direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×C5⋊2C8, C5⋊2C56, C35⋊5C8, C70.5C4, C28.4D5, C20.2C14, C140.6C2, C10.2C28, C14.2Dic5, C4.2(C7×D5), C2.(C7×Dic5), SmallGroup(280,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C7×C5⋊2C8 |
Generators and relations for C7×C5⋊2C8
G = < a,b,c | a7=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7×C5⋊2C8
►Regular action on 280 points
Generators in S280
(1 220 145 72 261 180 105)(2 221 146 65 262 181 106)(3 222 147 66 263 182 107)(4 223 148 67 264 183 108)(5 224 149 68 257 184 109)(6 217 150 69 258 177 110)(7 218 151 70 259 178 111)(8 219 152 71 260 179 112)(9 206 131 50 247 166 91)(10 207 132 51 248 167 92)(11 208 133 52 241 168 93)(12 201 134 53 242 161 94)(13 202 135 54 243 162 95)(14 203 136 55 244 163 96)(15 204 129 56 245 164 89)(16 205 130 49 246 165 90)(17 227 138 57 267 186 98)(18 228 139 58 268 187 99)(19 229 140 59 269 188 100)(20 230 141 60 270 189 101)(21 231 142 61 271 190 102)(22 232 143 62 272 191 103)(23 225 144 63 265 192 104)(24 226 137 64 266 185 97)(25 210 275 194 251 170 235)(26 211 276 195 252 171 236)(27 212 277 196 253 172 237)(28 213 278 197 254 173 238)(29 214 279 198 255 174 239)(30 215 280 199 256 175 240)(31 216 273 200 249 176 233)(32 209 274 193 250 169 234)(33 83 154 73 123 42 114)(34 84 155 74 124 43 115)(35 85 156 75 125 44 116)(36 86 157 76 126 45 117)(37 87 158 77 127 46 118)(38 88 159 78 128 47 119)(39 81 160 79 121 48 120)(40 82 153 80 122 41 113)
(1 225 27 91 84)(2 85 92 28 226)(3 227 29 93 86)(4 87 94 30 228)(5 229 31 95 88)(6 81 96 32 230)(7 231 25 89 82)(8 83 90 26 232)(9 155 220 144 212)(10 213 137 221 156)(11 157 222 138 214)(12 215 139 223 158)(13 159 224 140 216)(14 209 141 217 160)(15 153 218 142 210)(16 211 143 219 154)(17 239 168 36 107)(18 108 37 161 240)(19 233 162 38 109)(20 110 39 163 234)(21 235 164 40 111)(22 112 33 165 236)(23 237 166 34 105)(24 106 35 167 238)(41 259 190 251 56)(42 49 252 191 260)(43 261 192 253 50)(44 51 254 185 262)(45 263 186 255 52)(46 53 256 187 264)(47 257 188 249 54)(48 55 250 189 258)(57 279 208 76 147)(58 148 77 201 280)(59 273 202 78 149)(60 150 79 203 274)(61 275 204 80 151)(62 152 73 205 276)(63 277 206 74 145)(64 146 75 207 278)(65 125 132 197 266)(66 267 198 133 126)(67 127 134 199 268)(68 269 200 135 128)(69 121 136 193 270)(70 271 194 129 122)(71 123 130 195 272)(72 265 196 131 124)(97 181 116 248 173)(98 174 241 117 182)(99 183 118 242 175)(100 176 243 119 184)(101 177 120 244 169)(102 170 245 113 178)(103 179 114 246 171)(104 172 247 115 180)
(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)
G:=sub<Sym(280)| (1,220,145,72,261,180,105)(2,221,146,65,262,181,106)(3,222,147,66,263,182,107)(4,223,148,67,264,183,108)(5,224,149,68,257,184,109)(6,217,150,69,258,177,110)(7,218,151,70,259,178,111)(8,219,152,71,260,179,112)(9,206,131,50,247,166,91)(10,207,132,51,248,167,92)(11,208,133,52,241,168,93)(12,201,134,53,242,161,94)(13,202,135,54,243,162,95)(14,203,136,55,244,163,96)(15,204,129,56,245,164,89)(16,205,130,49,246,165,90)(17,227,138,57,267,186,98)(18,228,139,58,268,187,99)(19,229,140,59,269,188,100)(20,230,141,60,270,189,101)(21,231,142,61,271,190,102)(22,232,143,62,272,191,103)(23,225,144,63,265,192,104)(24,226,137,64,266,185,97)(25,210,275,194,251,170,235)(26,211,276,195,252,171,236)(27,212,277,196,253,172,237)(28,213,278,197,254,173,238)(29,214,279,198,255,174,239)(30,215,280,199,256,175,240)(31,216,273,200,249,176,233)(32,209,274,193,250,169,234)(33,83,154,73,123