direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C8×C5⋊2C8, C40⋊7C8, C5⋊2C82, C42.278D10, C4.18(C8×D5), (C4×C8).17D5, C10.12(C4×C8), C20.57(C2×C8), (C2×C40).44C4, (C4×C40).16C2, C2.1(C8×Dic5), (C2×C8).16Dic5, (C2×C10).35C42, (C4×C20).334C22, C22.13(C4×Dic5), C2.1(C4×C5⋊2C8), C4.10(C2×C5⋊2C8), (C4×C5⋊2C8).22C2, (C2×C5⋊2C8).37C4, (C2×C4).163(C4×D5), (C2×C20).410(C2×C4), (C2×C4).87(C2×Dic5), SmallGroup(320,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C8×C5⋊2C8 |
Generators and relations for C8×C5⋊2C8
G = < a,b,c | a8=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 134 in 74 conjugacy classes, 59 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C10, C42, C2×C8, C2×C8, C20, C2×C10, C4×C8, C4×C8, C5⋊2C8, C40, C2×C20, C2×C20, C82, C2×C5⋊2C8, C4×C20, C2×C40, C4×C5⋊2C8, C4×C40, C8×C5⋊2C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C42, C2×C8, Dic5, D10, C4×C8, C5⋊2C8, C4×D5, C2×Dic5, C82, C8×D5, C2×C5⋊2C8, C4×Dic5, C4×C5⋊2C8, C8×Dic5, C8×C5⋊2C8
►Regular action on 320 points
Generators in S320
(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)(281 282 283 284 285 286 287 288)(289 290 291 292 293 294 295 296)(297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312)(313 314 315 316 317 318 319 320)
(1 12 57 143 113)(2 13 58 144 114)(3 14 59 137 115)(4 15 60 138 116)(5 16 61 139 117)(6 9 62 140 118)(7 10 63 141 119)(8 11 64 142 120)(17 112 196 177 103)(18 105 197 178 104)(19 106 198 179 97)(20 107 199 180 98)(21 108 200 181 99)(22 109 193 182 100)(23 110 194 183 101)(24 111 195 184 102)(25 87 285 264 212)(26 88 286 257 213)(27 81 287 258 214)(28 82 288 259 215)(29 83 281 260 216)(30 84 282 261 209)(31 85 283 262 210)(32 86 284 263 211)(33 121 161 187 56)(34 122 162 188 49)(35 123 163 189 50)(36 124 164 190 51)(37 125 165 191 52)(38 126 166 192 53)(39 127 167 185 54)(40 128 168 186 55)(41 157 130 73 71)(42 158 131 74 72)(43 159 132 75 65)(44 160 133 76 66)(45 153 134 77 67)(46 154 135 78 68)(47 155 136 79 69)(48 156 129 80 70)(89 145 172 247 265)(90 146 173 248 266)(91 147 174 241 267)(92 148 175 242 268)(93 149 176 243 269)(94 150 169 244 270)(95 151 170 245 271)(96 152 171 246 272)(201 231 233 290 317)(202 232 234 291 318)(203 225 235 292 319)(204 226 236 293 320)(205 227 237 294 313)(206 228 238 295 314)(207 229 239 296 315)(208 230 240 289 316)(217 252 307 273 303)(218 253 308 274 304)(219 254 309 275 297)(220 255 310 276 298)(221 256 311 277 299)(222 249 312 278 300)(223 250 305 279 301)(224 251 306 280 302)
(1 19 243 295 136 257 39 306)(2 20 244 296 129 258 40 307)(3 21 245 289 130 259 33 308)(4 22 246 290 131 260 34 309)(5 23 247 291 132 261 35 310)(6 24 248 292 133 262 36 311)(7 17 241 293 134 263 37 312)(8 18 242 294 135 264 38 305)(9 102 266 235 76 283 124 256)(10 103 267 236 77 284 125 249)(11 104 268 237 78 285 126 250)(12 97 269 238 79 286 127 251)(13 98 270 239 80 287 128 252)(14 99 271 240 73 288 121 253)(15 100 272 233 74 281 122 254)(16 101 265 234 75 282 123 255)(25 192 301 142 197 148 205 46)(26 185 302 143 198 149 206 47)(27 186 303 144 199 150 207 48)(28 187 304 137 200 151 208 41)(29 188 297 138 193 152 201 42)(30 189 298 139 194 145 202 43)(31 190 299 140 195 146 203 44)(32 191 300 141 196 147 204 45)(49 275 116 109 171 317 158 216)(50 276 117 110 172 318 159 209)(51 277 118 111 173 319 160 210)(52 278 119 112 174 320 153 211)(53 279 120 105 175 313 154 212)(54 280 113 106 176 314 155 213)(55 273 114 107 169 315 156 214)(56 274 115 108 170 316 157 215)(57 179 93 228 69 88 167 224)(58 180 94 229 70 81 168 217)(59 181 95 230 71 82 161 218)(60 182 96 231 72 83 162 219)(61 183 89 232 65 84 163 220)(62 184 90 225 66 85 164 221)(63 177 91 226 67 86 165 222)(64 178 92 227 68 87 166 223)
