direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C13×D8, D4⋊C26, C8⋊1C26, C104⋊5C2, 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
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 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)]])
C13×D8 is a maximal subgroup of
C13⋊D16 D8.D13 D8⋊D13 D8⋊3D13
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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