,42,114)(34,84,155,74,124,43,115)(35,85,156,75,125,44,116)(36,86,157,76,126,45,117)(37,87,158,77,127,46,118)(38,88,159,78,128,47,119)(39,81,160,79,121,48,120)(40,82,153,80,122,41,113), (1,225,27,91,84)(2,85,92,28,226)(3,227,29,93,86)(4,87,94,30,228)(5,229,31,95,88)(6,81,96,32,230)(7,231,25,89,82)(8,83,90,26,232)(9,155,220,144,212)(10,213,137,221,156)(11,157,222,138,214)(12,215,139,223,158)(13,159,224,140,216)(14,209,141,217,160)(15,153,218,142,210)(16,211,143,219,154)(17,239,168,36,107)(18,108,37,161,240)(19,233,162,38,109)(20,110,39,163,234)(21,235,164,40,111)(22,112,33,165,236)(23,237,166,34,105)(24,106,35,167,238)(41,259,190,251,56)(42,49,252,191,260)(43,261,192,253,50)(44,51,254,185,262)(45,263,186,255,52)(46,53,256,187,264)(47,257,188,249,54)(48,55,250,189,258)(57,279,208,76,147)(58,148,77,201,280)(59,273,202,78,149)(60,150,79,203,274)(61,275,204,80,151)(62,152,73,205,276)(63,277,206,74,145)(64,146,75,207,278)(65,125,132,197,266)(66,267,198,133,126)(67,127,134,199,268)(68,269,200,135,128)(69,121,136,193,270)(70,271,194,129,122)(71,123,130,195,272)(72,265,196,131,124)(97,181,116,248,173)(98,174,241,117,182)(99,183,118,242,175)(100,176,243,119,184)(101,177,120,244,169)(102,170,245,113,178)(103,179,114,246,171)(104,172,247,115,180), (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)>;
G:=Group( (1,220,145,72,261,180,105)(2,221,146,65,262,181,106)(3,222,147,66,263,182,107)(4,223,148,67,264,183,108)(5,224,149,68,257,184,109)(6,217,150,69,258,177,110)(7,218,151,70,259,178,111)(8,219,152,71,260,179,112)(9,206,131,50,247,166,91)(10,207,132,51,248,167,92)(11,208,133,52,241,168,93)(12,201,134,53,242,161,94)(13,202,135,54,243,162,95)(14,203,136,55,244,163,96)(15,204,129,56,245,164,89)(16,205,130,49,246,165,90)(17,227,138,57,267,186,98)(18,228,139,58,268,187,99)(19,229,140,59,269,188,100)(20,230,141,60,270,189,101)(21,231,142,61,271,190,102)(22,232,143,62,272,191,103)(23,225,144,63,265,192,104)(24,226,137,64,266,185,97)(25,210,275,194,251,170,235)(26,211,276,195,252,171,236)(27,212,277,196,253,172,237)(28,213,278,197,254,173,238)(29,214,279,198,255,174,239)(30,215,280,199,256,175,240)(31,216,273,200,249,176,233)(32,209,274,193,250,169,234)(33,83,154,73,123,42,114)(34,84,155,74,124,43,115)(35,85,156,75,125,44,116)(36,86,157,76,126,45,117)(37,87,158,77,127,46,118)(38,88,159,78,128,47,119)(39,81,160,79,121,48,120)(40,82,153,80,122,41,113), (1,225,27,91,84)(2,85,92,28,226)(3,227,29,93,86)(4,87,94,30,228)(5,229,31,95,88)(6,81,96,32,230)(7,231,25,89,82)(8,83,90,26,232)(9,155,220,144,212)(10,213,137,221,156)(11,157,222,138,214)(12,215,139,223,158)(13,159,224,140,216)(14,209,141,217,160)(15,153,218,142,210)(16,211,143,219,154)(17,239,168,36,107)(18,108,37,161,240)(19,233,162,38,109)(20,110,39,163,234)(21,235,164,40,111)(22,112,33,165,236)(23,237,166,34,105)(24,106,35,167,238)(41,259,190,251,56)(42,49,252,191,260)(43,261,192,253,50)(44,51,254,185,262)(45,263,186,255,52)(46,53,256,187,264)(47,257,188,249,54)(48,55,250,189,258)(57,279,208,76,147)(58,148,77,201,280)(59,273,202,78,149)(60,150,79,203,274)(61,275,204,80,151)(62,152,73,205,276)(63,277,206,74,145)(64,146,75,207,278)(65,125,132,197,266)(66,267,198,133,126)(67,127,134,199,268)(68,269,200,135,128)(69,121,136,193,270)(70,271,194,129,122)(71,123,130,195,272)(72,265,196,131,124)(97,181,116,248,173)(98,174,241,117,182)(99,183,118,242,175)(100,176,243,119,184)(101,177,120,244,169)(102,170,245,113,178)(103,179,114,246,171)(104,172,247,115,180), (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) );