G:=sub<Sym(320)| (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)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,12,57,143,113)(2,13,58,144,114)(3,14,59,137,115)(4,15,60,138,116)(5,16,61,139,117)(6,9,62,140,118)(7,10,63,141,119)(8,11,64,142,120)(17,112,196,177,103)(18,105,197,178,104)(19,106,198,179,97)(20,107,199,180,98)(21,108,200,181,99)(22,109,193,182,100)(23,110,194,183,101)(24,111,195,184,102)(25,87,285,264,212)(26,88,286,257,213)(27,81,287,258,214)(28,82,288,259,215)(29,83,281,260,216)(30,84,282,261,209)(31,85,283,262,210)(32,86,284,263,211)(33,121,161,187,56)(34,122,162,188,49)(35,123,163,189,50)(36,124,164,190,51)(37,125,165,191,52)(38,126,166,192,53)(39,127,167,185,54)(40,128,168,186,55)(41,157,130,73,71)(42,158,131,74,72)(43,159,132,75,65)(44,160,133,76,66)(45,153,134,77,67)(46,154,135,78,68)(47,155,136,79,69)(48,156,129,80,70)(89,145,172,247,265)(90,146,173,248,266)(91,147,174,241,267)(92,148,175,242,268)(93,149,176,243,269)(94,150,169,244,270)(95,151,170,245,271)(96,152,171,246,272)(201,231,233,290,317)(202,232,234,291,318)(203,225,235,292,319)(204,226,236,293,320)(205,227,237,294,313)(206,228,238,295,314)(207,229,239,296,315)(208,230,240,289,316)(217,252,307,273,303)(218,253,308,274,304)(219,254,309,275,297)(220,255,310,276,298)(221,256,311,277,299)(222,249,312,278,300)(223,250,305,279,301)(224,251,306,280,302), (1,19,243,295,136,257,39,306)(2,20,244,296,129,258,40,307)(3,21,245,289,130,259,33,308)(4,22,246,290,131,260,34,309)(5,23,247,291,132,261,35,310)(6,24,248,292,133,262,36,311)(7,17,241,293,134,263,37,312)(8,18,242,294,135,264,38,305)(9,102,266,235,76,283,124,256)(10,103,267,236,77,284,125,249)(11,104,268,237,78,285,126,250)(12,97,269,238,79,286,127,251)(13,98,270,239,80,287,128,252)(14,99,271,240,73,288,121,253)(15,100,272,233,74,281,122,254)(16,101,265,234,75,282,123,255)(25,192,301,142,197,148,205,46)(26,185,302,143,198,149,206,47)(27,186,303,144,199,150,207,48)(28,187,304,137,200,151,208,41)(29,188,297,138,193,152,201,42)(30,189,298,139,194,145,202,43)(31,190,299,140,195,146,203,44)(32,191,300,141,196,147,204,45)(49,275,116,109,171,317,158,216)(50,276,117,110,172,318,159,209)(51,277,118,111,173,319,160,210)(52,278,119,112,174,320,153,211)(53,279,120,105,175,313,154,212)(54,280,113,106,176,314,155,213)(55,273,114,107,169,315,156,214)(56,274,115,108,170,316,157,215)(57,179,93,228,69,88,167,224)(58,180,94,229,70,81,168,217)(59,181,95,230,71,82,161,218)(60,182,96,231,72,83,162,219)(61,183,89,232,65,84,163,220)(62,184,90,225,66,85,164,221)(63,177,91,226,67,86,165,222)(64,178,92,227,68,87,166,223)>;
