direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D7×C29, C7⋊C58, C203⋊3C2, SmallGroup(406,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C29 |
Generators and relations for D7×C29
G = < a,b,c | a29=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C29
►On 203 points
Generators in S203
(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)(146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174)(175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203)
(1 183 50 64 151 93 138)(2 184 51 65 152 94 139)(3 185 52 66 153 95 140)(4 186 53 67 154 96 141)(5 187 54 68 155 97 142)(6 188 55 69 156 98 143)(7 189 56 70 157 99 144)(8 190 57 71 158 100 145)(9 191 58 72 159 101 117)(10 192 30 73 160 102 118)(11 193 31 74 161 103 119)(12 194 32 75 162 104 120)(13 195 33 76 163 105 121)(14 196 34 77 164 106 122)(15 197 35 78 165 107 123)(16 198 36 79 166 108 124)(17 199 37 80 167 109 125)(18 200 38 81 168 110 126)(19 201 39 82 169 111 127)(20 202 40 83 170 112 128)(21 203 41 84 171 113 129)(22 175 42 85 172 114 130)(23 176 43 86 173 115 131)(24 177 44 87 174 116 132)(25 178 45 59 146 88 133)(26 179 46 60 147 89 134)(27 180 47 61 148 90 135)(28 181 48 62 149 91 136)(29 182 49 63 150 92 137)
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 145)(9 117)(10 118)(11 119)(12 120)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 160)(31 161)(32 162)(33 163)(34 164)(35 165)(36 166)(37 167)(38 168)(39 169)(40 170)(41 171)(42 172)(43 173)(44 174)(45 146)(46 147)(47 148)(48 149)(49 150)(50 151)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(88 178)(89 179)(90 180)(91 181)(92 182)(93 183)(94 184)(95 185)(96 186)(97 187)(98 188)(99 189)(100 190)(101 191)(102 192)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 199)(110 200)(111 201)(112 202)(113 203)(114 175)(115 176)(116 177)
G:=sub<Sym(203)| (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)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203), (1,183,50,64,151,93,138)(2,184,51,65,152,94,139)(3,185,52,66,153,95,140)(4,186,53,67,154,96,141)(5,187,54,68,155,97,142)(6,188,55,69,156,98,143)(7,189,56,70,157,99,144)(8,190,57,71,158,100,145)(9,191,58,72,159,101,117)(10,192,30,73,160,102,118)(11,193,31,74,161,103,119)(12,194,32,75,162,104,120)(13,195,33,76,163,105,121)(14,196,34,77,164,106,122)(15,197,35,78,165,107,123)(16,198,36,79,166,108,124)(17,199,37,80,167,109,125)(18,200,38,81,168,110,126)(19,201,39,82,169,111,127)(20,202,40,83,170,112,128)(21,203,41,84,171,113,129)(22,175,42,85,172,114,130)(23,176,43,86,173,115,131)(24,177,44,87,174,116,132)(25,178,45,59,146,88,133)(26,179,46,60,147,89,134)(27,180,47,61,148,90,135)(28,181,48,62,149,91,136)(29,182,49,63,150,92,137), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(88,178)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(113,203)(114,175)(115,176)(116,177)>;
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)(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174)(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203), (1,183,50,64,151,93,138)(2,184,51,65,152,94,139)(3,185,52,66,153,95,140)(4,186,53,67,154,96,141)(5,187,54,68,155,97,142)(6,188,55,69,156,98,143)(7,189,56,70,157,99,144)(8,190,57,71,158,100,145)(9,191,58,72,159,101,117)(10,192,30,73,160,102,118)(11,193,31,74,161,103,119)(12,194,32,75,162,104,120)(13,195,33,76,163,105,121)(14,196,34,77,164,106,122)(15,197,35,78,165,107,123)(16,198,36,79,166,108,124)(17,199,37,80,167,109,125