direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C72, C35⋊3C14, C5⋊(C7×C14), (C7×C35)⋊5C2, SmallGroup(490,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C72 |
Generators and relations for D5×C72
G = < a,b,c,d | a7=b7=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D5×C72
►On 245 points
Generators in S245
(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)
(1 163 200 12 211 134 74)(2 164 201 13 212 135 75)(3 165 202 14 213 136 76)(4 166 203 8 214 137 77)(5 167 197 9 215 138 71)(6 168 198 10 216 139 72)(7 162 199 11 217 140 73)(15 221 109 237 81 242 174)(16 222 110 238 82 243 175)(17 223 111 232 83 244 169)(18 224 112 233 84 245 170)(19 218 106 234 78 239 171)(20 219 107 235 79 240 172)(21 220 108 236 80 241 173)(22 228 90 120 98 178 146)(23 229 91 121 92 179 147)(24 230 85 122 93 180 141)(25 231 86 123 94 181 142)(26 225 87 124 95 182 143)(27 226 88 125 96 176 144)(28 227 89 126 97 177 145)(29 151 119 207 157 184 57)(30 152 113 208 158 185 58)(31 153 114 209 159 186 59)(32 154 115 210 160 187 60)(33 148 116 204 161 188 61)(34 149 117 205 155 189 62)(35 150 118 206 156 183 63)(36 52 129 104 65 191 46)(37 53 130 105 66 192 47)(38 54 131 99 67 193 48)(39 55 132 100 68 194 49)(40 56 133 101 69 195 43)(41 50 127 102 70 196 44)(42 51 128 103 64 190 45)
(1 240 146 210 133)(2 241 147 204 127)(3 242 141 205 128)(4 243 142 206 129)(5 244 143 207 130)(6 245 144 208 131)(7 239 145 209 132)(8 222 86 63 191)(9 223 87 57 192)(10 224 88 58 193)(11 218 89 59 194)(12 219 90 60 195)(13 220 91 61 196)(14 221 85 62 190)(15 230 189 64 202)(16 231 183 65 203)(17 225 184 66 197)(18 226 185 67 198)(19 227 186 68 199)(20 228 187 69 200)(21 229 188 70 201)(22 160 101 163 172)(23 161 102 164 173)(24 155 103 165 174)(25 156 104 166 175)(26 157 105 167 169)(27 158 99 168 170)(28 159 100 162 171)(29 47 215 111 124)(30 48 216 112 125)(31 49 217 106 126)(32 43 211 107 120)(33 44 212 108 121)(34 45 213 109 122)(35 46 214 110 123)(36 137 238 94 150)(37 138 232 95 151)(38 139 233 96 152)(39 140 234 97 153)(40 134 235 98 154)(41 135 236 92 148)(42 136 237 93 149)(50 75 80 179 116)(51 76 81 180 117)(52 77 82 181 118)(53 71 83 182 119)(54 72 84 176 113)(55 73 78 177 114)(56 74 79 178 115)
(1 133)(2 127)(3 128)(4 129)(5 130)(6 131)(7 132)(8 191)(9 192)(10 193)(11 194)(12 195)(13 196)(14 190)(15 189)(16 183)(17 184)(18 185)(19 186)(20 187)(21 188)(29 111)(30 112)(31 106)(32 107)(33 108)(34 109)(35 110)(36 137)(37 138)(38 139)(39 140)(40 134)(41 135)(42 136)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 223)(58 224)(59 218)(60 219)(61 220)(62 221)(63 222)(64 202)(65 203)(66 197)(67 198)(68 199)(69 200)(70 201)(78 114)(79 115)(80 116)(81 117)(82 118)(83 119)(84 113)(99 168)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(148 236)(149 237)(150 238)(151 232)(152 233)(153 234)(154 235)(155 174)(156 175)(157 169)(158 170)(159 171)(160 172)(161 173)(204 241)(205 242)(206 243)(207 244)(208 245)(209 239)(210 240)
