direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C34, C14⋊C34, C238⋊3C2, C119⋊4C22, C7⋊(C2×C34), SmallGroup(476,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C34 |
Generators and relations for D7×C34
G = < a,b,c | a34=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D7×C34
►On 238 points
Generators in S238
(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)(171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204)(205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238)
(1 122 86 177 39 238 141)(2 123 87 178 40 205 142)(3 124 88 179 41 206 143)(4 125 89 180 42 207 144)(5 126 90 181 43 208 145)(6 127 91 182 44 209 146)(7 128 92 183 45 210 147)(8 129 93 184 46 211 148)(9 130 94 185 47 212 149)(10 131 95 186 48 213 150)(11 132 96 187 49 214 151)(12 133 97 188 50 215 152)(13 134 98 189 51 216 153)(14 135 99 190 52 217 154)(15 136 100 191 53 218 155)(16 103 101 192 54 219 156)(17 104 102 193 55 220 157)(18 105 69 194 56 221 158)(19 106 70 195 57 222 159)(20 107 71 196 58 223 160)(21 108 72 197 59 224 161)(22 109 73 198 60 225 162)(23 110 74 199 61 226 163)(24 111 75 200 62 227 164)(25 112 76 201 63 228 165)(26 113 77 202 64 229 166)(27 114 78 203 65 230 167)(28 115 79 204 66 231 168)(29 116 80 171 67 232 169)(30 117 81 172 68 233 170)(31 118 82 173 35 234 137)(32 119 83 174 36 235 138)(33 120 84 175 37 236 139)(34 121 85 176 38 237 140)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 163)(7 164)(8 165)(9 166)(10 167)(11 168)(12 169)(13 170)(14 137)(15 138)(16 139)(17 140)(18 141)(19 142)(20 143)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 99)(36 100)(37 101)(38 102)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(103 236)(104 237)(105 238)(106 205)(107 206)(108 207)(109 208)(110 209)(111 210)(112 211)(113 212)(114 213)(115 214)(116 215)(117 216)(118 217)(119 218)(120 219)(121 220)(122 221)(123 222)(124 223)(125 224)(126 225)(127 226)(128 227)(129 228)(130 229)(131 230)(132 231)(133 232)(134 233)(135 234)(136 235)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 197)(181 198)(182 199)(183 200)(184 201)(185 202)(186 203)(187 204)
G:=sub<Sym(238)| (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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238), (1,122,86,177,39,238,141)(2,123,87,178,40,205,142)(3,124,88,179,41,206,143)(4,125,89,180,42,207,144)(5,126,90,181,43,208,145)(6,127,91,182,44,209,146)(7,128,92,183,45,210,147)(8,129,93,184,46,211,148)(9,130,94,185,47,212,149)(10,131,95,186,48,213,150)(11,132,96,187,49,214,151)(12,133,97,188,50,215,152)(13,134,98,189,51,216,153)(14,135,99,190,52,217,154)(15,136,100,191,53,218,155)(16,103,101,192,54,219,156)(17,104,102,193,55,220,157)(18,105,69,194,56,221,158)(19,106,70,195,57,222,159)(20,107,71,196,58,223,160)(21,108,72,197,59,224,161)(22,109,73,198,60,225,162)(23,110,74,199,61,226,163)(24,111,75,200,62,227,164)(25,112,76,201,63,228,165)(26,113,77,202,64,229,166)(27,114,78,203,65,230,167)(28,115,79,204,66,231,168)(29,116,80,171,67,232,169)(30,117,81,172,68,233,170)(31,118,82,173,35,234,137)(32,119,83,174,36,235,138)(33,120,84,175,37,236,139)(34,121,85,176,38,237,140), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,99)(36,100)(37,101)(38,102)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(103,236)(104,237)(105,238)(106,205)(107,206)(108,207)(109,208)(110,209)(111,210)(112,211)(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)(120,219)(121,220)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)(128,227)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204)>;
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)(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204)(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238), (1,122,86,177,39,238,141)(2,123,87,178,40,205,142)(3,124,88,179,41,206,143)(4,125,89,180,42,207,144)(5,126,90,181,43,208,145)(6,127,91,182,44,209,146)(7,128,92,183,45,210,147)(8,129,93,184,46,211,148)(9,130,94,185,47,212,149)(10,131,95,186,48,213,150)(11,132,96,187,49,214,151)(12,133,97,188,50,215,152)(13,134,98,189,51,216,153)(14,135,99,190,52,217,154)(15,136,100,191,53,218,155)(16,103,101,192,54,219,156)(17,104,102,193,55,220,157)(18,105,69,194,56,221,158)(19,106,70,195,57,222,159)(20