direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×D11, C11⋊C26, C143⋊3C2, SmallGroup(286,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C13×D11 |
Generators and relations for C13×D11
G = < a,b,c | a13=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D11
►On 143 points
Generators in S143
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143)
(1 119 142 100 19 109 37 91 65 49 70)(2 120 143 101 20 110 38 79 53 50 71)(3 121 131 102 21 111 39 80 54 51 72)(4 122 132 103 22 112 27 81 55 52 73)(5 123 133 104 23 113 28 82 56 40 74)(6 124 134 92 24 114 29 83 57 41 75)(7 125 135 93 25 115 30 84 58 42 76)(8 126 136 94 26 116 31 85 59 43 77)(9 127 137 95 14 117 32 86 60 44 78)(10 128 138 96 15 105 33 87 61 45 66)(11 129 139 97 16 106 34 88 62 46 67)(12 130 140 98 17 107 35 89 63 47 68)(13 118 141 99 18 108 36 90 64 48 69)
(1 70)(2 71)(3 72)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 66)(11 67)(12 68)(13 69)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 27)(23 28)(24 29)(25 30)(26 31)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 118)(49 119)(50 120)(51 121)(52 122)(53 143)(54 131)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(79 101)(80 102)(81 103)(82 104)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(91 100)
G:=sub<Sym(143)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,119,142,100,19,109,37,91,65,49,70)(2,120,143,101,20,110,38,79,53,50,71)(3,121,131,102,21,111,39,80,54,51,72)(4,122,132,103,22,112,27,81,55,52,73)(5,123,133,104,23,113,28,82,56,40,74)(6,124,134,92,24,114,29,83,57,41,75)(7,125,135,93,25,115,30,84,58,42,76)(8,126,136,94,26,116,31,85,59,43,77)(9,127,137,95,14,117,32,86,60,44,78)(10,128,138,96,15,105,33,87,61,45,66)(11,129,139,97,16,106,34,88,62,46,67)(12,130,140,98,17,107,35,89,63,47,68)(13,118,141,99,18,108,36,90,64,48,69), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,66)(11,67)(12,68)(13,69)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,27)(23,28)(24,29)(25,30)(26,31)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,118)(49,119)(50,120)(51,121)(52,122)(53,143)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(79,101)(80,102)(81,103)(82,104)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100)>;
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)(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)(131,132,133,134,135,136,137,138,139,140,141,142,143), (1,119,142,100,19,109,37,91,65,49,70)(2,120,143,101,20,110,38,79,53,50,71)(3,121,131,102,21,111,39,80,54,51,72)(4,122,132,103,22,112,27,81,55,52,73)(5,123,133,104,23,113,28,82,56,40,74)(6,124,134,92,24,114,29,83,57,41,75)(7,125,135,93,25,115,30,84,58,42,76)(8,126,136,94,26,116,31,85,59,43,77)(9,127,137,95,14,117,32,86,60,44,78)(10,128,138,96,15,105,33,87,61,45,66)(11,129,139,97,16,106,34,88,62,46,67)(12,130,140,98,17,107,35,89,63,47,68)(13,118,141,99,18,108,36,90,64,48,69), (1,70)(2,71)(3,72)(4,73)(5,74)(6,75)(7,76)(8,77)(9,78)(10,66)(11,67)(12,68)(13,69)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,27)(23,28)(24,29)(25,30)(26,31)(40,123)(41,124)(42,125)(43,126)(44,127)(45,128)(46,129)(47,130)(48,118)(49,119)(50,120)(51,121)(52,122)(53,143)(54,131)(55,132)(56,133)(57,134)(58,135)(59,136)(60,137)(61,138)(62,139)(63,140)(64,141)(65,142)(79,101)(80,102)(81,103)(82,104)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(91,100) );
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),(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),(131,132,133,134,135,136,137,138,139,140,141,142,143)], [(1,119,142,100,19,109,37,91,65,49,70),(2,120,143,101,20,110,38,79,53,50,71),(3,121,131,102,21,111,39,80,54,51,72),(4,122,132,103,22,112,27,81,55,52,73),(5,123,133,104,23,113,28,82,56,40,74),(6,124,134,92,24,114,29,83,57,41,75),(7,125,135,93,25,115,30,84,58,42,76),(8,126,136,94,26,116,31,85,59,43,77),(9,127,137,95,14,117,32,86,60,44,78),(10,128,138,96,15,105,33,87,61,45,66),(11,129,139,97,16,106,34,88,62,46,67),(12,130,140,98,17,107,35,89,63,47,68),(13,118,141,99,18,108,36,90,64,48,69)], [(1,70),(2,71),(3,72),(4,73),(5,74),(6,75),(7,76),(8,77),(9,78),(10,66),(11,67),(12,68),(13,69),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,27),(23,28),(24,29),(25,30),(26,31),(40,123),(41,124),(42,125),(43,126),(44,127),(45,128),(46,129),(47,130),(48,118),(49,119),(50,120),(51,121),(52,122),(53,143),(54,131),(55,132),(56,133),(57,134),(58,135),(59,136),(60,137),(61,138),(62,139),(63,140),(64,141),(65,142),(79,101),(80,102),(81,103),(82,104),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(91,100)]])
91 conjugacy classes
| class | 1 | 2 | 11A | ··· | 11E | 13A | ··· | 13L | 26A | ··· | 26L | 143A | ··· | 143BH |
| order | 1 | 2 | 11 | ··· | 11 | 13 | ··· | 13 | 26 | ··· | 26 | 143 | ··· | 143 |
| size | 1 | 11 | 2 | ··· | 2 | 1 | ··· | 1 | 11 | ··· | 11 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | D11 | C13×D11 |
| kernel | C13×D11 | C143 | D11 | C11 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 5 | 60 |
Matrix representation of C13×D11 ►in GL2(𝔽859) generated by
| 478 | 0 |
| 0 | 478 |
| 0 | 1 |
| 858 | 748 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(859))| [478,0,0,478],[0,858,1,748],[0,1,1,0] >;
C13×D11 in GAP, Magma, Sage, TeX
C_{13}\times D_{11} % in TeX
G:=Group("C13xD11"); // GroupNames label
G:=SmallGroup(286,1);
// by ID
G=gap.SmallGroup(286,1);
# by ID
G:=PCGroup([3,-2,-13,-11,2342]);
// Polycyclic
G:=Group<a,b,c|a^13=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export