direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C7×Dic10, C35⋊4Q8, C28.3D5, C140.4C2, C20.1C14, C14.13D10, C70.18C22, Dic5.1C14, C5⋊(C7×Q8), C4.(C7×D5), C2.3(D5×C14), C10.1(C2×C14), (C7×Dic5).3C2, SmallGroup(280,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Dic10
G = < a,b,c | a7=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×Dic10
►Regular action on 280 points
Generators in S280
(1 202 158 90 256 226 129)(2 203 159 91 257 227 130)(3 204 160 92 258 228 131)(4 205 141 93 259 229 132)(5 206 142 94 260 230 133)(6 207 143 95 241 231 134)(7 208 144 96 242 232 135)(8 209 145 97 243 233 136)(9 210 146 98 244 234 137)(10 211 147 99 245 235 138)(11 212 148 100 246 236 139)(12 213 149 81 247 237 140)(13 214 150 82 248 238 121)(14 215 151 83 249 239 122)(15 216 152 84 250 240 123)(16 217 153 85 251 221 124)(17 218 154 86 252 222 125)(18 219 155 87 253 223 126)(19 220 156 88 254 224 127)(20 201 157 89 255 225 128)(21 59 191 75 274 180 101)(22 60 192 76 275 161 102)(23 41 193 77 276 162 103)(24 42 194 78 277 163 104)(25 43 195 79 278 164 105)(26 44 196 80 279 165 106)(27 45 197 61 280 166 107)(28 46 198 62 261 167 108)(29 47 199 63 262 168 109)(30 48 200 64 263 169 110)(31 49 181 65 264 170 111)(32 50 182 66 265 171 112)(33 51 183 67 266 172 113)(34 52 184 68 267 173 114)(35 53 185 69 268 174 115)(36 54 186 70 269 175 116)(37 55 187 71 270 176 117)(38 56 188 72 271 177 118)(39 57 189 73 272 178 119)(40 58 190 74 273 179 120)
(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 27 11 37)(2 26 12 36)(3 25 13 35)(4 24 14 34)(5 23 15 33)(6 22 16 32)(7 21 17 31)(8 40 18 30)(9 39 19 29)(10 38 20 28)(41 216 51 206)(42 215 52 205)(43 214 53 204)(44 213 54 203)(45 212 55 202)(46 211 56 201)(47 210 57 220)(48 209 58 219)(49 208 59 218)(50 207 60 217)(61 100 71 90)(62 99 72 89)(63 98 73 88)(64 97 74 87)(65 96 75 86)(66 95 76 85)(67 94 77 84)(68 93 78 83)(69 92 79 82)(70 91 80 81)(101 125 111 135)(102 124 112 134)(103 123 113 133)(104 122 114 132)(105 121 115 131)(106 140 116 130)(107 139 117 129)(108 138 118 128)(109 137 119 127)(110 136 120 126)(141 194 151 184)(142 193 152 183)(143 192 153 182)(144 191 154 181)(145 190 155 200)(146 189 156 199)(147 188 157 198)(148 187 158 197)(149 186 159 196)(150 185 160 195)(161 221 171 231)(162 240 172 230)(163 239 173 229)(164 238 174 228)(165 237 175 227)(166 236 176 226)(167 235 177 225)(168 234 178 224)(169 233 179 223)(170 232 180 222)(241 275 251 265)(242 274 252 264)(243 273 253 263)(244 272 254 262)(245 271 255 261)(246 270 256 280)(247 269 257 279)(248 268 258 278)(249 267 259 277)(250 266 260 276)
G:=sub<Sym(280)| (1,202,158,90,256,226,129)(2,203,159,91,257,227,130)(3,204,160,92,258,228,131)(4,205,141,93,259,229,132)(5,206,142,94,260,230,133)(6,207,143,95,241,231,134)(7,208,144,96,242,232,135)(8,209,145,97,243,233,136)(9,210,146,98,244,234,137)(10,211,147,99,245,235,138)(11,212,148,100,246,236,139)(12,213,149,81,247,237,140)(13,214,150,82,248,238,121)(14,215,151,83,249,239,122)(15,216,152,84,250,240,123)(16,217,153,85,251,221,124)(17,218,154,86,252,222,125)(18,219,155,87,253,223,126)(19,220,156,88,254,224,127)(20,201,157,89,255,225,128)(21,59,191,75,274,180,101)(22,60,192,76,275,161,102)(23,41,193,77,276,162,103)(24,42,194,78,277,163,104)(25,43,195,79,278,164,105)(26,44,196,80,279,165,106)(27,45,197,61,280,166,107)(28,46,198,62,261,167,108)(29,47,199,63,262,168,109)(30,48,200,64,263,169,110)(31,49,181,65,264,170,111)(32,50,182,66,265,171,