,107,71,196,58,223,160)(21,108,72,197,59,224,161)(22,109,73,198,60,225,162)(23,110,74,199,61,226,163)(24,111,75,200,62,227,164)(25,112,76,201,63,228,165)(26,113,77,202,64,229,166)(27,114,78,203,65,230,167)(28,115,79,204,66,231,168)(29,116,80,171,67,232,169)(30,117,81,172,68,233,170)(31,118,82,173,35,234,137)(32,119,83,174,36,235,138)(33,120,84,175,37,236,139)(34,121,85,176,38,237,140), (1,158)(2,159)(3,160)(4,161)(5,162)(6,163)(7,164)(8,165)(9,166)(10,167)(11,168)(12,169)(13,170)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,99)(36,100)(37,101)(38,102)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(103,236)(104,237)(105,238)(106,205)(107,206)(108,207)(109,208)(110,209)(111,210)(112,211)(113,212)(114,213)(115,214)(116,215)(117,216)(118,217)(119,218)(120,219)(121,220)(122,221)(123,222)(124,223)(125,224)(126,225)(127,226)(128,227)(129,228)(130,229)(131,230)(132,231)(133,232)(134,233)(135,234)(136,235)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)(184,201)(185,202)(186,203)(187,204) );
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),(171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204),(205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238)], [(1,122,86,177,39,238,141),(2,123,87,178,40,205,142),(3,124,88,179,41,206,143),(4,125,89,180,42,207,144),(5,126,90,181,43,208,145),(6,127,91,182,44,209,146),(7,128,92,183,45,210,147),(8,129,93,184,46,211,148),(9,130,94,185,47,212,149),(10,131,95,186,48,213,150),(11,132,96,187,49,214,151),(12,133,97,188,50,215,152),(13,134,98,189,51,216,153),(14,135,99,190,52,217,154),(15,136,100,191,53,218,155),(16,103,101,192,54,219,156),(17,104,102,193,55,220,157),(18,105,69,194,56,221,158),(19,106,70,195,57,222,159),(20,107,71,196,58,223,160),(21,108,72,197,59,224,161),(22,109,73,198,60,225,162),(23,110,74,199,61,226,163),(24,111,75,200,62,227,164),(25,112,76,201,63,228,165),(26,113,77,202,64,229,166),(27,114,78,203,65,230,167),(28,115,79,204,66,231,168),(29,116,80,171,67,232,169),(30,117,81,172,68,233,170),(31,118,82,173,35,234,137),(32,119,83,174,36,235,138),(33,120,84,175,37,236,139),(34,121,85,176,38,237,140)], [(1,158),(2,159),(3,160),(4,161),(5,162),(6,163),(7,164),(8,165),(9,166),(10,167),(11,168),(12,169),(13,170),(14,137),(15,138),(16,139),(17,140),(18,141),(19,142),(20,143),(21,144),(22,145),(23,146),(24,147),(25,148),(26,149),(27,150),(28,151),(29,152),(30,153),(31,154),(32,155),(33,156),(34,157),(35,99),(36,100),(37,101),(38,102),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(103,236),(104,237),(105,238),(106,205),(107,206),(108,207),(109,208),(110,209),(111,210),(112,211),(113,212),(114,213),(115,214),(116,215),(117,216),(118,217),(119,218),(120,219),(121,220),(122,221),(123,222),(124,223),(125,224),(126,225),(127,226),(128,227),(129,228),(130,229),(131,230),(132,231),(133,232),(134,233),(135,234),(136,235),(171,188),(172,189),(173,190),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,197),(181,198),(182,199),(183,200),(184,201),(185,202),(186,203),(187,204)]])
170 conjugacy classes
| class | 1 | 2A | 2B | 2C | 7A | 7B | 7C | 14A | 14B | 14C | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 119A | ··· | 119AV | 238A | ··· | 238AV |
| order | 1 | 2 | 2 | 2 | 7 | 7 | 7 | 14 | 14 | 14 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 119 | ··· | 119 | 238 | ··· | 238 |
| size | 1 | 1 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
170 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | D7 | D14 | D7×C17 | D7×C34 |
| kernel | D7×C34 | D7×C17 | C238 | D14 | D7 | C14 | C34 | C17 | C2 | C1 |
| # reps | 1 | 2 | 1 | 16 | 32 | 16 | 3 | 3 | 48 | 48 |
Matrix representation of D7×C34 ►in GL3(𝔽239) generated by
| 238 | 0 | 0 |
| 0 | 163 | 0 |
| 0 | 0 | 163 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 238 | 198 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(239))| [238,0,0,0,163,0,0,0,163],[1,0,0,0,0,238,0,1,198],[1,0,0,0,0,1,0,1,0] >;
D7×C34 in GAP, Magma, Sage, TeX
D_7\times C_{34} % in TeX
G:=Group("D7xC34"); // GroupNames label
G:=SmallGroup(476,9);
// by ID
G=gap.SmallGroup(476,9);
# by ID
G:=PCGroup([4,-2,-2,-17,-7,6531]);
// Polycyclic
G:=Group<a,b,c|a^34=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export