direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic35, C70⋊3C4, C14⋊Dic5, C2.2D70, C22.D35, C10⋊2Dic7, C10.11D14, C14.11D10, C70.11C22, (C2×C10).D7, (C2×C14).D5, C35⋊10(C2×C4), C5⋊3(C2×Dic7), C7⋊2(C2×Dic5), (C2×C70).1C2, SmallGroup(280,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C2×Dic35 |
Generators and relations for C2×Dic35
G = < a,b,c | a2=b70=1, c2=b35, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic35
►Regular action on 280 points
Generators in S280
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 85)(68 86)(69 87)(70 88)(141 225)(142 226)(143 227)(144 228)(145 229)(146 230)(147 231)(148 232)(149 233)(150 234)(151 235)(152 236)(153 237)(154 238)(155 239)(156 240)(157 241)(158 242)(159 243)(160 244)(161 245)(162 246)(163 247)(164 248)(165 249)(166 250)(167 251)(168 252)(169 253)(170 254)(171 255)(172 256)(173 257)(174 258)(175 259)(176 260)(177 261)(178 262)(179 263)(180 264)(181 265)(182 266)(183 267)(184 268)(185 269)(186 270)(187 271)(188 272)(189 273)(190 274)(191 275)(192 276)(193 277)(194 278)(195 279)(196 280)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280)
(1 274 36 239)(2 273 37 238)(3 272 38 237)(4 271 39 236)(5 270 40 235)(6 269 41 234)(7 268 42 233)(8 267 43 232)(9 266 44 231)(10 265 45 230)(11 264 46 229)(12 263 47 228)(13 262 48 227)(14 261 49 226)(15 260 50 225)(16 259 51 224)(17 258 52 223)(18 257 53 222)(19 256 54 221)(20 255 55 220)(21 254 56 219)(22 253 57 218)(23 252 58 217)(24 251 59 216)(25 250 60 215)(26 249 61 214)(27 248 62 213)(28 247 63 212)(29 246 64 211)(30 245 65 280)(31 244 66 279)(32 243 67 278)(33 242 68 277)(34 241 69 276)(35 240 70 275)(71 208 106 173)(72 207 107 172)(73 206 108 171)(74 205 109 170)(75 204 110 169)(76 203 111 168)(77 202 112 167)(78 201 113 166)(79 200 114 165)(80 199 115 164)(81 198 116 163)(82 197 117 162)(83 196 118 161)(84 195 119 160)(85 194 120 159)(86 193 121 158)(87 192 122 157)(88 191 123 156)(89 190 124 155)(90 189 125 154)(91 188 126 153)(92 187 127 152)(93 186 128 151)(94 185 129 150)(95 184 130 149)(96 183 131 148)(97 182 132 147)(98 181 133 146)(99 180 134 145)(100 179 135 144)(101 178 136 143)(102 177 137 142)(103 176 138 141)(104 175 139 210)(105 174 140 209)
G:=sub<Sym(280)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,225)(142,226)(143,227)(144,228)(145,229)(146,230)(147,231)(148,232)(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280), (1,274,36,239)(2,273,37,238)(3,272,38,237)(4,271,39,236)(5,270,40,235)(6,269,41,234)(7,268,42,233)(8,267,43,232)(9,266,44,231)(10,265,45,230)(11,264,46,229)(12,263,47,228)(13,262,48,227)(14,261,49,226)(15,260,50,225)(16,259,51,224)(17,258,52,223)(18,257,53,222)(19,256,54,221)(20,255,55,220)(21,254,56,219)(22,253,57,218)(23,252,58,217)(24,251,59,216)(25,250,60,215)(26,249,61,214)(27,248,62,213)(28,247,63,212)(29,246,64,211)(30,245,65,280)(31,244,66,279)(32,243,67,278)(33,242,68,277)(34,241,69,276)(35,240,70,275)(71,208,106,173)(72,207,107,172)(73,206,108,171)(74,205,109,170)(75,204,110,169)(76,203,111,168)(77,202,112,167)(78,201,113,166)(79,200,114,165)(80,199,115,164)(81,198,116,163)(82,197,117,162)(83,196,118,161)(84,195,119,160)(85,194,120,159)(86,193,121,158)(87,192,122,157)(88,191,123,156)(89,190,124,155)(90,189,125,154)(91,188,126,153)(92,187,127,152)(93,186,128,151)(94,185,129,150)(95,184,130,149)(96,183,131,148)(97,182,132,147)(98,181,133,146)(99,180,134,145)(100,179,135,144)(101,178,136,143)(102,177,137,142)(103,176,138,141)(104,175,139,210)(105,174,140,209)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,71)(54,72)(55,73)(56,74)(57,75)(58,76)(59,77)(60,78)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,85)(68,86)(69,87)(70,88)(141,225)(142,226)(143,227)(144,228)(145,229)(146,230)(147,231)(148,232)(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)(165,249)(166,250)(167,251)(168,252)(169,253)(170,254)(171,255)(172,256)(173,257)(174,258)(175,259)(176,260)(177,261)(178,262)(179,263)(180,264)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,277)(194,278)(195,279)(196,280)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280), (1,274,36,239)(2,273,37,238)(3,272,38,237)(4,271,39,236)(5,270,40,235)(6,269,41,234)(7,268,42,233)(8,267,43,