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## G = S3×C13⋊C4order 312 = 23·3·13

### Direct product of S3 and C13⋊C4

Aliases: S3×C13⋊C4, D39⋊C4, D13.1D6, C13⋊(C4×S3), C39⋊(C2×C4), C39⋊C4⋊C2, (S3×C13)⋊C4, (S3×D13).C2, (C3×D13).C22, (C3×C13⋊C4)⋊C2, C31(C2×C13⋊C4), SmallGroup(312,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — S3×C13⋊C4
 Chief series C1 — C13 — C39 — C3×D13 — C3×C13⋊C4 — S3×C13⋊C4
 Lower central C39 — S3×C13⋊C4
 Upper central C1

Generators and relations for S3×C13⋊C4
G = < a,b,c,d | a3=b2=c13=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Character table of S3×C13⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 12A 12B 13A 13B 13C 26A 26B 26C 39A 39B 39C size 1 3 13 39 2 13 13 39 39 26 26 26 4 4 4 12 12 12 8 8 8 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 1 -1 -1 1 1 -i i i -i -1 -i i 1 1 1 -1 -1 -1 1 1 1 linear of order 4 ρ6 1 -1 -1 1 1 i -i -i i -1 i -i 1 1 1 -1 -1 -1 1 1 1 linear of order 4 ρ7 1 1 -1 -1 1 -i i -i i -1 -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 -1 -1 1 i -i i -i -1 i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 0 2 0 -1 2 2 0 0 -1 -1 -1 2 2 2 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ10 2 0 2 0 -1 -2 -2 0 0 -1 1 1 2 2 2 0 0 0 -1 -1 -1 orthogonal lifted from D6 ρ11 2 0 -2 0 -1 2i -2i 0 0 1 -i i 2 2 2 0 0 0 -1 -1 -1 complex lifted from C4×S3 ρ12 2 0 -2 0 -1 -2i 2i 0 0 1 i -i 2 2 2 0 0 0 -1 -1 -1 complex lifted from C4×S3 ρ13 4 4 0 0 4 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C13⋊C4 ρ14 4 -4 0 0 4 0 0 0 0 0 0 0 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 orthogonal lifted from C2×C13⋊C4 ρ15 4 -4 0 0 4 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C2×C13⋊C4 ρ16 4 4 0 0 4 0 0 0 0 0 0 0 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 orthogonal lifted from C13⋊C4 ρ17 4 4 0 0 4 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 ζ139+ζ137+ζ136+ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C13⋊C4 ρ18 4 -4 0 0 4 0 0 0 0 0 0 0 ζ139+ζ137+ζ136+ζ134 ζ1311+ζ1310+ζ133+ζ132 ζ1312+ζ138+ζ135+ζ13 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 ζ139+ζ137+ζ136+ζ134 ζ1312+ζ138+ζ135+ζ13 ζ1311+ζ1310+ζ133+ζ132 orthogonal lifted from C2×C13⋊C4 ρ19 8 0 0 0 -4 0 0 0 0 0 0 0 2ζ1311+2ζ1310+2ζ133+2ζ132 2ζ1312+2ζ138+2ζ135+2ζ13 2ζ139+2ζ137+2ζ136+2ζ134 0 0 0 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 orthogonal faithful ρ20 8 0 0 0 -4 0 0 0 0 0 0 0 2ζ139+2ζ137+2ζ136+2ζ134 2ζ1311+2ζ1310+2ζ133+2ζ132 2ζ1312+2ζ138+2ζ135+2ζ13 0 0 0 -ζ139-ζ137-ζ136-ζ134 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 orthogonal faithful ρ21 8 0 0 0 -4 0 0 0 0 0 0 0 2ζ1312+2ζ138+2ζ135+2ζ13 2ζ139+2ζ137+2ζ136+2ζ134 2ζ1311+2ζ1310+2ζ133+2ζ132 0 0 0 -ζ1312-ζ138-ζ135-ζ13 -ζ1311-ζ1310-ζ133-ζ132 -ζ139-ζ137-ζ136-ζ134 orthogonal faithful

Smallest permutation representation of S3×C13⋊C4
On 39 points
Generators in S39
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)
(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)
(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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)

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

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

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

Matrix representation of S3×C13⋊C4 in GL6(𝔽157)

 1 19 0 0 0 0 99 155 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 0 1
,
 156 138 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 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 33 135 33 156 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 129 0 0 0 0 0 0 129 0 0 0 0 0 0 1 0 0 0 0 0 124 146 92 125 0 0 135 65 10 32 0 0 0 0 1 0

G:=sub<GL(6,GF(157))| [1,99,0,0,0,0,19,155,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[156,0,0,0,0,0,138,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,1,0,0,0,0,135,0,1,0,0,0,33,0,0,1,0,0,156,0,0,0],[129,0,0,0,0,0,0,129,0,0,0,0,0,0,1,124,135,0,0,0,0,146,65,0,0,0,0,92,10,1,0,0,0,125,32,0] >;

S3×C13⋊C4 in GAP, Magma, Sage, TeX

S_3\times C_{13}\rtimes C_4
% in TeX

G:=Group("S3xC13:C4");
// GroupNames label

G:=SmallGroup(312,46);
// by ID

G=gap.SmallGroup(312,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-13,20,168,4804,1814]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^13=d^4=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^-1=c^5>;
// generators/relations

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