)(18,200,38,81,168,110,126)(19,201,39,82,169,111,127)(20,202,40,83,170,112,128)(21,203,41,84,171,113,129)(22,175,42,85,172,114,130)(23,176,43,86,173,115,131)(24,177,44,87,174,116,132)(25,178,45,59,146,88,133)(26,179,46,60,147,89,134)(27,180,47,61,148,90,135)(28,181,48,62,149,91,136)(29,182,49,63,150,92,137), (1,138)(2,139)(3,140)(4,141)(5,142)(6,143)(7,144)(8,145)(9,117)(10,118)(11,119)(12,120)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,160)(31,161)(32,162)(33,163)(34,164)(35,165)(36,166)(37,167)(38,168)(39,169)(40,170)(41,171)(42,172)(43,173)(44,174)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(88,178)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,185)(96,186)(97,187)(98,188)(99,189)(100,190)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(111,201)(112,202)(113,203)(114,175)(115,176)(116,177) );
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),(146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174),(175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203)], [(1,183,50,64,151,93,138),(2,184,51,65,152,94,139),(3,185,52,66,153,95,140),(4,186,53,67,154,96,141),(5,187,54,68,155,97,142),(6,188,55,69,156,98,143),(7,189,56,70,157,99,144),(8,190,57,71,158,100,145),(9,191,58,72,159,101,117),(10,192,30,73,160,102,118),(11,193,31,74,161,103,119),(12,194,32,75,162,104,120),(13,195,33,76,163,105,121),(14,196,34,77,164,106,122),(15,197,35,78,165,107,123),(16,198,36,79,166,108,124),(17,199,37,80,167,109,125),(18,200,38,81,168,110,126),(19,201,39,82,169,111,127),(20,202,40,83,170,112,128),(21,203,41,84,171,113,129),(22,175,42,85,172,114,130),(23,176,43,86,173,115,131),(24,177,44,87,174,116,132),(25,178,45,59,146,88,133),(26,179,46,60,147,89,134),(27,180,47,61,148,90,135),(28,181,48,62,149,91,136),(29,182,49,63,150,92,137)], [(1,138),(2,139),(3,140),(4,141),(5,142),(6,143),(7,144),(8,145),(9,117),(10,118),(11,119),(12,120),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,160),(31,161),(32,162),(33,163),(34,164),(35,165),(36,166),(37,167),(38,168),(39,169),(40,170),(41,171),(42,172),(43,173),(44,174),(45,146),(46,147),(47,148),(48,149),(49,150),(50,151),(51,152),(52,153),(53,154),(54,155),(55,156),(56,157),(57,158),(58,159),(88,178),(89,179),(90,180),(91,181),(92,182),(93,183),(94,184),(95,185),(96,186),(97,187),(98,188),(99,189),(100,190),(101,191),(102,192),(103,193),(104,194),(105,195),(106,196),(107,197),(108,198),(109,199),(110,200),(111,201),(112,202),(113,203),(114,175),(115,176),(116,177)]])
145 conjugacy classes
| class | 1 | 2 | 7A | 7B | 7C | 29A | ··· | 29AB | 58A | ··· | 58AB | 203A | ··· | 203CF |
| order | 1 | 2 | 7 | 7 | 7 | 29 | ··· | 29 | 58 | ··· | 58 | 203 | ··· | 203 |
| size | 1 | 7 | 2 | 2 | 2 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 |
145 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C29 | C58 | D7 | D7×C29 |
| kernel | D7×C29 | C203 | D7 | C7 | C29 | C1 |
| # reps | 1 | 1 | 28 | 28 | 3 | 84 |
Matrix representation of D7×C29 ►in GL2(𝔽2437) generated by
| 1703 | 0 |
| 0 | 1703 |
| 990 | 1 |
| 2011 | 1415 |
| 1415 | 2405 |
| 426 | 1022 |
G:=sub<GL(2,GF(2437))| [1703,0,0,1703],[990,2011,1,1415],[1415,426,2405,1022] >;
D7×C29 in GAP, Magma, Sage, TeX
D_7\times C_{29} % in TeX
G:=Group("D7xC29"); // GroupNames label
G:=SmallGroup(406,3);
// by ID
G=gap.SmallGroup(406,3);
# by ID
G:=PCGroup([3,-2,-29,-7,3134]);
// Polycyclic
G:=Group<a,b,c|a^29=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export