232)(9,266,44,231)(10,265,45,230)(11,264,46,229)(12,263,47,228)(13,262,48,227)(14,261,49,226)(15,260,50,225)(16,259,51,224)(17,258,52,223)(18,257,53,222)(19,256,54,221)(20,255,55,220)(21,254,56,219)(22,253,57,218)(23,252,58,217)(24,251,59,216)(25,250,60,215)(26,249,61,214)(27,248,62,213)(28,247,63,212)(29,246,64,211)(30,245,65,280)(31,244,66,279)(32,243,67,278)(33,242,68,277)(34,241,69,276)(35,240,70,275)(71,208,106,173)(72,207,107,172)(73,206,108,171)(74,205,109,170)(75,204,110,169)(76,203,111,168)(77,202,112,167)(78,201,113,166)(79,200,114,165)(80,199,115,164)(81,198,116,163)(82,197,117,162)(83,196,118,161)(84,195,119,160)(85,194,120,159)(86,193,121,158)(87,192,122,157)(88,191,123,156)(89,190,124,155)(90,189,125,154)(91,188,126,153)(92,187,127,152)(93,186,128,151)(94,185,129,150)(95,184,130,149)(96,183,131,148)(97,182,132,147)(98,181,133,146)(99,180,134,145)(100,179,135,144)(101,178,136,143)(102,177,137,142)(103,176,138,141)(104,175,139,210)(105,174,140,209) );
G=PermutationGroup([[(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,71),(54,72),(55,73),(56,74),(57,75),(58,76),(59,77),(60,78),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,85),(68,86),(69,87),(70,88),(141,225),(142,226),(143,227),(144,228),(145,229),(146,230),(147,231),(148,232),(149,233),(150,234),(151,235),(152,236),(153,237),(154,238),(155,239),(156,240),(157,241),(158,242),(159,243),(160,244),(161,245),(162,246),(163,247),(164,248),(165,249),(166,250),(167,251),(168,252),(169,253),(170,254),(171,255),(172,256),(173,257),(174,258),(175,259),(176,260),(177,261),(178,262),(179,263),(180,264),(181,265),(182,266),(183,267),(184,268),(185,269),(186,270),(187,271),(188,272),(189,273),(190,274),(191,275),(192,276),(193,277),(194,278),(195,279),(196,280),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280)], [(1,274,36,239),(2,273,37,238),(3,272,38,237),(4,271,39,236),(5,270,40,235),(6,269,41,234),(7,268,42,233),(8,267,43,232),(9,266,44,231),(10,265,45,230),(11,264,46,229),(12,263,47,228),(13,262,48,227),(14,261,49,226),(15,260,50,225),(16,259,51,224),(17,258,52,223),(18,257,53,222),(19,256,54,221),(20,255,55,220),(21,254,56,219),(22,253,57,218),(23,252,58,217),(24,251,59,216),(25,250,60,215),(26,249,61,214),(27,248,62,213),(28,247,63,212),(29,246,64,211),(30,245,65,280),(31,244,66,279),(32,243,67,278),(33,242,68,277),(34,241,69,276),(35,240,70,275),(71,208,106,173),(72,207,107,172),(73,206,108,171),(74,205,109,170),(75,204,110,169),(76,203,111,168),(77,202,112,167),(78,201,113,166),(79,200,114,165),(80,199,115,164),(81,198,116,163),(82,197,117,162),(83,196,118,161),(84,195,119,160),(85,194,120,159),(86,193,121,158),(87,192,122,157),(88,191,123,156),(89,190,124,155),(90,189,125,154),(91,188,126,153),(92,187,127,152),(93,186,128,151),(94,185,129,150),(95,184,130,149),(96,183,131,148),(97,182,132,147),(98,181,133,146),(99,180,134,145),(100,179,135,144),(101,178,136,143),(102,177,137,142),(103,176,138,141),(104,175,139,210),(105,174,140,209)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 7A | 7B | 7C | 10A | ··· | 10F | 14A | ··· | 14I | 35A | ··· | 35L | 70A | ··· | 70AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | ··· | 10 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 1 | 1 | 35 | 35 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D5 | D7 | Dic5 | D10 | Dic7 | D14 | D35 | Dic35 | D70 |
| kernel | C2×Dic35 | Dic35 | C2×C70 | C70 | C2×C14 | C2×C10 | C14 | C14 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 2 | 3 | 4 | 2 | 6 | 3 | 12 | 24 | 12 |
Matrix representation of C2×Dic35 ►in GL3(𝔽281) generated by
| 280 | 0 | 0 |
| 0 | 280 | 0 |
| 0 | 0 | 280 |
| 1 | 0 | 0 |
| 0 | 11 | 104 |
| 0 | 177 | 243 |
| 280 | 0 | 0 |
| 0 | 73 | 168 |
| 0 | 266 | 208 |
G:=sub<GL(3,GF(281))| [280,0,0,0,280,0,0,0,280],[1,0,0,0,11,177,0,104,243],[280,0,0,0,73,266,0,168,208] >;
C2×Dic35 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{35} % in TeX
G:=Group("C2xDic35"); // GroupNames label
G:=SmallGroup(280,27);
// by ID
G=gap.SmallGroup(280,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,643,6004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^70=1,c^2=b^35,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export