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)(241,242,243,244,245,246,247,248)(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)(281,282,283,284,285,286,287,288)(289,290,291,292,293,294,295,296)(297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312)(313,314,315,316,317,318,319,320), (1,12,57,143,113)(2,13,58,144,114)(3,14,59,137,115)(4,15,60,138,116)(5,16,61,139,117)(6,9,62,140,118)(7,10,63,141,119)(8,11,64,142,120)(17,112,196,177,103)(18,105,197,178,104)(19,106,198,179,97)(20,107,199,180,98)(21,108,200,181,99)(22,109,193,182,100)(23,110,194,183,101)(24,111,195,184,102)(25,87,285,264,212)(26,88,286,257,213)(27,81,287,258,214)(28,82,288,259,215)(29,83,281,260,216)(30,84,282,261,209)(31,85,283,262,210)(32,86,284,263,211)(33,121,161,187,56)(34,122,162,188,49)(35,123,163,189,50)(36,124,164,190,51)(37,125,165,191,52)(38,126,166,192,53)(39,127,167,185,54)(40,128,168,186,55)(41,157,130,73,71)(42,158,131,74,72)(43,159,132,75,65)(44,160,133,76,66)(45,153,134,77,67)(46,154,135,78,68)(47,155,136,79,69)(48,156,129,80,70)(89,145,172,247,265)(90,146,173,248,266)(91,147,174,241,267)(92,148,175,242,268)(93,149,176,243,269)(94,150,169,244,270)(95,151,170,245,271)(96,152,171,246,272)(201,231,233,290,317)(202,232,234,291,318)(203,225,235,292,319)(204,226,236,293,320)(205,227,237,294,313)(206,228,238,295,314)(207,229,239,296,315)(208,230,240,289,316)(217,252,307,273,303)(218,253,308,274,304)(219,254,309,275,297)(220,255,310,276,298)(221,256,311,277,299)(222,249,312,278,300)(223,250,305,279,301)(224,251,306,280,302), (1,19,243,295,136,257,39,306)(2,20,244,296,129,258,40,307)(3,21,245,289,130,259,33,308)(4,22,246,290,131,260,34,309)(5,23,247,291,132,261,35,310)(6,24,248,292,133,262,36,311)(7,17,241,293,134,263,37,312)(8,18,242,294,135,264,38,305)(9,102,266,235,76,283,124,256)(10,103,267,236,77,284,125,249)(11,104,268,237,78,285,126,250)(12,97,269,238,79,286,127,251)(13,98,270,239,80,287,128,252)(14,99,271,240,73,288,121,253)(15,100,272,233,74,281,122,254)(16,101,265,234,75,282,123,255)(25,192,301,142,197,148,205,46)(26,185,302,143,198,149,206,47)(27,186,303,144,199,150,207,48)(28,187,304,137,200,151,208,41)(29,188,297,138,193,152,201,42)(30,189,298,139,194,145,202,43)(31,190,299,140,195,146,203,44)(32,191,300,141,196,147,204,45)(49,275,116,109,171,317,158,216)(50,276,117,110,172,318,159,209)(51,277,118,111,173,319,160,210)(52,278,119,112,174,320,153,211)(53,279,120,105,175,313,154,212)(54,280,113,106,176,314,155,213)(55,273,114,107,169,315,156,214)(56,274,115,108,170,316,157,215)(57,179,93,228,69,88,167,224)(58,180,94,229,70,81,168,217)(59,181,95,230,71,82,161,218)(60,182,96,231,72,83,162,219)(61,183,89,232,65,84,163,220)(62,184,90,225,66,85,164,221)(63,177,91,226,67,86,165,222)(64,178,92,227,68,87,166,223) );
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),(241,242,243,244,245,246,247,248),(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),(281,282,283,284,285,286,287,288),(289,290,291,292,293,294,295,296),(297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312),(313,314,315,316,317,318,319,320)], [(1,12,57,143,113),(2,13,58,144,114),(3,14,59,137,115),(4,15,60,138,116),(5,16,61,139,117),(6,9,62,140,118),(7,10,63,141,119),(8,11,64,142,120),(17,112,196,177,103),(18,105,197,178,104),(19,106,198,179,97),(20,107,199,180,98),(21,108,200,181,99),(22,109,193,182,100),(23,110,194,183,101),(24,111,195,184,102),