26 32)
(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 34 35 36 37 38 39)
(1 13)(2 12)(3 11)(4 10)(5 9)(6 8)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)

G:=sub<Sym(39)| (1,21,27)(2,22,28)(3,23,29)(4,24,30)(5,25,31)(6,26,32)(7,14,33)(8,15,34)(9,16,35)(10,17,36)(11,18,37)(12,19,38)(13,20,39), (14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)>;

G:=Group( (1,21,27)(2,22,28)(3,23,29)(4,24,30)(5,25,31)(6,26,32)(7,14,33)(8,15,34)(9,16,35)(10,17,36)(11,18,37)(12,19,38)(13,20,39), (14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,27)(22,28)(23,29)(24,30)(25,31)(26,32), (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)(27,28,29,30,31,32,33,34,35,36,37,38,39), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34) );

G=PermutationGroup([[(1,21,27),(2,22,28),(3,23,29),(4,24,30),(5,25,31),(6,26,32),(7,14,33),(8,15,34),(9,16,35),(10,17,36),(11,18,37),(12,19,38),(13,20,39)], [(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,27),(22,28),(23,29),(24,30),(25,31),(26,32)], [(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),(27,28,29,30,31,32,33,34,35,36,37,38,39)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(15,26),(16,25),(17,24),(18,23),(19,22),(20,21),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)]])

S3×D13 is a maximal subgroup of   D39⋊S3
S3×D13 is a maximal quotient of   D78.C2  C39⋊D4  C3⋊D52  C13⋊D12  C39⋊Q8  D39⋊S3

Matrix representation of S3×D13 in GL4(𝔽79) generated by

 1 0 0 0 0 1 0 0 0 0 77 9 0 0 26 1
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 53 78
,
 39 1 0 0 13 51 0 0 0 0 1 0 0 0 0 1
,
 26 38 0 0 55 53 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(79))| [1,0,0,0,0,1,0,0,0,0,77,26,0,0,9,1],[1,0,0,0,0,1,0,0,0,0,1,53,0,0,0,78],[39,13,0,0,1,51,0,0,0,0,1,0,0,0,0,1],[26,55,0,0,38,53,0,0,0,0,1,0,0,0,0,1] >;

S3×D13 in GAP, Magma, Sage, TeX

S_3\times D_{13}
% in TeX

G:=Group("S3xD13");
// GroupNames label

G:=SmallGroup(156,11);
// by ID

G=gap.SmallGroup(156,11);
# by ID

G:=PCGroup([4,-2,-2,-3,-13,54,2307]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^13=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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