G:=sub<Sym(245)| (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), (1,163,200,12,211,134,74)(2,164,201,13,212,135,75)(3,165,202,14,213,136,76)(4,166,203,8,214,137,77)(5,167,197,9,215,138,71)(6,168,198,10,216,139,72)(7,162,199,11,217,140,73)(15,221,109,237,81,242,174)(16,222,110,238,82,243,175)(17,223,111,232,83,244,169)(18,224,112,233,84,245,170)(19,218,106,234,78,239,171)(20,219,107,235,79,240,172)(21,220,108,236,80,241,173)(22,228,90,120,98,178,146)(23,229,91,121,92,179,147)(24,230,85,122,93,180,141)(25,231,86,123,94,181,142)(26,225,87,124,95,182,143)(27,226,88,125,96,176,144)(28,227,89,126,97,177,145)(29,151,119,207,157,184,57)(30,152,113,208,158,185,58)(31,153,114,209,159,186,59)(32,154,115,210,160,187,60)(33,148,116,204,161,188,61)(34,149,117,205,155,189,62)(35,150,118,206,156,183,63)(36,52,129,104,65,191,46)(37,53,130,105,66,192,47)(38,54,131,99,67,193,48)(39,55,132,100,68,194,49)(40,56,133,101,69,195,43)(41,50,127,102,70,196,44)(42,51,128,103,64,190,45), (1,240,146,210,133)(2,241,147,204,127)(3,242,141,205,128)(4,243,142,206,129)(5,244,143,207,130)(6,245,144,208,131)(7,239,145,209,132)(8,222,86,63,191)(9,223,87,57,192)(10,224,88,58,193)(11,218,89,59,194)(12,219,90,60,195)(13,220,91,61,196)(14,221,85,62,190)(15,230,189,64,202)(16,231,183,65,203)(17,225,184,66,197)(18,226,185,67,198)(19,227,186,68,199)(20,228,187,69,200)(21,229,188,70,201)(22,160,101,163,172)(23,161,102,164,173)(24,155,103,165,174)(25,156,104,166,175)(26,157,105,167,169)(27,158,99,168,170)(28,159,100,162,171)(29,47,215,111,124)(30,48,216,112,125)(31,49,217,106,126)(32,43,211,107,120)(33,44,212,108,121)(34,45,213,109,122)(35,46,214,110,123)(36,137,238,94,150)(37,138,232,95,151)(38,139,233,96,152)(39,140,234,97,153)(40,134,235,98,154)(41,135,236,92,148)(42,136,237,93,149)(50,75,80,179,116)(51,76,81,180,117)(52,77,82,181,118)(53,71,83,182,119)(54,72,84,176,113)(55,73,78,177,114)(56,74,79,178,115), (1,133)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,191)(9,192)(10,193)(11,194)(12,195)(13,196)(14,190)(15,189)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(29,111)(30,112)(31,106)(32,107)(33,108)(34,109)(35,110)(36,137)(37,138)(38,139)(39,140)(40,134)(41,135)(42,136)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,223)(58,224)(59,218)(60,219)(61,220)(62,221)(63,222)(64,202)(65,203)(66,197)(67,198)(68,199)(69,200)(70,201)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,113)(99,168)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(148,236)(149,237)(150,238)(151,232)(152,233)(153,234)(154,235)(155,174)(156,175)(157,169)(158,170)(159,171)(160,172)(161,173)(204,241)(205,242)(206,243)(207,244)(208,245)(209,239)(210,240)>;
