direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C7⋊D7, C35⋊2D7, C72⋊2C10, C7⋊(C5×D7), (C7×C35)⋊4C2, SmallGroup(490,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — C5×C7⋊D7 |
Generators and relations for C5×C7⋊D7
G = < a,b,c,d | a5=b7=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Smallest permutation representation of C5×C7⋊D7
►On 245 points
Generators in S245
(1 29 224 175 126)(2 30 218 169 120)(3 31 219 170 121)(4 32 220 171 122)(5 33 221 172 123)(6 34 222 173 124)(7 35 223 174 125)(8 203 154 105 56)(9 197 148 99 50)(10 198 149 100 51)(11 199 150 101 52)(12 200 151 102 53)(13 201 152 103 54)(14 202 153 104 55)(15 210 161 112 63)(16 204 155 106 57)(17 205 156 107 58)(18 206 157 108 59)(19 207 158 109 60)(20 208 159 110 61)(21 209 160 111 62)(22 217 168 119 70)(23 211 162 113 64)(24 212 163 114 65)(25 213 164 115 66)(26 214 165 116 67)(27 215 166 117 68)(28 216 167 118 69)(36 231 182 133 84)(37 225 176 127 78)(38 226 177 128 79)(39 227 178 129 80)(40 228 179 130 81)(41 229 180 131 82)(42 230 181 132 83)(43 238 189 140 91)(44 232 183 134 85)(45 233 184 135 86)(46 234 185 136 87)(47 235 186 137 88)(48 236 187 138 89)(49 237 188 139 90)(71 242 193 144 95)(72 243 194 145 96)(73 244 195 146 97)(74 245 196 147 98)(75 239 190 141 92)(76 240 191 142 93)(77 241 192 143 94)
(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)(239 240 241 242 243 244 245)
(1 74 50 58 66 87 82)(2 75 51 59 67 88 83)(3 76 52 60 68 89 84)(4 77 53 61 69 90 78)(5 71 54 62 70 91 79)(6 72 55 63 64 85 80)(7 73 56 57 65 86 81)(8 16 24 45 40 35 244)(9 17 25 46 41 29 245)(10 18 26 47 42 30 239)(11 19 27 48 36 31 240)(12 20 28 49 37 32 241)(13 21 22 43 38 33 242)(14 15 23 44 39 34 243)(92 100 108 116 137 132 120)(93 101 109 117 138 133 121)(94 102 110 118 139 127 122)(95 103 111 119 140 128 123)(96 104 112 113 134 129 124)(97 105 106 114 135 130 125)(98 99 107 115 136 131 126)(141 149 157 165 186 181 169)(142 150 158 166 187 182 170)(143 151 159 167 188 176 171)(144 152 160 168 189 177 172)(145 153 161 162 183 178 173)(146 154 155 163 184 179 174)(147 148 156 164 185 180 175)(190 198 206 214 235 230 218)(191 199 207 215 236 231 219)(192 200 208 216 237 225 220)(193 201 209 217 238 226 221)(194 202 210 211 232 227 222)(195 203 204 212 233 228 223)(196 197 205 213 234 229 224)
(1 82)(2 81)(3 80)(4 79)(5 78)(6 84)(7 83)(8 26)(9 25)(10 24)(11 23)(12 22)(13 28)(14 27)(15 19)(16 18)(20 21)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(35 42)(43 241)(44 240)(45 239)(46 245)(47 244)(48 243)(49 242)(50 66)(51 65)(52 64)(53 70)(54 69)(55 68)(56 67)(57 59)(60 63)(61 62)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 91)(92 135)(93 134)(94 140)(95 139)(96 138)(97 137)(98 136)(99 115)(100 114)(101 113)(102 119)(103 118)(104 117)(105 116)(106 108)(109 112)(110 111)(120 130)(121 129)(122 128)(123 127)(124 133)(125 132)(126 131)(141 184)(142 183)(143 189)(144 188)(145 187)(146 186)(147 185)(148 164)(149 163)(150 162)(151 168)(152 167)(153 166)(154 165)(155 157)(158 161)(159 160)(169 179)(170 178)(171 177)(172 176)(173 182)(174 181)(175 180)(190 233)(191 232)(192 238)(193 237)(194 236)(195 235)(196 234)(197 213)(198 212)(199 211)(200 217)(201 216)(202 215)(203 214)(204 206)(207 210)(208 209)(218 228)(219 227)(220 226)(221 225)(222 231)(223 230)(224 229)
