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## G = C13×D7order 182 = 2·7·13

### Direct product of C13 and D7

Aliases: C13×D7, C7⋊C26, C913C2, SmallGroup(182,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C13×D7
 Chief series C1 — C7 — C91 — C13×D7
 Lower central C7 — C13×D7
 Upper central C1 — C13

Generators and relations for C13×D7
G = < a,b,c | a13=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C13×D7
On 91 points
Generators in S91
(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)(53 54 55 56 57 58 59 60 61 62 63 64 65)(66 67 68 69 70 71 72 73 74 75 76 77 78)(79 80 81 82 83 84 85 86 87 88 89 90 91)
(1 54 89 35 19 66 44)(2 55 90 36 20 67 45)(3 56 91 37 21 68 46)(4 57 79 38 22 69 47)(5 58 80 39 23 70 48)(6 59 81 27 24 71 49)(7 60 82 28 25 72 50)(8 61 83 29 26 73 51)(9 62 84 30 14 74 52)(10 63 85 31 15 75 40)(11 64 86 32 16 76 41)(12 65 87 33 17 77 42)(13 53 88 34 18 78 43)
(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 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 79)(23 80)(24 81)(25 82)(26 83)(53 78)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)

G:=sub<Sym(91)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,54,89,35,19,66,44)(2,55,90,36,20,67,45)(3,56,91,37,21,68,46)(4,57,79,38,22,69,47)(5,58,80,39,23,70,48)(6,59,81,27,24,71,49)(7,60,82,28,25,72,50)(8,61,83,29,26,73,51)(9,62,84,30,14,74,52)(10,63,85,31,15,75,40)(11,64,86,32,16,76,41)(12,65,87,33,17,77,42)(13,53,88,34,18,78,43), (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)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,79)(23,80)(24,81)(25,82)(26,83)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65)(66,67,68,69,70,71,72,73,74,75,76,77,78)(79,80,81,82,83,84,85,86,87,88,89,90,91), (1,54,89,35,19,66,44)(2,55,90,36,20,67,45)(3,56,91,37,21,68,46)(4,57,79,38,22,69,47)(5,58,80,39,23,70,48)(6,59,81,27,24,71,49)(7,60,82,28,25,72,50)(8,61,83,29,26,73,51)(9,62,84,30,14,74,52)(10,63,85,31,15,75,40)(11,64,86,32,16,76,41)(12,65,87,33,17,77,42)(13,53,88,34,18,78,43), (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)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,79)(23,80)(24,81)(25,82)(26,83)(53,78)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72)(61,73)(62,74)(63,75)(64,76)(65,77) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65),(66,67,68,69,70,71,72,73,74,75,76,77,78),(79,80,81,82,83,84,85,86,87,88,89,90,91)], [(1,54,89,35,19,66,44),(2,55,90,36,20,67,45),(3,56,91,37,21,68,46),(4,57,79,38,22,69,47),(5,58,80,39,23,70,48),(6,59,81,27,24,71,49),(7,60,82,28,25,72,50),(8,61,83,29,26,73,51),(9,62,84,30,14,74,52),(10,63,85,31,15,75,40),(11,64,86,32,16,76,41),(12,65,87,33,17,77,42),(13,53,88,34,18,78,43)], [(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),(14,84),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,79),(23,80),(24,81),(25,82),(26,83),(53,78),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72),(61,73),(62,74),(63,75),(64,76),(65,77)])

65 conjugacy classes

 class 1 2 7A 7B 7C 13A ··· 13L 26A ··· 26L 91A ··· 91AJ order 1 2 7 7 7 13 ··· 13 26 ··· 26 91 ··· 91 size 1 7 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2

65 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C13 C26 D7 C13×D7 kernel C13×D7 C91 D7 C7 C13 C1 # reps 1 1 12 12 3 36

Matrix representation of C13×D7 in GL2(𝔽547) generated by

 509 0 0 509
,
 0 1 546 179
,
 0 1 1 0
G:=sub<GL(2,GF(547))| [509,0,0,509],[0,546,1,179],[0,1,1,0] >;

C13×D7 in GAP, Magma, Sage, TeX

C_{13}\times D_7
% in TeX

G:=Group("C13xD7");
// GroupNames label

G:=SmallGroup(182,1);
// by ID

G=gap.SmallGroup(182,1);
# by ID

G:=PCGroup([3,-2,-13,-7,1406]);
// Polycyclic

G:=Group<a,b,c|a^13=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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