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## G = C13×S4order 312 = 23·3·13

### Direct product of C13 and S4

Aliases: C13×S4, A4⋊C26, (C2×C26)⋊1S3, C22⋊(S3×C13), (A4×C13)⋊3C2, SmallGroup(312,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4 — C13×S4
 Chief series C1 — C22 — A4 — A4×C13 — C13×S4
 Lower central A4 — C13×S4
 Upper central C1 — C13

Generators and relations for C13×S4
G = < a,b,c,d,e | a13=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >

Smallest permutation representation of C13×S4
On 52 points
Generators in S52
(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)(40 41 42 43 44 45 46 47 48 49 50 51 52)
(1 25)(2 26)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(27 50)(28 51)(29 52)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)
(1 39)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 51)(15 52)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)
(14 28 51)(15 29 52)(16 30 40)(17 31 41)(18 32 42)(19 33 43)(20 34 44)(21 35 45)(22 36 46)(23 37 47)(24 38 48)(25 39 49)(26 27 50)
(14 51)(15 52)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)

G:=sub<Sym(52)| (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)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,50)(28,51)(29,52)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49), (1,39)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (14,28,51)(15,29,52)(16,30,40)(17,31,41)(18,32,42)(19,33,43)(20,34,44)(21,35,45)(22,36,46)(23,37,47)(24,38,48)(25,39,49)(26,27,50), (14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)>;

G:=Group( (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)(40,41,42,43,44,45,46,47,48,49,50,51,52), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,50)(28,51)(29,52)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(37,47)(38,48)(39,49), (1,39)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50), (14,28,51)(15,29,52)(16,30,40)(17,31,41)(18,32,42)(19,33,43)(20,34,44)(21,35,45)(22,36,46)(23,37,47)(24,38,48)(25,39,49)(26,27,50), (14,51)(15,52)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50) );

G=PermutationGroup([[(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),(40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,25),(2,26),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(27,50),(28,51),(29,52),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(37,47),(38,48),(39,49)], [(1,39),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,51),(15,52),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50)], [(14,28,51),(15,29,52),(16,30,40),(17,31,41),(18,32,42),(19,33,43),(20,34,44),(21,35,45),(22,36,46),(23,37,47),(24,38,48),(25,39,49),(26,27,50)], [(14,51),(15,52),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50)]])

65 conjugacy classes

 class 1 2A 2B 3 4 13A ··· 13L 26A ··· 26L 26M ··· 26X 39A ··· 39L 52A ··· 52L order 1 2 2 3 4 13 ··· 13 26 ··· 26 26 ··· 26 39 ··· 39 52 ··· 52 size 1 3 6 8 6 1 ··· 1 3 ··· 3 6 ··· 6 8 ··· 8 6 ··· 6

65 irreducible representations

 dim 1 1 1 1 2 2 3 3 type + + + + image C1 C2 C13 C26 S3 S3×C13 S4 C13×S4 kernel C13×S4 A4×C13 S4 A4 C2×C26 C22 C13 C1 # reps 1 1 12 12 1 12 2 24

Matrix representation of C13×S4 in GL3(𝔽157) generated by

 99 0 0 0 99 0 0 0 99
,
 0 0 1 156 156 156 1 0 0
,
 156 156 156 0 0 1 0 1 0
,
 1 0 0 156 156 156 0 1 0
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(157))| [99,0,0,0,99,0,0,0,99],[0,156,1,0,156,0,1,156,0],[156,0,0,156,0,1,156,1,0],[1,156,0,0,156,1,0,156,0],[1,0,0,0,0,1,0,1,0] >;

C13×S4 in GAP, Magma, Sage, TeX

C_{13}\times S_4
% in TeX

G:=Group("C13xS4");
// GroupNames label

G:=SmallGroup(312,47);
// by ID

G=gap.SmallGroup(312,47);
# by ID

G:=PCGroup([5,-2,-13,-3,-2,2,782,3123,133,1954,239]);
// Polycyclic

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

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