G:=sub<Sym(245)| (1,29,224,175,126)(2,30,218,169,120)(3,31,219,170,121)(4,32,220,171,122)(5,33,221,172,123)(6,34,222,173,124)(7,35,223,174,125)(8,203,154,105,56)(9,197,148,99,50)(10,198,149,100,51)(11,199,150,101,52)(12,200,151,102,53)(13,201,152,103,54)(14,202,153,104,55)(15,210,161,112,63)(16,204,155,106,57)(17,205,156,107,58)(18,206,157,108,59)(19,207,158,109,60)(20,208,159,110,61)(21,209,160,111,62)(22,217,168,119,70)(23,211,162,113,64)(24,212,163,114,65)(25,213,164,115,66)(26,214,165,116,67)(27,215,166,117,68)(28,216,167,118,69)(36,231,182,133,84)(37,225,176,127,78)(38,226,177,128,79)(39,227,178,129,80)(40,228,179,130,81)(41,229,180,131,82)(42,230,181,132,83)(43,238,189,140,91)(44,232,183,134,85)(45,233,184,135,86)(46,234,185,136,87)(47,235,186,137,88)(48,236,187,138,89)(49,237,188,139,90)(71,242,193,144,95)(72,243,194,145,96)(73,244,195,146,97)(74,245,196,147,98)(75,239,190,141,92)(76,240,191,142,93)(77,241,192,143,94), (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)(239,240,241,242,243,244,245), (1,74,50,58,66,87,82)(2,75,51,59,67,88,83)(3,76,52,60,68,89,84)(4,77,53,61,69,90,78)(5,71,54,62,70,91,79)(6,72,55,63,64,85,80)(7,73,56,57,65,86,81)(8,16,24,45,40,35,244)(9,17,25,46,41,29,245)(10,18,26,47,42,30,239)(11,19,27,48,36,31,240)(12,20,28,49,37,32,241)(13,21,22,43,38,33,242)(14,15,23,44,39,34,243)(92,100,108,116,137,132,120)(93,101,109,117,138,133,121)(94,102,110,118,139,127,122)(95,103,111,119,140,128,123)(96,104,112,113,134,129,124)(97,105,106,114,135,130,125)(98,99,107,115,136,131,126)(141,149,157,165,186,181,169)(142,150,158,166,187,182,170)(143,151,159,167,188,176,171)(144,152,160,168,189,177,172)(145,153,161,162,183,178,173)(146,154,155,163,184,179,174)(147,148,156,164,185,180,175)(190,198,206,214,235,230,218)(191,199,207,215,236,231,219)(192,200,208,216,237,225,220)(193,201,209,217,238,226,221)(194,202,210,211,232,227,222)(195,203,204,212,233,228,223)(196,197,205,213,234,229,224), (1,82)(2,81)(3,80)(4,79)(5,78)(6,84)(7,83)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(15,19)(16,18)(20,21)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42)(43,241)(44,240)(45,239)(46,245)(47,244)(48,243)(49,242)(50,66)(51,65)(52,64)(53,70)(54,69)(55,68)(56,67)(57,59)(60,63)(61,62)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,91)(92,135)(93,134)(94,140)(95,139)(96,138)(97,137)(98,136)(99,115)(100,114)(101,113)(102,119)(103,118)(104,117)(105,116)(106,108)(109,112)(110,111)(120,130)(121,129)(122,128)(123,127)(124,133)(125,132)(126,131)(141,184)(142,183)(143,189)(144,188)(145,187)(146,186)(147,185)(148,164)(149,163)(150,162)(151,168)(152,167)(153,166)(154,165)(155,157)(158,161)(159,160)(169,179)(170,178)(171,177)(172,176)(173,182)(174,181)(175,180)(190,233)(191,232)(192,238)(193,237)(194,236)(195,235)(196,234)(197,213)(198,212)(199,211)(200,217)(201,216)(202,215)(203,214)(204,206)(207,210)(208,209)(218,228)(219,227)(220,226)(221,225)(222,231)(223,230)(224,229)>;
G:=Group( (1,29,224,175,126)(2,30,218,169,120)(3,31,219,170,121)(4,32,220,171,122)(5,33,221,172,123)(6,34,222,173,124)(7,35,223,174,125)(8,203,154,105,56)(9,197,148,99,50)(10,198,149,100,51)(11,199,150,101,52)(12,200,151,102,53)(13,201,152,103,54)(14,202,153,104,55)(15,210,161,112,63)(16,204,155,106,57)(17,205,156,107,58)(18,206,157,108,59)(19,207,158,109,60)(20,208,159,110,61)(21,209,160,111,62)(22,217,168,119,70)(23,211,162,113,64)(24,212,163,114,65)(25,213,164,115,66)(26,214,165,116,67)(27,215,166,117,68)(28,216,167,118,69)(36,231,182,133,84)(37,225,176,127,78)(38,226,177