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## G = C13×D8order 208 = 24·13

### Direct product of C13 and D8

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C13×D8, D4⋊C26, C81C26, C1045C2, C26.14D4, C52.17C22, (D4×C13)⋊4C2, C4.1(C2×C26), C2.3(D4×C13), SmallGroup(208,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C13×D8
 Chief series C1 — C2 — C4 — C52 — D4×C13 — C13×D8
 Lower central C1 — C2 — C4 — C13×D8
 Upper central C1 — C26 — C52 — C13×D8

Generators and relations for C13×D8
G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C13×D8
On 104 points
Generators in S104
(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)(92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 38 17 57 66 83 40 93)(2 39 18 58 67 84 41 94)(3 27 19 59 68 85 42 95)(4 28 20 60 69 86 43 96)(5 29 21 61 70 87 44 97)(6 30 22 62 71 88 45 98)(7 31 23 63 72 89 46 99)(8 32 24 64 73 90 47 100)(9 33 25 65 74 91 48 101)(10 34 26 53 75 79 49 102)(11 35 14 54 76 80 50 103)(12 36 15 55 77 81 51 104)(13 37 16 56 78 82 52 92)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 92)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 79)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 40)(39 41)(53 75)(54 76)(55 77)(56 78)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 73)(65 74)

G:=sub<Sym(104)| (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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,38,17,57,66,83,40,93)(2,39,18,58,67,84,41,94)(3,27,19,59,68,85,42,95)(4,28,20,60,69,86,43,96)(5,29,21,61,70,87,44,97)(6,30,22,62,71,88,45,98)(7,31,23,63,72,89,46,99)(8,32,24,64,73,90,47,100)(9,33,25,65,74,91,48,101)(10,34,26,53,75,79,49,102)(11,35,14,54,76,80,50,103)(12,36,15,55,77,81,51,104)(13,37,16,56,78,82,52,92), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,92)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,79)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,40)(39,41)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74)>;

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)(92,93,94,95,96,97,98,99,100,101,102,103,104), (1,38,17,57,66,83,40,93)(2,39,18,58,67,84,41,94)(3,27,19,59,68,85,42,95)(4,28,20,60,69,86,43,96)(5,29,21,61,70,87,44,97)(6,30,22,62,71,88,45,98)(7,31,23,63,72,89,46,99)(8,32,24,64,73,90,47,100)(9,33,25,65,74,91,48,101)(10,34,26,53,75,79,49,102)(11,35,14,54,76,80,50,103)(12,36,15,55,77,81,51,104)(13,37,16,56,78,82,52,92), (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,92)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,79)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,40)(39,41)(53,75)(54,76)(55,77)(56,78)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,73)(65,74) );

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),(92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,38,17,57,66,83,40,93),(2,39,18,58,67,84,41,94),(3,27,19,59,68,85,42,95),(4,28,20,60,69,86,43,96),(5,29,21,61,70,87,44,97),(6,30,22,62,71,88,45,98),(7,31,23,63,72,89,46,99),(8,32,24,64,73,90,47,100),(9,33,25,65,74,91,48,101),(10,34,26,53,75,79,49,102),(11,35,14,54,76,80,50,103),(12,36,15,55,77,81,51,104),(13,37,16,56,78,82,52,92)], [(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,92),(14,80),(15,81),(16,82),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,91),(26,79),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,40),(39,41),(53,75),(54,76),(55,77),(56,78),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,73),(65,74)])

C13×D8 is a maximal subgroup of   C13⋊D16  D8.D13  D8⋊D13  D83D13

91 conjugacy classes

 class 1 2A 2B 2C 4 8A 8B 13A ··· 13L 26A ··· 26L 26M ··· 26AJ 52A ··· 52L 104A ··· 104X order 1 2 2 2 4 8 8 13 ··· 13 26 ··· 26 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 4 4 2 2 2 1 ··· 1 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2

91 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C13 C26 C26 D4 D8 D4×C13 C13×D8 kernel C13×D8 C104 D4×C13 D8 C8 D4 C26 C13 C2 C1 # reps 1 1 2 12 12 24 1 2 12 24

Matrix representation of C13×D8 in GL2(𝔽313) generated by

 48 0 0 48
,
 193 193 60 0
,
 193 193 60 120
G:=sub<GL(2,GF(313))| [48,0,0,48],[193,60,193,0],[193,60,193,120] >;

C13×D8 in GAP, Magma, Sage, TeX

C_{13}\times D_8
% in TeX

G:=Group("C13xD8");
// GroupNames label

G:=SmallGroup(208,25);
// by ID

G=gap.SmallGroup(208,25);
# by ID

G:=PCGroup([5,-2,-2,-13,-2,-2,541,3123,1568,58]);
// Polycyclic

G:=Group<a,b,c|a^13=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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