112)(33,51,183,67,266,172,113)(34,52,184,68,267,173,114)(35,53,185,69,268,174,115)(36,54,186,70,269,175,116)(37,55,187,71,270,176,117)(38,56,188,72,271,177,118)(39,57,189,73,272,178,119)(40,58,190,74,273,179,120), (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), (1,27,11,37)(2,26,12,36)(3,25,13,35)(4,24,14,34)(5,23,15,33)(6,22,16,32)(7,21,17,31)(8,40,18,30)(9,39,19,29)(10,38,20,28)(41,216,51,206)(42,215,52,205)(43,214,53,204)(44,213,54,203)(45,212,55,202)(46,211,56,201)(47,210,57,220)(48,209,58,219)(49,208,59,218)(50,207,60,217)(61,100,71,90)(62,99,72,89)(63,98,73,88)(64,97,74,87)(65,96,75,86)(66,95,76,85)(67,94,77,84)(68,93,78,83)(69,92,79,82)(70,91,80,81)(101,125,111,135)(102,124,112,134)(103,123,113,133)(104,122,114,132)(105,121,115,131)(106,140,116,130)(107,139,117,129)(108,138,118,128)(109,137,119,127)(110,136,120,126)(141,194,151,184)(142,193,152,183)(143,192,153,182)(144,191,154,181)(145,190,155,200)(146,189,156,199)(147,188,157,198)(148,187,158,197)(149,186,159,196)(150,185,160,195)(161,221,171,231)(162,240,172,230)(163,239,173,229)(164,238,174,228)(165,237,175,227)(166,236,176,226)(167,235,177,225)(168,234,178,224)(169,233,179,223)(170,232,180,222)(241,275,251,265)(242,274,252,264)(243,273,253,263)(244,272,254,262)(245,271,255,261)(246,270,256,280)(247,269,257,279)(248,268,258,278)(249,267,259,277)(250,266,260,276)>;
G:=Group( (1,202,158,90,256,226,129)(2,203,159,91,257,227,130)(3,204,160,92,258,228,131)(4,205,141,93,259,229,132)(5,206,142,94,260,230,133)(6,207,143,95,241,231,134)(7,208,144,96,242,232,135)(8,209,145,97,243,233,136)(9,210,146,98,244,234,137)(10,211,147,99,245,235,138)(11,212,148,100,246,236,139)(12,213,149,81,247,237,140)(13,214,150,82,248,238,121)(14,215,151,83,249,239,122)(15,216,152,84,250,240,123)(16,217,153,85,251,221,124)(17,218,154,86,252,222,125)(18,219,155,87,253,223,126)(19,220,156,88,254,224,127)(20,201,157,89,255,225,128)(21,59,191,75,274,180,101)(22,60,192,76,275,161,102)(23,41,193,77,276,162,103)(24,42,194,78,277,163,104)(25,43,195,79,278,164,105)(26,44,196,80,279,165,106)(27,45,197,61,280,166,107)(28,46,198,62,261,167,108)(29,47,199,63,262,168,109)(30,48,200,64,263,169,110)(31,49,181,65,264,170,111)(32,50,182,66,265,171,112)(33,51,183,67,266,172,113)(34,52,184,68,267,173,114)(35,53,185,69,268,174,115)(36,54,186,70,269,175,116)(37,55,187,71,270,176,117)(38,56,188,72,271,177,118)(39,57,189,73,272,178,119)(40,58,190,74,273,179,120), (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), (1,27,11,37)(2,26,12,36)(3,25,13,35)(4,24,14,34)(5,23,15,33)(6,22,16,32)(7,21,17,31)(8,40,18,30)(9,39,19,29)(10,38,20,28)(41,216,51,206)(42,215,52,205)(43,214,53,204)(44,213,54,203)(45,212,55,202)(46,211,56,201)(47,210,57,220)(48,209,58,219)(49,208,59,218)(50,207,60,217)(61,100,71,90)(62,99,72,89)(63,98,73,88)(64,97,74,87)(65,96,75,86)(66,95,76,85)(67,94,77,84)(68,93,78,83)(69,92,79,82)(70,91,80,81)(101,125,111,135)(102,124,112,134)(103,123,113,133)(104,122,114,132)(105,121,115,131)(106,140,116,130)(107,139,117,129)(108,138,118,128)(109,137,119,127)(110,136,120,126)(141,194,151,184)(142,193,152,183)(143,192,153,182)(144,191,154,181)(145,190,155,200)(146,189,156,199)(147,188,157,198)(148,187,158,197)(149,186,159,196)(150,185,160,195)(161,221,171,231)(162,240,172,230)(163,239,173,229)(164,238,174,228)(165,237,175,227)(166,236,176,226)(167,235,177,225)(168,234,178,224)(169,233,179,223)(170,232,180,222)(241,275,251,265)(242,274,252,264)(243,273,253,263)(244,272,254,262)(245,271,255,261)(246,270,256,280)(247,269,257,279)(248,268,258,278)(249,267,259,277)(250,266,260,276) );