G=PermutationGroup([[(1,220,145,72,261,180,105),(2,221,146,65,262,181,106),(3,222,147,66,263,182,107),(4,223,148,67,264,183,108),(5,224,149,68,257,184,109),(6,217,150,69,258,177,110),(7,218,151,70,259,178,111),(8,219,152,71,260,179,112),(9,206,131,50,247,166,91),(10,207,132,51,248,167,92),(11,208,133,52,241,168,93),(12,201,134,53,242,161,94),(13,202,135,54,243,162,95),(14,203,136,55,244,163,96),(15,204,129,56,245,164,89),(16,205,130,49,246,165,90),(17,227,138,57,267,186,98),(18,228,139,58,268,187,99),(19,229,140,59,269,188,100),(20,230,141,60,270,189,101),(21,231,142,61,271,190,102),(22,232,143,62,272,191,103),(23,225,144,63,265,192,104),(24,226,137,64,266,185,97),(25,210,275,194,251,170,235),(26,211,276,195,252,171,236),(27,212,277,196,253,172,237),(28,213,278,197,254,173,238),(29,214,279,198,255,174,239),(30,215,280,199,256,175,240),(31,216,273,200,249,176,233),(32,209,274,193,250,169,234),(33,83,154,73,123,42,114),(34,84,155,74,124,43,115),(35,85,156,75,125,44,116),(36,86,157,76,126,45,117),(37,87,158,77,127,46,118),(38,88,159,78,128,47,119),(39,81,160,79,121,48,120),(40,82,153,80,122,41,113)], [(1,225,27,91,84),(2,85,92,28,226),(3,227,29,93,86),(4,87,94,30,228),(5,229,31,95,88),(6,81,96,32,230),(7,231,25,89,82),(8,83,90,26,232),(9,155,220,144,212),(10,213,137,221,156),(11,157,222,138,214),(12,215,139,223,158),(13,159,224,140,216),(14,209,141,217,160),(15,153,218,142,210),(16,211,143,219,154),(17,239,168,36,107),(18,108,37,161,240),(19,233,162,38,109),(20,110,39,163,234),(21,235,164,40,111),(22,112,33,165,236),(23,237,166,34,105),(24,106,35,167,238),(41,259,190,251,56),(42,49,252,191,260),(43,261,192,253,50),(44,51,254,185,262),(45,263,186,255,52),(46,53,256,187,264),(47,257,188,249,54),(48,55,250,189,258),(57,279,208,76,147),(58,148,77,201,280),(59,273,202,78,149),(60,150,79,203,274),(61,275,204,80,151),(62,152,73,205,276),(63,277,206,74,145),(64,146,75,207,278),(65,125,132,197,266),(66,267,198,133,126),(67,127,134,199,268),(68,269,200,135,128),(69,121,136,193,270),(70,271,194,129,122),(71,123,130,195,272),(72,265,196,131,124),(97,181,116,248,173),(98,174,241,117,182),(99,183,118,242,175),(100,176,243,119,184),(101,177,120,244,169),(102,170,245,113,178),(103,179,114,246,171),(104,172,247,115,180)], [(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)]])
112 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 10A | 10B | 14A | ··· | 14F | 20A | 20B | 20C | 20D | 28A | ··· | 28L | 35A | ··· | 35L | 56A | ··· | 56X | 70A | ··· | 70L | 140A | ··· | 140X |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 10 | 10 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 28 | ··· | 28 | 35 | ··· | 35 | 56 | ··· | 56 | 70 | ··· | 70 | 140 | ··· | 140 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 | D5 | Dic5 | C5⋊2C8 | C7×D5 | C7×Dic5 | C7×C5⋊2C8 |
| kernel | C7×C5⋊2C8 | C140 | C70 | C5⋊2C8 | C35 | C20 | C10 | C5 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 | 2 | 2 | 4 | 12 | 12 | 24 |
Matrix representation of C7×C5⋊2C8 ►in GL2(𝔽29) generated by
| 16 | 0 |
| 0 | 16 |
| 1 | 27 |
| 4 | 22 |
| 0 | 8 |
| 16 | 0 |
G:=sub<GL(2,GF(29))| [16,0,0,16],[1,4,27,22],[0,16,8,0] >;
C7×C5⋊2C8 in GAP, Magma, Sage, TeX
C_7\times C_5\rtimes_2C_8
% in TeX
G:=Group("C7xC5:2C8"); // GroupNames label
G:=SmallGroup(280,1);
// by ID
G=gap.SmallGroup(280,1);
# by ID
G:=PCGroup([5,-2,-7,-2,-2,-5,70,42,5604]);
// Polycyclic
G:=Group<a,b,c|a^7=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export