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)(239,240,241,242,243,244,245), (1,163,200,12,211,134,74)(2,164,201,13,212,135,75)(3,165,202,14,213,136,76)(4,166,203,8,214,137,77)(5,167,197,9,215,138,71)(6,168,198,10,216,139,72)(7,162,199,11,217,140,73)(15,221,109,237,81,242,174)(16,222,110,238,82,243,175)(17,223,111,232,83,244,169)(18,224,112,233,84,245,170)(19,218,106,234,78,239,171)(20,219,107,235,79,240,172)(21,220,108,236,80,241,173)(22,228,90,120,98,178,146)(23,229,91,121,92,179,147)(24,230,85,122,93,180,141)(25,231,86,123,94,181,142)(26,225,87,124,95,182,143)(27,226,88,125,96,176,144)(28,227,89,126,97,177,145)(29,151,119,207,157,184,57)(30,152,113,208,158,185,58)(31,153,114,209,159,186,59)(32,154,115,210,160,187,60)(33,148,116,204,161,188,61)(34,149,117,205,155,189,62)(35,150,118,206,156,183,63)(36,52,129,104,65,191,46)(37,53,130,105,66,192,47)(38,54,131,99,67,193,48)(39,55,132,100,68,194,49)(40,56,133,101,69,195,43)(41,50,127,102,70,196,44)(42,51,128,103,64,190,45), (1,240,146,210,133)(2,241,147,204,127)(3,242,141,205,128)(4,243,142,206,129)(5,244,143,207,130)(6,245,144,208,131)(7,239,145,209,132)(8,222,86,63,191)(9,223,87,57,192)(10,224,88,58,193)(11,218,89,59,194)(12,219,90,60,195)(13,220,91,61,196)(14,221,85,62,190)(15,230,189,64,202)(16,231,183,65,203)(17,225,184,66,197)(18,226,185,67,198)(19,227,186,68,199)(20,228,187,69,200)(21,229,188,70,201)(22,160,101,163,172)(23,161,102,164,173)(24,155,103,165,174)(25,156,104,166,175)(26,157,105,167,169)(27,158,99,168,170)(28,159,100,162,171)(29,47,215,111,124)(30,48,216,112,125)(31,49,217,106,126)(32,43,211,107,120)(33,44,212,108,121)(34,45,213,109,122)(35,46,214,110,123)(36,137,238,94,150)(37,138,232,95,151)(38,139,233,96,152)(39,140,234,97,153)(40,134,235,98,154)(41,135,236,92,148)(42,136,237,93,149)(50,75,80,179,116)(51,76,81,180,117)(52,77,82,181,118)(53,71,83,182,119)(54,72,84,176,113)(55,73,78,177,114)(56,74,79,178,115), (1,133)(2,127)(3,128)(4,129)(5,130)(6,131)(7,132)(8,191)(9,192)(10,193)(11,194)(12,195)(13,196)(14,190)(15,189)(16,183)(17,184)(18,185)(19,186)(20,187)(21,188)(29,111)(30,112)(31,106)(32,107)(33,108)(34,109)(35,110)(36,137)(37,138)(38,139)(39,140)(40,134)(41,135)(42,136)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,223)(58,224)(59,218)(60,219)(61,220)(62,221)(63,222)(64,202)(65,203)(66,197)(67,198)(68,199)(69,200)(70,201)(78,114)(79,115)(80,116)(81,117)(82,118)(83,119)(84,113)(99,168)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(148,236)(149,237)(150,238)(151,232)(152,233)(153,234)(154,235)(155,174)(156,175)(157,169)(158,170)(159,171)(160,172)(161,173)(204,241)(205,242)(206,243)(207,244)(208,245)(209,239)(210,240) );
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),(239,240,241,242,243,244,245)], [(1,163,200,12,211,134,74),(2,164,201,13,212,135,75),(3,165,202,14,213,136,76),(4,166,203,8,214,137,77),(5,167,197,9,215,138,71),(6,168,198,10,216,139,72),(7,162,199,11,217,140,73),(15,221,109,237,81,242,174),(16,222,110,238,82,243,175),(17,223,111,232,83,244,169),(18,224,112,233,84,245,170),(19,218,106,234,78,239,171),(20,219,107,235,79,240,172),(21,220,108,236,80,241,173),(22,228,90,120,98,178,146),(23,229,91,121,92,179,147),(24,230,85,122,93,180,141),(25,231,86,123,94,181,142),(26,225,87,124,95,182,143),(27,226,88,125,96,176,144),(28,227,89,126,97,177,145),(29,151,119,207,157,184,57),(30,152,113,208,158,185,58),(31,153,114,209,159,186,59),(32,154,115,210,160,187,60),(33,148,116,204,161,188,61),(34,149,117,205,155,189,62),(