,128,79)(39,227,178,129,80)(40,228,179,130,81)(41,229,180,131,82)(42,230,181,132,83)(43,238,189,140,91)(44,232,183,134,85)(45,233,184,135,86)(46,234,185,136,87)(47,235,186,137,88)(48,236,187,138,89)(49,237,188,139,90)(71,242,193,144,95)(72,243,194,145,96)(73,244,195,146,97)(74,245,196,147,98)(75,239,190,141,92)(76,240,191,142,93)(77,241,192,143,94), (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)(239,240,241,242,243,244,245), (1,74,50,58,66,87,82)(2,75,51,59,67,88,83)(3,76,52,60,68,89,84)(4,77,53,61,69,90,78)(5,71,54,62,70,91,79)(6,72,55,63,64,85,80)(7,73,56,57,65,86,81)(8,16,24,45,40,35,244)(9,17,25,46,41,29,245)(10,18,26,47,42,30,239)(11,19,27,48,36,31,240)(12,20,28,49,37,32,241)(13,21,22,43,38,33,242)(14,15,23,44,39,34,243)(92,100,108,116,137,132,120)(93,101,109,117,138,133,121)(94,102,110,118,139,127,122)(95,103,111,119,140,128,123)(96,104,112,113,134,129,124)(97,105,106,114,135,130,125)(98,99,107,115,136,131,126)(141,149,157,165,186,181,169)(142,150,158,166,187,182,170)(143,151,159,167,188,176,171)(144,152,160,168,189,177,172)(145,153,161,162,183,178,173)(146,154,155,163,184,179,174)(147,148,156,164,185,180,175)(190,198,206,214,235,230,218)(191,199,207,215,236,231,219)(192,200,208,216,237,225,220)(193,201,209,217,238,226,221)(194,202,210,211,232,227,222)(195,203,204,212,233,228,223)(196,197,205,213,234,229,224), (1,82)(2,81)(3,80)(4,79)(5,78)(6,84)(7,83)(8,26)(9,25)(10,24)(11,23)(12,22)(13,28)(14,27)(15,19)(16,18)(20,21)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42)(43,241)(44,240)(45,239)(46,245)(47,244)(48,243)(49,242)(50,66)(51,65)(52,64)(53,70)(54,69)(55,68)(56,67)(57,59)(60,63)(61,62)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,91)(92,135)(93,134)(94,140)(95,139)(96,138)(97,137)(98,136)(99,115)(100,114)(101,113)(102,119)(103,118)(104,117)(105,116)(106,108)(109,112)(110,111)(120,130)(121,129)(122,128)(123,127)(124,133)(125,132)(126,131)(141,184)(142,183)(143,189)(144,188)(145,187)(146,186)(147,185)(148,164)(149,163)(150,162)(151,168)(152,167)(153,166)(154,165)(155,157)(158,161)(159,160)(169,179)(170,178)(171,177)(172,176)(173,182)(174,181)(175,180)(190,233)(191,232)(192,238)(193,237)(194,236)(195,235)(196,234)(197,213)(198,212)(199,211)(200,217)(201,216)(202,215)(203,214)(204,206)(207,210)(208,209)(218,228)(219,227)(220,226)(221,225)(222,231)(223,230)(224,229) );
G=PermutationGroup([[(1,29,224,175,126),(2,30,218,169,120),(3,31,219,170,121),(4,32,220,171,122),(5,33,221,172,123),(6,34,222,173,124),(7,35,223,174,125),(8,203,154,105,56),(9,197,148,99,50),(10,198,149,100,51),(11,199,150,101,52),(12,200,151,102,53),(13,201,152,103,54),(14,202,153,104,55),(15,210,161,112,63),(16,204,155,106,57),(17,205,156,107,58),(18,206,157,108,59),(19,207,158,109,60),(20,208,159,110,61),(21,209,160,111,62),(22,217,168,119,70),(23,211,162,113,64),(24,212,163,114,65),(25,213,164,115,66),(26,214,165,116,67),(27,215,166,117,68),(28,216,167,118,69),(36,231,182,133,84),(37,225,176,127,78),(38,226,177,128,79),(39,227,178,129,80),(40,228,179,130,81),(41,229,180,131,82),(42,230,181,132,83),(43,238,189,140,91),(44,232,183,134,85),(45,233,184,135,86),(46,234,185,136,87),(47,235,186,137,88),(48,236,187,138,89),(49,237,188,139,90),(71,242,193,144,95),(72,243,194,145,96),(73,244,195,146,97),(74,245,196,147,98),(75,239,190,141,92),(76,240,191,142,93),(77,241,192,143,94)], [(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),(239,240,241