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## G = A4×C13order 156 = 22·3·13

### Direct product of C13 and A4

Aliases: A4×C13, C22⋊C39, (C2×C26)⋊1C3, SmallGroup(156,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C13
 Chief series C1 — C22 — C2×C26 — A4×C13
 Lower central C22 — A4×C13
 Upper central C1 — C13

Generators and relations for A4×C13
G = < a,b,c,d | a13=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Smallest permutation representation of A4×C13
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 35)(2 36)(3 37)(4 38)(5 39)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 49)(15 50)(16 51)(17 52)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 40)(11 41)(12 42)(13 43)(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)
(14 27 49)(15 28 50)(16 29 51)(17 30 52)(18 31 40)(19 32 41)(20 33 42)(21 34 43)(22 35 44)(23 36 45)(24 37 46)(25 38 47)(26 39 48)

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

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

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

A4×C13 is a maximal subgroup of   C13⋊S4

52 conjugacy classes

 class 1 2 3A 3B 13A ··· 13L 26A ··· 26L 39A ··· 39X order 1 2 3 3 13 ··· 13 26 ··· 26 39 ··· 39 size 1 3 4 4 1 ··· 1 3 ··· 3 4 ··· 4

52 irreducible representations

 dim 1 1 1 1 3 3 type + + image C1 C3 C13 C39 A4 A4×C13 kernel A4×C13 C2×C26 A4 C22 C13 C1 # reps 1 2 12 24 1 12

Matrix representation of A4×C13 in GL3(𝔽79) generated by

 46 0 0 0 46 0 0 0 46
,
 78 0 0 78 0 1 78 1 0
,
 0 1 78 1 0 78 0 0 78
,
 0 1 0 0 0 1 1 0 0
G:=sub<GL(3,GF(79))| [46,0,0,0,46,0,0,0,46],[78,78,78,0,0,1,0,1,0],[0,1,0,1,0,0,78,78,78],[0,0,1,1,0,0,0,1,0] >;

A4×C13 in GAP, Magma, Sage, TeX

A_4\times C_{13}
% in TeX

G:=Group("A4xC13");
// GroupNames label

G:=SmallGroup(156,13);
// by ID

G=gap.SmallGroup(156,13);
# by ID

G:=PCGroup([4,-3,-13,-2,2,938,1875]);
// Polycyclic

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

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