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## G = S3×C13⋊C6order 468 = 22·32·13

### Direct product of S3 and C13⋊C6

Aliases: S3×C13⋊C6, D39⋊C6, C39⋊(C2×C6), (S3×C13)⋊C6, D39⋊C3⋊C2, C131(S3×C6), D13⋊(C3×S3), (S3×D13)⋊C3, (C3×D13)⋊C6, C13⋊C31D6, (C3×C13⋊C6)⋊C2, (S3×C13⋊C3)⋊C2, C31(C2×C13⋊C6), (C3×C13⋊C3)⋊C22, SmallGroup(468,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — S3×C13⋊C6
 Chief series C1 — C13 — C39 — C3×C13⋊C3 — C3×C13⋊C6 — S3×C13⋊C6
 Lower central C39 — S3×C13⋊C6
 Upper central C1

Generators and relations for S3×C13⋊C6
G = < a,b,c,d | a3=b2=c13=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >

Character table of S3×C13⋊C6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 6A 6B 6C 6D 6E 6F 6G 6H 6I 13A 13B 26A 26B 39A 39B size 1 3 13 39 2 13 13 26 26 13 13 26 26 26 39 39 39 39 6 6 18 18 12 12 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 1 1 -1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ65 ζ32 ζ6 ζ3 1 1 1 1 1 1 linear of order 6 ρ6 1 -1 -1 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 ζ6 ζ65 -1 ζ3 ζ6 ζ32 ζ65 1 1 -1 -1 1 1 linear of order 6 ρ7 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ8 1 -1 1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ65 ζ6 ζ6 ζ65 1 1 -1 -1 1 1 linear of order 6 ρ9 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ10 1 -1 -1 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ32 ζ65 ζ3 ζ6 1 1 -1 -1 1 1 linear of order 6 ρ11 1 1 -1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 ζ65 ζ6 -1 ζ6 ζ3 ζ65 ζ32 1 1 1 1 1 1 linear of order 6 ρ12 1 -1 1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ6 ζ65 ζ65 ζ6 1 1 -1 -1 1 1 linear of order 6 ρ13 2 0 2 0 -1 2 2 -1 -1 2 2 -1 -1 -1 0 0 0 0 2 2 0 0 -1 -1 orthogonal lifted from S3 ρ14 2 0 -2 0 -1 2 2 -1 -1 -2 -2 1 1 1 0 0 0 0 2 2 0 0 -1 -1 orthogonal lifted from D6 ρ15 2 0 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 -1+√-3 -1-√-3 ζ65 ζ6 -1 0 0 0 0 2 2 0 0 -1 -1 complex lifted from C3×S3 ρ16 2 0 -2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 1+√-3 1-√-3 ζ32 ζ3 1 0 0 0 0 2 2 0 0 -1 -1 complex lifted from S3×C6 ρ17 2 0 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 -1-√-3 -1+√-3 ζ6 ζ65 -1 0 0 0 0 2 2 0 0 -1 -1 complex lifted from C3×S3 ρ18 2 0 -2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 1-√-3 1+√-3 ζ3 ζ32 1 0 0 0 0 2 2 0 0 -1 -1 complex lifted from S3×C6 ρ19 6 -6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1+√13/2 1-√13/2 -1+√13/2 -1-√13/2 orthogonal lifted from C2×C13⋊C6 ρ20 6 -6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1-√13/2 1+√13/2 -1-√13/2 -1+√13/2 orthogonal lifted from C2×C13⋊C6 ρ21 6 6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 orthogonal lifted from C13⋊C6 ρ22 6 6 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 orthogonal lifted from C13⋊C6 ρ23 12 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√13 -1-√13 0 0 1+√13/2 1-√13/2 orthogonal faithful ρ24 12 0 0 0 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√13 -1+√13 0 0 1-√13/2 1+√13/2 orthogonal faithful

Smallest permutation representation of S3×C13⋊C6
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 5 4 13 10 11)(3 9 7 12 6 8)(15 18 17 26 23 24)(16 22 20 25 19 21)(28 31 30 39 36 37)(29 35 33 38 32 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), (2,5,4,13,10,11)(3,9,7,12,6,8)(15,18,17,26,23,24)(16,22,20,25,19,21)(28,31,30,39,36,37)(29,35,33,38,32,34)>;

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

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

Matrix representation of S3×C13⋊C6 in GL8(𝔽79)

 0 78 0 0 0 0 0 0 1 78 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 78 0 0 0 0 0 0 78 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 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 63 77 64 77 63 78 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
,
 24 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 61 46 77 63 60 62 0 0 16 18 16 1 17 17 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 17 17 1 16 18 16

G:=sub<GL(8,GF(79))| [0,1,0,0,0,0,0,0,78,78,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,63,1,0,0,0,0,0,0,77,0,1,0,0,0,0,0,64,0,0,1,0,0,0,0,77,0,0,0,1,0,0,0,63,0,0,0,0,1,0,0,78,0,0,0,0,0],[24,0,0,0,0,0,0,0,0,24,0,0,0,0,0,0,0,0,1,61,16,0,0,17,0,0,0,46,18,0,1,17,0,0,0,77,16,0,0,1,0,0,0,63,1,0,0,16,0,0,0,60,17,1,0,18,0,0,0,62,17,0,0,16] >;

S3×C13⋊C6 in GAP, Magma, Sage, TeX

S_3\times C_{13}\rtimes C_6
% in TeX

G:=Group("S3xC13:C6");
// GroupNames label

G:=SmallGroup(468,31);
// by ID

G=gap.SmallGroup(468,31);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,248,10804,2039]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^13=d^6=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^10>;
// generators/relations

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