35,150,118,206,156,183,63),(36,52,129,104,65,191,46),(37,53,130,105,66,192,47),(38,54,131,99,67,193,48),(39,55,132,100,68,194,49),(40,56,133,101,69,195,43),(41,50,127,102,70,196,44),(42,51,128,103,64,190,45)], [(1,240,146,210,133),(2,241,147,204,127),(3,242,141,205,128),(4,243,142,206,129),(5,244,143,207,130),(6,245,144,208,131),(7,239,145,209,132),(8,222,86,63,191),(9,223,87,57,192),(10,224,88,58,193),(11,218,89,59,194),(12,219,90,60,195),(13,220,91,61,196),(14,221,85,62,190),(15,230,189,64,202),(16,231,183,65,203),(17,225,184,66,197),(18,226,185,67,198),(19,227,186,68,199),(20,228,187,69,200),(21,229,188,70,201),(22,160,101,163,172),(23,161,102,164,173),(24,155,103,165,174),(25,156,104,166,175),(26,157,105,167,169),(27,158,99,168,170),(28,159,100,162,171),(29,47,215,111,124),(30,48,216,112,125),(31,49,217,106,126),(32,43,211,107,120),(33,44,212,108,121),(34,45,213,109,122),(35,46,214,110,123),(36,137,238,94,150),(37,138,232,95,151),(38,139,233,96,152),(39,140,234,97,153),(40,134,235,98,154),(41,135,236,92,148),(42,136,237,93,149),(50,75,80,179,116),(51,76,81,180,117),(52,77,82,181,118),(53,71,83,182,119),(54,72,84,176,113),(55,73,78,177,114),(56,74,79,178,115)], [(1,133),(2,127),(3,128),(4,129),(5,130),(6,131),(7,132),(8,191),(9,192),(10,193),(11,194),(12,195),(13,196),(14,190),(15,189),(16,183),(17,184),(18,185),(19,186),(20,187),(21,188),(29,111),(30,112),(31,106),(32,107),(33,108),(34,109),(35,110),(36,137),(37,138),(38,139),(39,140),(40,134),(41,135),(42,136),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,223),(58,224),(59,218),(60,219),(61,220),(62,221),(63,222),(64,202),(65,203),(66,197),(67,198),(68,199),(69,200),(70,201),(78,114),(79,115),(80,116),(81,117),(82,118),(83,119),(84,113),(99,168),(100,162),(101,163),(102,164),(103,165),(104,166),(105,167),(148,236),(149,237),(150,238),(151,232),(152,233),(153,234),(154,235),(155,174),(156,175),(157,169),(158,170),(159,171),(160,172),(161,173),(204,241),(205,242),(206,243),(207,244),(208,245),(209,239),(210,240)]])
196 conjugacy classes
| class | 1 | 2 | 5A | 5B | 7A | ··· | 7AV | 14A | ··· | 14AV | 35A | ··· | 35CR |
| order | 1 | 2 | 5 | 5 | 7 | ··· | 7 | 14 | ··· | 14 | 35 | ··· | 35 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C7 | C14 | D5 | C7×D5 |
| kernel | D5×C72 | C7×C35 | C7×D5 | C35 | C72 | C7 |
| # reps | 1 | 1 | 48 | 48 | 2 | 96 |
Matrix representation of D5×C72 ►in GL3(𝔽71) generated by
| 45 | 0 | 0 |
| 0 | 48 | 0 |
| 0 | 0 | 48 |
| 1 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 63 | 64 |
| 0 | 70 | 70 |
| 70 | 0 | 0 |
| 0 | 70 | 7 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(71))| [45,0,0,0,48,0,0,0,48],[1,0,0,0,32,0,0,0,32],[1,0,0,0,63,70,0,64,70],[70,0,0,0,70,0,0,7,1] >;
D5×C72 in GAP, Magma, Sage, TeX
D_5\times C_7^2
% in TeX
G:=Group("D5xC7^2"); // GroupNames label
G:=SmallGroup(490,7);
// by ID
G=gap.SmallGroup(490,7);
# by ID
G:=PCGroup([4,-2,-7,-7,-5,6275]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^7=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export