direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D13, C26⋊C10, C130⋊2C2, C65⋊3C22, C13⋊(C2×C10), SmallGroup(260,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C10×D13 |
Generators and relations for C10×D13
G = < a,b,c | a10=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D13
►On 130 points
Generators in S130
(1 102 57 81 51 66 28 123 25 111)(2 103 58 82 52 67 29 124 26 112)(3 104 59 83 40 68 30 125 14 113)(4 92 60 84 41 69 31 126 15 114)(5 93 61 85 42 70 32 127 16 115)(6 94 62 86 43 71 33 128 17 116)(7 95 63 87 44 72 34 129 18 117)(8 96 64 88 45 73 35 130 19 105)(9 97 65 89 46 74 36 118 20 106)(10 98 53 90 47 75 37 119 21 107)(11 99 54 91 48 76 38 120 22 108)(12 100 55 79 49 77 39 121 23 109)(13 101 56 80 50 78 27 122 24 110)
(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)(105 106 107 108 109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126 127 128 129 130)
(1 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 91)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 102)(28 101)(29 100)(30 99)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 104)(39 103)(40 108)(41 107)(42 106)(43 105)(44 117)(45 116)(46 115)(47 114)(48 113)(49 112)(50 111)(51 110)(52 109)(53 126)(54 125)(55 124)(56 123)(57 122)(58 121)(59 120)(60 119)(61 118)(62 130)(63 129)(64 128)(65 127)
G:=sub<Sym(130)| (1,102,57,81,51,66,28,123,25,111)(2,103,58,82,52,67,29,124,26,112)(3,104,59,83,40,68,30,125,14,113)(4,92,60,84,41,69,31,126,15,114)(5,93,61,85,42,70,32,127,16,115)(6,94,62,86,43,71,33,128,17,116)(7,95,63,87,44,72,34,129,18,117)(8,96,64,88,45,73,35,130,19,105)(9,97,65,89,46,74,36,118,20,106)(10,98,53,90,47,75,37,119,21,107)(11,99,54,91,48,76,38,120,22,108)(12,100,55,79,49,77,39,121,23,109)(13,101,56,80,50,78,27,122,24,110), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,104)(39,103)(40,108)(41,107)(42,106)(43,105)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,120)(60,119)(61,118)(62,130)(63,129)(64,128)(65,127)>;
G:=Group( (1,102,57,81,51,66,28,123,25,111)(2,103,58,82,52,67,29,124,26,112)(3,104,59,83,40,68,30,125,14,113)(4,92,60,84,41,69,31,126,15,114)(5,93,61,85,42,70,32,127,16,115)(6,94,62,86,43,71,33,128,17,116)(7,95,63,87,44,72,34,129,18,117)(8,96,64,88,45,73,35,130,19,105)(9,97,65,89,46,74,36,118,20,106)(10,98,53,90,47,75,37,119,21,107)(11,99,54,91,48,76,38,120,22,108)(12,100,55,79,49,77,39,121,23,109)(13,101,56,80,50,78,27,122,24,110), (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)(105,106,107,108,109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126,127,128,129,130), (1,78)(2,77)(3,76)(4,75)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,102)(28,101)(29,100)(30,99)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,104)(39,103)(40,108)(41,107)(42,106)(43,105)(44,117)(45,116)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,126)(54,125)(55,124)(56,123)(57,122)(58,121)(59,120)(60,119)(61,118)(62,130)(63,129)(64,128)(65,127) );
G=PermutationGroup([[(1,102,57,81,51,66,28,123,25,111),(2,103,58,82,52,67,29,124,26,112),(3,104,59,83,40,68,30,125,14,113),(4,92,60,84,41,69,31,126,15,114),(5,93,61,85,42,70,32,127,16,115),(6,94,62,86,43,71,33,128,17,116),(7,95,63,87,44,72,34,129,18,117),(8,96,64,88,45,73,35,130,19,105),(9,97,65,89,46,74,36,118,20,106),(10,98,53,90,47,75,37,119,21,107),(11,99,54,91,48,76,38,120,22,108),(12,100,55,79,49,77,39,121,23,109),(13,101,56,80,50,78,27,122,24,110)], [(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),(105,106,107,108,109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126,127,128,129,130)], [(1,78),(2,77),(3,76),(4,75),(5,74),(6,73),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,91),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,84),(22,83),(23,82),(24,81),(25,80),(26,79),(27,102),(28,101),(29,100),(30,99),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,104),(39,103),(40,108),(41,107),(42,106),(43,105),(44,117),(45,116),(46,115),(47,114),(48,113),(49,112),(50,111),(51,110),(52,109),(53,126),(54,125),(55,124),(56,123),(57,122),(58,121),(59,120),(60,119),(61,118),(62,130),(63,129),(64,128),(65,127)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 13A | ··· | 13F | 26A | ··· | 26F | 65A | ··· | 65X | 130A | ··· | 130X |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 13 | 13 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 13 | ··· | 13 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D13 | D26 | C5×D13 | C10×D13 |
| kernel | C10×D13 | C5×D13 | C130 | D26 | D13 | C26 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 6 | 6 | 24 | 24 |
Matrix representation of C10×D13 ►in GL3(𝔽131) generated by
| 130 | 0 | 0 |
| 0 | 89 | 0 |
| 0 | 0 | 89 |
| 1 | 0 | 0 |
| 0 | 119 | 1 |
| 0 | 92 | 25 |
| 1 | 0 | 0 |
| 0 | 57 | 110 |
| 0 | 111 | 74 |
G:=sub<GL(3,GF(131))| [130,0,0,0,89,0,0,0,89],[1,0,0,0,119,92,0,1,25],[1,0,0,0,57,111,0,110,74] >;
C10×D13 in GAP, Magma, Sage, TeX
C_{10}\times D_{13} % in TeX
G:=Group("C10xD13"); // GroupNames label
G:=SmallGroup(260,12);
// by ID
G=gap.SmallGroup(260,12);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,3843]);
// Polycyclic
G:=Group<a,b,c|a^10=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export