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G = C13xD8order 208 = 24·13

Direct product of C13 and D8

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

Aliases: C13xD8, D4:C26, C8:1C26, C104:5C2, C26.14D4, C52.17C22, (D4xC13):4C2, C4.1(C2xC26), C2.3(D4xC13), SmallGroup(208,25)

Series: Derived Chief Lower central Upper central

C1C4 — C13xD8
C1C2C4C52D4xC13 — C13xD8
C1C2C4 — C13xD8
C1C26C52 — C13xD8

Generators and relations for C13xD8
 G = < a,b,c | a13=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 38 in 22 conjugacy classes, 14 normal (10 characteristic)
Quotients: C1, C2, C22, D4, C13, D8, C26, C2xC26, D4xC13, C13xD8
4C2
4C2
2C22
2C22
4C26
4C26
2C2xC26
2C2xC26

Smallest permutation representation of C13xD8
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 23 101 56 32 80 49 66)(2 24 102 57 33 81 50 67)(3 25 103 58 34 82 51 68)(4 26 104 59 35 83 52 69)(5 14 92 60 36 84 40 70)(6 15 93 61 37 85 41 71)(7 16 94 62 38 86 42 72)(8 17 95 63 39 87 43 73)(9 18 96 64 27 88 44 74)(10 19 97 65 28 89 45 75)(11 20 98 53 29 90 46 76)(12 21 99 54 30 91 47 77)(13 22 100 55 31 79 48 78)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 64)(28 65)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(79 100)(80 101)(81 102)(82 103)(83 104)(84 92)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 99)

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

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

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

C13xD8 is a maximal subgroup of   C13:D16  D8.D13  D8:D13  D8:3D13

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+++++
imageC1C2C2C13C26C26D4D8D4xC13C13xD8
kernelC13xD8C104D4xC13D8C8D4C26C13C2C1
# reps112121224121224

Matrix representation of C13xD8 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] >;

C13xD8 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 C13xD8 in TeX

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