direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C17×D9, C9⋊C34, C153⋊3C2, C51.2S3, C3.(S3×C17), SmallGroup(306,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C17×D9 |
Generators and relations for C17×D9
G = < a,b,c | a17=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C17×D9
►On 153 points
Generators in S153
(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)(137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153)
(1 63 39 141 112 28 72 132 98)(2 64 40 142 113 29 73 133 99)(3 65 41 143 114 30 74 134 100)(4 66 42 144 115 31 75 135 101)(5 67 43 145 116 32 76 136 102)(6 68 44 146 117 33 77 120 86)(7 52 45 147 118 34 78 121 87)(8 53 46 148 119 18 79 122 88)(9 54 47 149 103 19 80 123 89)(10 55 48 150 104 20 81 124 90)(11 56 49 151 105 21 82 125 91)(12 57 50 152 106 22 83 126 92)(13 58 51 153 107 23 84 127 93)(14 59 35 137 108 24 85 128 94)(15 60 36 138 109 25 69 129 95)(16 61 37 139 110 26 70 130 96)(17 62 38 140 111 27 71 131 97)
(1 98)(2 99)(3 100)(4 101)(5 102)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 148)(19 149)(20 150)(21 151)(22 152)(23 153)(24 137)(25 138)(26 139)(27 140)(28 141)(29 142)(30 143)(31 144)(32 145)(33 146)(34 147)(35 85)(36 69)(37 70)(38 71)(39 72)(40 73)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 121)(53 122)(54 123)(55 124)(56 125)(57 126)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 120)
G:=sub<Sym(153)| (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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153), (1,63,39,141,112,28,72,132,98)(2,64,40,142,113,29,73,133,99)(3,65,41,143,114,30,74,134,100)(4,66,42,144,115,31,75,135,101)(5,67,43,145,116,32,76,136,102)(6,68,44,146,117,33,77,120,86)(7,52,45,147,118,34,78,121,87)(8,53,46,148,119,18,79,122,88)(9,54,47,149,103,19,80,123,89)(10,55,48,150,104,20,81,124,90)(11,56,49,151,105,21,82,125,91)(12,57,50,152,106,22,83,126,92)(13,58,51,153,107,23,84,127,93)(14,59,35,137,108,24,85,128,94)(15,60,36,138,109,25,69,129,95)(16,61,37,139,110,26,70,130,96)(17,62,38,140,111,27,71,131,97), (1,98)(2,99)(3,100)(4,101)(5,102)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,85)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,120)>;
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)(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153), (1,63,39,141,112,28,72,132,98)(2,64,40,142,113,29,73,133,99)(3,65,41,143,114,30,74,134,100)(4,66,42,144,115,31,75,135,101)(5,67,43,145,116,32,76,136,102)(6,68,44,146,117,33,77,120,86)(7,52,45,147,118,34,78,121,87)(8,53,46,148,119,18,79,122,88)(9,54,47,149,103,19,80,123,89)(10,55,48,150,104,20,81,124,90)(11,56,49,151,105,21,82,125,91)(12,57,50,152,106,22,83,126,92)(13,58,51,153,107,23,84,127,93)(14,59,35,137,108,24,85,128,94)(15,60,36,138,109,25,69,129,95)(16,61,37,139,110,26,70,130,96)(17,62,38,140,111,27,71,131,97), (1,98)(2,99)(3,100)(4,101)(5,102)(6,86)(7,87)(8,88)(9,89)(10,90)(11,91)(12,92)(13,93)(14,94)(15,95)(16,96)(17,97)(18,148)(19,149)(20,150)(21,151)(22,152)(23,153)(24,137)(25,138)(26,139)(27,140)(28,141)(29,142)(30,143)(31,144)(32,145)(33,146)(34,147)(35,85)(36,69)(37,70)(38,71)(39,72)(40,73)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,121)(53,122)(54,123)(55,124)(56,125)(57,126)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,120) );
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),(137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153)], [(1,63,39,141,112,28,72,132,98),(2,64,40,142,113,29,73,133,99),(3,65,41,143,114,30,74,134,100),(4,66,42,144,115,31,75,135,101),(5,67,43,145,116,32,76,136,102),(6,68,44,146,117,33,77,120,86),(7,52,45,147,118,34,78,121,87),(8,53,46,148,119,18,79,122,88),(9,54,47,149,103,19,80,123,89),(10,55,48,150,104,20,81,124,90),(11,56,49,151,105,21,82,125,91),(12,57,50,152,106,22,83,126,92),(13,58,51,153,107,23,84,127,93),(14,59,35,137,108,24,85,128,94),(15,60,36,138,109,25,69,129,95),(16,61,37,139,110,26,70,130,96),(17,62,38,140,111,27,71,131,97)], [(1,98),(2,99),(3,100),(4,101),(5,102),(6,86),(7,87),(8,88),(9,89),(10,90),(11,91),(12,92),(13,93),(14,94),(15,95),(16,96),(17,97),(18,148),(19,149),(20,150),(21,151),(22,152),(23,153),(24,137),(25,138),(26,139),(27,140),(28,141),(29,142),(30,143),(31,144),(32,145),(33,146),(34,147),(35,85),(36,69),(37,70),(38,71),(39,72),(40,73),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,121),(53,122),(54,123),(55,124),(56,125),(57,126),(58,127),(59,128),(60,129),(61,130),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,120)]])
102 conjugacy classes
| class | 1 | 2 | 3 | 9A | 9B | 9C | 17A | ··· | 17P | 34A | ··· | 34P | 51A | ··· | 51P | 153A | ··· | 153AV |
| order | 1 | 2 | 3 | 9 | 9 | 9 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 153 | ··· | 153 |
| size | 1 | 9 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C17 | C34 | S3 | D9 | S3×C17 | C17×D9 |
| kernel | C17×D9 | C153 | D9 | C9 | C51 | C17 | C3 | C1 |
| # reps | 1 | 1 | 16 | 16 | 1 | 3 | 16 | 48 |
Matrix representation of C17×D9 ►in GL2(𝔽307) generated by
| 273 | 0 |
| 0 | 273 |
| 91 | 156 |
| 151 | 242 |
| 151 | 242 |
| 91 | 156 |
G:=sub<GL(2,GF(307))| [273,0,0,273],[91,151,156,242],[151,91,242,156] >;
C17×D9 in GAP, Magma, Sage, TeX
C_{17}\times D_9 % in TeX
G:=Group("C17xD9"); // GroupNames label
G:=SmallGroup(306,1);
// by ID
G=gap.SmallGroup(306,1);
# by ID
G:=PCGroup([4,-2,-17,-3,-3,2042,82,3267]);
// Polycyclic
G:=Group<a,b,c|a^17=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export