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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

C1C4 — C13×D8
C1C2C4C52D4×C13 — C13×D8
C1C2C4 — C13×D8
C1C26C52 — C13×D8

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

4C2
4C2
2C22
2C22
4C26
4C26
2C2×C26
2C2×C26

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 2A2B2C 4 8A8B13A···13L26A···26L26M···26AJ52A···52L104A···104X
order122248813···1326···2626···2652···52104···104
size11442221···11···14···42···22···2

91 irreducible representations

dim1111112222
type+++++
imageC1C2C2C13C26C26D4D8D4×C13C13×D8
kernelC13×D8C104D4×C13D8C8D4C26C13C2C1
# reps112121224121224

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

480
048
,
193193
600
,
193193
60120
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

Export

Subgroup lattice of C13×D8 in TeX

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