G=PermutationGroup([[(1,202,158,90,256,226,129),(2,203,159,91,257,227,130),(3,204,160,92,258,228,131),(4,205,141,93,259,229,132),(5,206,142,94,260,230,133),(6,207,143,95,241,231,134),(7,208,144,96,242,232,135),(8,209,145,97,243,233,136),(9,210,146,98,244,234,137),(10,211,147,99,245,235,138),(11,212,148,100,246,236,139),(12,213,149,81,247,237,140),(13,214,150,82,248,238,121),(14,215,151,83,249,239,122),(15,216,152,84,250,240,123),(16,217,153,85,251,221,124),(17,218,154,86,252,222,125),(18,219,155,87,253,223,126),(19,220,156,88,254,224,127),(20,201,157,89,255,225,128),(21,59,191,75,274,180,101),(22,60,192,76,275,161,102),(23,41,193,77,276,162,103),(24,42,194,78,277,163,104),(25,43,195,79,278,164,105),(26,44,196,80,279,165,106),(27,45,197,61,280,166,107),(28,46,198,62,261,167,108),(29,47,199,63,262,168,109),(30,48,200,64,263,169,110),(31,49,181,65,264,170,111),(32,50,182,66,265,171,112),(33,51,183,67,266,172,113),(34,52,184,68,267,173,114),(35,53,185,69,268,174,115),(36,54,186,70,269,175,116),(37,55,187,71,270,176,117),(38,56,188,72,271,177,118),(39,57,189,73,272,178,119),(40,58,190,74,273,179,120)], [(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)], [(1,27,11,37),(2,26,12,36),(3,25,13,35),(4,24,14,34),(5,23,15,33),(6,22,16,32),(7,21,17,31),(8,40,18,30),(9,39,19,29),(10,38,20,28),(41,216,51,206),(42,215,52,205),(43,214,53,204),(44,213,54,203),(45,212,55,202),(46,211,56,201),(47,210,57,220),(48,209,58,219),(49,208,59,218),(50,207,60,217),(61,100,71,90),(62,99,72,89),(63,98,73,88),(64,97,74,87),(65,96,75,86),(66,95,76,85),(67,94,77,84),(68,93,78,83),(69,92,79,82),(70,91,80,81),(101,125,111,135),(102,124,112,134),(103,123,113,133),(104,122,114,132),(105,121,115,131),(106,140,116,130),(107,139,117,129),(108,138,118,128),(109,137,119,127),(110,136,120,126),(141,194,151,184),(142,193,152,183),(143,192,153,182),(144,191,154,181),(145,190,155,200),(146,189,156,199),(147,188,157,198),(148,187,158,197),(149,186,159,196),(150,185,160,195),(161,221,171,231),(162,240,172,230),(163,239,173,229),(164,238,174,228),(165,237,175,227),(166,236,176,226),(167,235,177,225),(168,234,178,224),(169,233,179,223),(170,232,180,222),(241,275,251,265),(242,274,252,264),(243,273,253,263),(244,272,254,262),(245,271,255,261),(246,270,256,280),(247,269,257,279),(248,268,258,278),(249,267,259,277),(250,266,260,276)]])
91 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 20A | 20B | 20C | 20D | 28A | ··· | 28F | 28G | ··· | 28R | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
91 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | Q8 | D5 | D10 | Dic10 | C7×Q8 | C7×D5 | D5×C14 | C7×Dic10 |
| kernel | C7×Dic10 | C7×Dic5 | C140 | Dic10 | Dic5 | C20 | C35 | C28 | C14 | C7 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 1 | 2 | 2 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C7×Dic10 ►in GL2(𝔽281) generated by
| 165 | 0 |
| 0 | 165 |
| 231 | 84 |
| 214 | 17 |
| 126 | 142 |
| 187 | 155 |
G:=sub<GL(2,GF(281))| [165,0,0,165],[231,214,84,17],[126,187,142,155] >;
C7×Dic10 in GAP, Magma, Sage, TeX
C_7\times {\rm Dic}_{10} % in TeX
G:=Group("C7xDic10"); // GroupNames label
G:=SmallGroup(280,19);
// by ID
G=gap.SmallGroup(280,19);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-5,140,301,146,5604]);
// Polycyclic
G:=Group<a,b,c|a^7=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export