,242,243,244,245)], [(1,74,50,58,66,87,82),(2,75,51,59,67,88,83),(3,76,52,60,68,89,84),(4,77,53,61,69,90,78),(5,71,54,62,70,91,79),(6,72,55,63,64,85,80),(7,73,56,57,65,86,81),(8,16,24,45,40,35,244),(9,17,25,46,41,29,245),(10,18,26,47,42,30,239),(11,19,27,48,36,31,240),(12,20,28,49,37,32,241),(13,21,22,43,38,33,242),(14,15,23,44,39,34,243),(92,100,108,116,137,132,120),(93,101,109,117,138,133,121),(94,102,110,118,139,127,122),(95,103,111,119,140,128,123),(96,104,112,113,134,129,124),(97,105,106,114,135,130,125),(98,99,107,115,136,131,126),(141,149,157,165,186,181,169),(142,150,158,166,187,182,170),(143,151,159,167,188,176,171),(144,152,160,168,189,177,172),(145,153,161,162,183,178,173),(146,154,155,163,184,179,174),(147,148,156,164,185,180,175),(190,198,206,214,235,230,218),(191,199,207,215,236,231,219),(192,200,208,216,237,225,220),(193,201,209,217,238,226,221),(194,202,210,211,232,227,222),(195,203,204,212,233,228,223),(196,197,205,213,234,229,224)], [(1,82),(2,81),(3,80),(4,79),(5,78),(6,84),(7,83),(8,26),(9,25),(10,24),(11,23),(12,22),(13,28),(14,27),(15,19),(16,18),(20,21),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(35,42),(43,241),(44,240),(45,239),(46,245),(47,244),(48,243),(49,242),(50,66),(51,65),(52,64),(53,70),(54,69),(55,68),(56,67),(57,59),(60,63),(61,62),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,91),(92,135),(93,134),(94,140),(95,139),(96,138),(97,137),(98,136),(99,115),(100,114),(101,113),(102,119),(103,118),(104,117),(105,116),(106,108),(109,112),(110,111),(120,130),(121,129),(122,128),(123,127),(124,133),(125,132),(126,131),(141,184),(142,183),(143,189),(144,188),(145,187),(146,186),(147,185),(148,164),(149,163),(150,162),(151,168),(152,167),(153,166),(154,165),(155,157),(158,161),(159,160),(169,179),(170,178),(171,177),(172,176),(173,182),(174,181),(175,180),(190,233),(191,232),(192,238),(193,237),(194,236),(195,235),(196,234),(197,213),(198,212),(199,211),(200,217),(201,216),(202,215),(203,214),(204,206),(207,210),(208,209),(218,228),(219,227),(220,226),(221,225),(222,231),(223,230),(224,229)]])
130 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 7A | ··· | 7X | 10A | 10B | 10C | 10D | 35A | ··· | 35CR |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 35 | ··· | 35 |
| size | 1 | 49 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 49 | 49 | 49 | 49 | 2 | ··· | 2 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | D7 | C5×D7 |
| kernel | C5×C7⋊D7 | C7×C35 | C7⋊D7 | C72 | C35 | C7 |
| # reps | 1 | 1 | 4 | 4 | 24 | 96 |
Matrix representation of C5×C7⋊D7 ►in GL4(𝔽71) generated by
| 57 | 0 | 0 | 0 |
| 0 | 57 | 0 | 0 |
| 0 | 0 | 54 | 0 |
| 0 | 0 | 0 | 54 |
| 4 | 52 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 52 | 70 |
| 0 | 0 | 1 | 0 |
| 67 | 1 | 0 | 0 |
| 59 | 56 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 70 | 52 |
| 0 | 52 | 0 | 0 |
| 56 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(71))| [57,0,0,0,0,57,0,0,0,0,54,0,0,0,0,54],[4,15,0,0,52,0,0,0,0,0,52,1,0,0,70,0],[67,59,0,0,1,56,0,0,0,0,0,70,0,0,1,52],[0,56,0,0,52,0,0,0,0,0,0,1,0,0,1,0] >;
C5×C7⋊D7 in GAP, Magma, Sage, TeX
C_5\times C_7\rtimes D_7
% in TeX
G:=Group("C5xC7:D7"); // GroupNames label
G:=SmallGroup(490,6);
// by ID
G=gap.SmallGroup(490,6);
# by ID
G:=PCGroup([4,-2,-5,-7,-7,722,6723]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^7=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export