(25,87,285,264,212),(26,88,286,257,213),(27,81,287,258,214),(28,82,288,259,215),(29,83,281,260,216),(30,84,282,261,209),(31,85,283,262,210),(32,86,284,263,211),(33,121,161,187,56),(34,122,162,188,49),(35,123,163,189,50),(36,124,164,190,51),(37,125,165,191,52),(38,126,166,192,53),(39,127,167,185,54),(40,128,168,186,55),(41,157,130,73,71),(42,158,131,74,72),(43,159,132,75,65),(44,160,133,76,66),(45,153,134,77,67),(46,154,135,78,68),(47,155,136,79,69),(48,156,129,80,70),(89,145,172,247,265),(90,146,173,248,266),(91,147,174,241,267),(92,148,175,242,268),(93,149,176,243,269),(94,150,169,244,270),(95,151,170,245,271),(96,152,171,246,272),(201,231,233,290,317),(202,232,234,291,318),(203,225,235,292,319),(204,226,236,293,320),(205,227,237,294,313),(206,228,238,295,314),(207,229,239,296,315),(208,230,240,289,316),(217,252,307,273,303),(218,253,308,274,304),(219,254,309,275,297),(220,255,310,276,298),(221,256,311,277,299),(222,249,312,278,300),(223,250,305,279,301),(224,251,306,280,302)], [(1,19,243,295,136,257,39,306),(2,20,244,296,129,258,40,307),(3,21,245,289,130,259,33,308),(4,22,246,290,131,260,34,309),(5,23,247,291,132,261,35,310),(6,24,248,292,133,262,36,311),(7,17,241,293,134,263,37,312),(8,18,242,294,135,264,38,305),(9,102,266,235,76,283,124,256),(10,103,267,236,77,284,125,249),(11,104,268,237,78,285,126,250),(12,97,269,238,79,286,127,251),(13,98,270,239,80,287,128,252),(14,99,271,240,73,288,121,253),(15,100,272,233,74,281,122,254),(16,101,265,234,75,282,123,255),(25,192,301,142,197,148,205,46),(26,185,302,143,198,149,206,47),(27,186,303,144,199,150,207,48),(28,187,304,137,200,151,208,41),(29,188,297,138,193,152,201,42),(30,189,298,139,194,145,202,43),(31,190,299,140,195,146,203,44),(32,191,300,141,196,147,204,45),(49,275,116,109,171,317,158,216),(50,276,117,110,172,318,159,209),(51,277,118,111,173,319,160,210),(52,278,119,112,174,320,153,211),(53,279,120,105,175,313,154,212),(54,280,113,106,176,314,155,213),(55,273,114,107,169,315,156,214),(56,274,115,108,170,316,157,215),(57,179,93,228,69,88,167,224),(58,180,94,229,70,81,168,217),(59,181,95,230,71,82,161,218),(60,182,96,231,72,83,162,219),(61,183,89,232,65,84,163,220),(62,184,90,225,66,85,164,221),(63,177,91,226,67,86,165,222),(64,178,92,227,68,87,166,223)]])
128 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 8A | ··· | 8P | 8Q | ··· | 8AV | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
128 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D5 | D10 | Dic5 | C5⋊2C8 | C4×D5 | C8×D5 |
| kernel | C8×C5⋊2C8 | C4×C5⋊2C8 | C4×C40 | C2×C5⋊2C8 | C2×C40 | C5⋊2C8 | C40 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 32 | 16 | 2 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C8×C5⋊2C8 ►in GL3(𝔽41) generated by
| 38 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 16 |
| 14 | 0 | 0 |
| 0 | 0 | 40 |
| 0 | 32 | 0 |
G:=sub<GL(3,GF(41))| [38,0,0,0,40,0,0,0,40],[1,0,0,0,18,0,0,0,16],[14,0,0,0,0,32,0,40,0] >;
C8×C5⋊2C8 in GAP, Magma, Sage, TeX
C_8\times C_5\rtimes_2C_8
% in TeX
G:=Group("C8xC5:2C8"); // GroupNames label
G:=SmallGroup(320,11);
// by ID
G=gap.SmallGroup(320,11);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,64,100,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations