direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic14, C35⋊3Q8, C20.3D7, C140.3C2, C28.1C10, Dic7.C10, C10.13D14, C70.13C22, C7⋊(C5×Q8), C4.(C5×D7), C2.3(C10×D7), C14.1(C2×C10), (C5×Dic7).2C2, SmallGroup(280,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic14
G = < a,b,c | a5=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic14
►Regular action on 280 points
Generators in S280
(1 40 248 173 153)(2 41 249 174 154)(3 42 250 175 155)(4 43 251 176 156)(5 44 252 177 157)(6 45 225 178 158)(7 46 226 179 159)(8 47 227 180 160)(9 48 228 181 161)(10 49 229 182 162)(11 50 230 183 163)(12 51 231 184 164)(13 52 232 185 165)(14 53 233 186 166)(15 54 234 187 167)(16 55 235 188 168)(17 56 236 189 141)(18 29 237 190 142)(19 30 238 191 143)(20 31 239 192 144)(21 32 240 193 145)(22 33 241 194 146)(23 34 242 195 147)(24 35 243 196 148)(25 36 244 169 149)(26 37 245 170 150)(27 38 246 171 151)(28 39 247 172 152)(57 111 120 256 206)(58 112 121 257 207)(59 85 122 258 208)(60 86 123 259 209)(61 87 124 260 210)(62 88 125 261 211)(63 89 126 262 212)(64 90 127 263 213)(65 91 128 264 214)(66 92 129 265 215)(67 93 130 266 216)(68 94 131 267 217)(69 95 132 268 218)(70 96 133 269 219)(71 97 134 270 220)(72 98 135 271 221)(73 99 136 272 222)(74 100 137 273 223)(75 101 138 274 224)(76 102 139 275 197)(77 103 140 276 198)(78 104 113 277 199)(79 105 114 278 200)(80 106 115 279 201)(81 107 116 280 202)(82 108 117 253 203)(83 109 118 254 204)(84 110 119 255 205)
(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)
(1 222 15 208)(2 221 16 207)(3 220 17 206)(4 219 18 205)(5 218 19 204)(6 217 20 203)(7 216 21 202)(8 215 22 201)(9 214 23 200)(10 213 24 199)(11 212 25 198)(12 211 26 197)(13 210 27 224)(14 209 28 223)(29 84 43 70)(30 83 44 69)(31 82 45 68)(32 81 46 67)(33 80 47 66)(34 79 48 65)(35 78 49 64)(36 77 50 63)(37 76 51 62)(38 75 52 61)(39 74 53 60)(40 73 54 59)(41 72 55 58)(42 71 56 57)(85 248 99 234)(86 247 100 233)(87 246 101 232)(88 245 102 231)(89 244 103 230)(90 243 104 229)(91 242 105 228)(92 241 106 227)(93 240 107 226)(94 239 108 225)(95 238 109 252)(96 237 110 251)(97 236 111 250)(98 235 112 249)(113 182 127 196)(114 181 128 195)(115 180 129 194)(116 179 130 193)(117 178 131 192)(118 177 132 191)(119 176 133 190)(120 175 134 189)(121 174 135 188)(122 173 136 187)(123 172 137 186)(124 171 138 185)(125 170 139 184)(126 169 140 183)(141 256 155 270)(142 255 156 269)(143 254 157 268)(144 253 158 267)(145 280 159 266)(146 279 160 265)(147 278 161 264)(148 277 162 263)(149 276 163 262)(150 275 164 261)(151 274 165 260)(152 273 166 259)(153 272 167 258)(154 271 168 257)
G:=sub<Sym(280)| (1,40,248,173,153)(2,41,249,174,154)(3,42,250,175,155)(4,43,251,176,156)(5,44,252,177,157)(6,45,225,178,158)(7,46,226,179,159)(8,47,227,180,160)(9,48,228,181,161)(10,49,229,182,162)(11,50,230,183,163)(12,51,231,184,164)(13,52,232,185,165)(14,53,233,186,166)(15,54,234,187,167)(16,55,235,188,168)(17,56,236,189,141)(18,29,237,190,142)(19,30,238,191,143)(20,31,239,192,144)(21,32,240,193,145)(22,33,241,194,146)(23,34,242,195,147)(24,35,243,196,148)(25,36,244,169,149)(26,37,245,170,150)(27,38,246,171,151)(28,39,247,172,152)(57,111,120,256,206)(58,112,121,257,207)(59,85,122,258,208)(60,86,123,259,209)(61,87,124,260,210)(62,88,125,261,211)(63,89,126,262,212)(64,90,127,263,213)(65,91,128,264,214)(66,92,129,265,215)(67,93,130,266,216)(68,94,131,267,217)(69,95,132,268,218)(70,96,133,269,219)(71,97,134,270,220)(72,98,135,271,221)(73,99,136,272,222)(74,100,137,273,223)(75,101,138,274,224)(76,102,139,275,197)(77,103,140,276,198)(78,104,113,277,199)(79,105,114,278,200)(80,106,115,279,201)(81,107,116,280,202)(82,108,117,253,203)(83,109,118,254,204)(84,110,119,255,205), (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), (1,222,15,208)(2,221,16,207)(3,220,17,206)(4,219,18,205)(5,218,19,204)(6,217,20,203)(7,216,21,202)(8,215,22,201)(9,214,23,200)(10,213,24,199)(11,212,25,198)(12,211,26,197)(13,210,27,224)(14,209,28,223)(29,84,43,70)(30,83,44,69)(31,82,45,68)(32,81,46,67)(33,80,47,66)(34,79,48,65)(35,78,49,64)(36,77,50,63)(37,76,51,62)(38,75,52,61)(39,74,53,60)(40,73,54,59)(41,72,55,58)(42,71,56,57)(85,248,99,234)(86,247,100,233)(87,246,101,232)(88,245,102,231)(89,244,103,230)(90,243,104,229)(91,242,105,228)(92,241,106,227)(93,240,107,226)(94,239,108,225)(95,238,109,252)(96,237,110,251)(97,236,111,250)(98,235,112,249)(113,182,127,196)(114,181,128,195)(115,180,129,194)(116,179,130,193)(117,178,131,192)(118,177,132,191)(119,176,133,190)(120,175,134,189)(121,174,135,188)(122,173,136,187)(123,172,137,186)(124,171,138,185)(125,170,139,184)(126,169,140,183)(141,256,155,270)(142,255,156,269)(143,254,157,268)(144,253,158,267)(145,280,159,266)(146,279,160,265)(147,278,161,264)(148,277,162,263)(149,276,163,262)(150,275,164,261)(151,274,165,260)(152,273,166,259)(153,272,167,258)(154,271,168,257)>;
G:=Group( (1,40,248,173,153)(2,41,249,174,154)(3,42,250,175,155)(4,43,251,176,156)(5,44,252,177,157)(6,45,225,178,158)(7,46,226,179,159)(8,47,227,180,160)(9,48,228,181,161)(10,49,229,182,162)(11,50,230,183,163)(12,51,231,184,164)(13,52,232,185,165)(14,53,233,186,166)(15,54,234,187,167)(16,55,235,188,168)(17,56,236,189,141)(18,29,237,190,142)(19,30,238,191,143)(20,31,239,192,144)(21,32,240,193,145)(22,33,241,194,146)(23,34,242,195,147)(24,35,243,196,148)(25,36,244,169,149)(26,37,245,170,150)(27,38,246,171,151)(28,39,247,172,152)(57,111,120,256,206)(58,112,121,257,207)(59,85,122,258,208)(60,86,123,259,209)(61,87,124,260,210)(62,88,125,261,211)(63,89,126,262,212)(64,90,127,263,213)(65,91,128,264,214)(66,92,129,265,215)(67,93,130,266,216)(68,94,131,267,217)(69,95,132,268,218)(70,96,133,269,219)(71,97,134,270,220)(72,98,135,271,221)(73,99,136,272,222)(74,100,137,273,223)(75,101,138,274,224)(76,102,139,275,197)(77,103,140,276,198)(78,104,113,277,199)(79,105,114,278,200)(80,106,115,279,201)(81,107,116,280,202)(82,108,117,253,203)(83,109,118,254,204)(84,110,119,255,205), (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), (1,222,15,208)(2,221,16,207)(3,220,17,206)(4,219,18,205)(5,218,19,204)(6,217,20,203)(7,216,21,202)(8,215,22,201)(9,214,23,200)(10,213,24,199)(11,212,25,198)(12,211,26,197)(13,210,27,224)(14,209,28,223)(29,84,43,70)(30,83,44,69)(31,82,45,68)(32,81,46,67)(33,80,47,66)(34,79,48,65)(35,78,49,64)(36,77,50,63)(37,76,51,62)(38,75,52,61)(39,74,53,60)(40,73,54,59)(41,72,55,58)(42,71,56,57)(85,248,99,234)(86,247,100,233)(87,246,101,232)(88,245,102,231)(89,244,103,230)(90,243,104,229)(91,242,105,228)(92,241,106,227)(93,240,107,226)(94,239,108,225)(95,238,109,252)(96,237,110,251)(97,236,111,250)(98,235,112,249)(113,182,127,196)(114,181,128,195)(115,180,129,194)(116,179,130,193)(117,178,131,192)(118,177,132,191)(119,176,133,190)(120,175,134,189)(121,174,135,188)(122,173,136,187)(123,172,137,186)(124,171,138,185)(125,170,139,184)(126,169,140,183)(141,256,155,270)(142,255,156,269)(143,254,157,268)(144,253,158,267)(145,280,159,266)(146,279,160,265)(147,278,161,264)(148,277,162,263)(149,276,163,262)(150,275,164,261)(151,274,165,260)(152,273,166,259)(153,272,167,258)(154,271,168,257) );
G=PermutationGroup([[(1,40,248,173,153),(2,41,249,174,154),(3,42,250,175,155),(4,43,251,176,156),(5,44,252,177,157),(6,45,225,178,158),(7,46,226,179,159),(8,47,227,180,160),(9,48,228,181,161),(10,49,229,182,162),(11,50,230,183,163),(12,51,231,184,164),(13,52,232,185,165),(14,53,233,186,166),(15,54,234,187,167),(16,55,235,188,168),(17,56,236,189,141),(18,29,237,190,142),(19,30,238,191,143),(20,31,239,192,144),(21,32,240,193,145),(22,33,241,194,146),(23,34,242,195,147),(24,35,243,196,148),(25,36,244,169,149),(26,37,245,170,150),(27,38,246,171,151),(28,39,247,172,152),(57,111,120,256,206),(58,112,121,257,207),(59,85,122,258,208),(60,86,123,259,209),(61,87,124,260,210),(62,88,125,261,211),(63,89,126,262,212),(64,90,127,263,213),(65,91,128,264,214),(66,92,129,265,215),(67,93,130,266,216),(68,94,131,267,217),(69,95,132,268,218),(70,96,133,269,219),(71,97,134,270,220),(72,98,135,271,221),(73,99,136,272,222),(74,100,137,273,223),(75,101,138,274,224),(76,102,139,275,197),(77,103,140,276,198),(78,104,113,277,199),(79,105,114,278,200),(80,106,115,279,201),(81,107,116,280,202),(82,108,117,253,203),(83,109,118,254,204),(84,110,119,255,205)], [(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)], [(1,222,15,208),(2,221,16,207),(3,220,17,206),(4,219,18,205),(5,218,19,204),(6,217,20,203),(7,216,21,202),(8,215,22,201),(9,214,23,200),(10,213,24,199),(11,212,25,198),(12,211,26,197),(13,210,27,224),(14,209,28,223),(29,84,43,70),(30,83,44,69),(31,82,45,68),(32,81,46,67),(33,80,47,66),(34,79,48,65),(35,78,49,64),(36,77,50,63),(37,76,51,62),(38,75,52,61),(39,74,53,60),(40,73,54,59),(41,72,55,58),(42,71,56,57),(85,248,99,234),(86,247,100,233),(87,246,101,232),(88,245,102,231),(89,244,103,230),(90,243,104,229),(91,242,105,228),(92,241,106,227),(93,240,107,226),(94,239,108,225),(95,238,109,252),(96,237,110,251),(97,236,111,250),(98,235,112,249),(113,182,127,196),(114,181,128,195),(115,180,129,194),(116,179,130,193),(117,178,131,192),(118,177,132,191),(119,176,133,190),(120,175,134,189),(121,174,135,188),(122,173,136,187),(123,172,137,186),(124,171,138,185),(125,170,139,184),(126,169,140,183),(141,256,155,270),(142,255,156,269),(143,254,157,268),(144,253,158,267),(145,280,159,266),(146,279,160,265),(147,278,161,264),(148,277,162,263),(149,276,163,262),(150,275,164,261),(151,274,165,260),(152,273,166,259),(153,272,167,258),(154,271,168,257)]])
85 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 14A | 14B | 14C | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 28A | ··· | 28F | 35A | ··· | 35L | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | Q8 | D7 | D14 | C5×Q8 | Dic14 | C5×D7 | C10×D7 | C5×Dic14 |
| kernel | C5×Dic14 | C5×Dic7 | C140 | Dic14 | Dic7 | C28 | C35 | C20 | C10 | C7 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 3 | 3 | 4 | 6 | 12 | 12 | 24 |
Matrix representation of C5×Dic14 ►in GL2(𝔽281) generated by
| 232 | 0 |
| 0 | 232 |
| 32 | 112 |
| 169 | 91 |
| 205 | 76 |
| 46 | 76 |
G:=sub<GL(2,GF(281))| [232,0,0,232],[32,169,112,91],[205,46,76,76] >;
C5×Dic14 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{14} % in TeX
G:=Group("C5xDic14"); // GroupNames label
G:=SmallGroup(280,14);
// by ID
G=gap.SmallGroup(280,14);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-7,100,221,106,6004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export