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G = C2×C408C4order 320 = 26·5

Direct product of C2 and C408C4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C408C4, C20.56C42, (C2×C40)⋊23C4, C4042(C2×C4), (C2×C8)⋊8Dic5, C89(C2×Dic5), C104(C8⋊C4), (C2×C8).327D10, C4.21(C4×Dic5), C23.62(C4×D5), (C22×C8).15D5, (C22×C40).24C2, (C2×C10).47C42, C10.43(C2×C42), (C4×Dic5).24C4, (C2×C20).855C23, (C2×C40).414C22, C20.197(C22×C4), (C22×C4).462D10, C10.45(C2×M4(2)), (C2×C10).30M4(2), C4.32(C22×Dic5), C22.21(C4×Dic5), C22.10(C8⋊D5), (C22×Dic5).18C4, (C22×C20).558C22, (C4×Dic5).311C22, C56(C2×C8⋊C4), (C2×C52C8)⋊15C4, C4.112(C2×C4×D5), C52C833(C2×C4), C2.4(C2×C8⋊D5), C2.11(C2×C4×Dic5), C22.58(C2×C4×D5), (C2×C4).183(C4×D5), (C2×C4×Dic5).39C2, (C2×C20).426(C2×C4), (C22×C52C8).20C2, (C2×C4).103(C2×Dic5), (C2×C4).797(C22×D5), (C22×C10).158(C2×C4), (C2×C10).226(C22×C4), (C2×C52C8).326C22, (C2×Dic5).153(C2×C4), SmallGroup(320,727)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C408C4
C1C5C10C2×C10C2×C20C4×Dic5C2×C4×Dic5 — C2×C408C4
C5C10 — C2×C408C4
C1C22×C4C22×C8

Generators and relations for C2×C408C4
 G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=b29 >

Subgroups: 334 in 146 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], C23, C10, C10 [×6], C42 [×4], C2×C8 [×6], C2×C8 [×6], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C8⋊C4 [×4], C2×C42, C22×C8, C22×C8, C52C8 [×4], C40 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×C8⋊C4, C2×C52C8 [×6], C4×Dic5 [×4], C2×C40 [×6], C22×Dic5 [×2], C22×C20, C408C4 [×4], C22×C52C8, C2×C4×Dic5, C22×C40, C2×C408C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], M4(2) [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C8⋊C4 [×4], C2×C42, C2×M4(2) [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C2×C8⋊C4, C8⋊D5 [×4], C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C408C4 [×4], C2×C8⋊D5 [×2], C2×C4×Dic5, C2×C408C4

Smallest permutation representation of C2×C408C4
Regular action on 320 points
Generators in S320
(1 159)(2 160)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 127)(10 128)(11 129)(12 130)(13 131)(14 132)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 141)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 225)(42 226)(43 227)(44 228)(45 229)(46 230)(47 231)(48 232)(49 233)(50 234)(51 235)(52 236)(53 237)(54 238)(55 239)(56 240)(57 201)(58 202)(59 203)(60 204)(61 205)(62 206)(63 207)(64 208)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(73 217)(74 218)(75 219)(76 220)(77 221)(78 222)(79 223)(80 224)(81 190)(82 191)(83 192)(84 193)(85 194)(86 195)(87 196)(88 197)(89 198)(90 199)(91 200)(92 161)(93 162)(94 163)(95 164)(96 165)(97 166)(98 167)(99 168)(100 169)(101 170)(102 171)(103 172)(104 173)(105 174)(106 175)(107 176)(108 177)(109 178)(110 179)(111 180)(112 181)(113 182)(114 183)(115 184)(116 185)(117 186)(118 187)(119 188)(120 189)(241 300)(242 301)(243 302)(244 303)(245 304)(246 305)(247 306)(248 307)(249 308)(250 309)(251 310)(252 311)(253 312)(254 313)(255 314)(256 315)(257 316)(258 317)(259 318)(260 319)(261 320)(262 281)(263 282)(264 283)(265 284)(266 285)(267 286)(268 287)(269 288)(270 289)(271 290)(272 291)(273 292)(274 293)(275 294)(276 295)(277 296)(278 297)(279 298)(280 299)
(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)
(1 247 227 110)(2 276 228 99)(3 265 229 88)(4 254 230 117)(5 243 231 106)(6 272 232 95)(7 261 233 84)(8 250 234 113)(9 279 235 102)(10 268 236 91)(11 257 237 120)(12 246 238 109)(13 275 239 98)(14 264 240 87)(15 253 201 116)(16 242 202 105)(17 271 203 94)(18 260 204 83)(19 249 205 112)(20 278 206 101)(21 267 207 90)(22 256 208 119)(23 245 209 108)(24 274 210 97)(25 263 211 86)(26 252 212 115)(27 241 213 104)(28 270 214 93)(29 259 215 82)(30 248 216 111)(31 277 217 100)(32 266 218 89)(33 255 219 118)(34 244 220 107)(35 273 221 96)(36 262 222 85)(37 251 223 114)(38 280 224 103)(39 269 225 92)(40 258 226 81)(41 161 157 288)(42 190 158 317)(43 179 159 306)(44 168 160 295)(45 197 121 284)(46 186 122 313)(47 175 123 302)(48 164 124 291)(49 193 125 320)(50 182 126 309)(51 171 127 298)(52 200 128 287)(53 189 129 316)(54 178 130 305)(55 167 131 294)(56 196 132 283)(57 185 133 312)(58 174 134 301)(59 163 135 290)(60 192 136 319)(61 181 137 308)(62 170 138 297)(63 199 139 286)(64 188 140 315)(65 177 141 304)(66 166 142 293)(67 195 143 282)(68 184 144 311)(69 173 145 300)(70 162 146 289)(71 191 147 318)(72 180 148 307)(73 169 149 296)(74 198 150 285)(75 187 151 314)(76 176 152 303)(77 165 153 292)(78 194 154 281)(79 183 155 310)(80 172 156 299)

G:=sub<Sym(320)| (1,159)(2,160)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,225)(42,226)(43,227)(44,228)(45,229)(46,230)(47,231)(48,232)(49,233)(50,234)(51,235)(52,236)(53,237)(54,238)(55,239)(56,240)(57,201)(58,202)(59,203)(60,204)(61,205)(62,206)(63,207)(64,208)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,217)(74,218)(75,219)(76,220)(77,221)(78,222)(79,223)(80,224)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,161)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,168)(100,169)(101,170)(102,171)(103,172)(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)(111,180)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(241,300)(242,301)(243,302)(244,303)(245,304)(246,305)(247,306)(248,307)(249,308)(250,309)(251,310)(252,311)(253,312)(254,313)(255,314)(256,315)(257,316)(258,317)(259,318)(260,319)(261,320)(262,281)(263,282)(264,283)(265,284)(266,285)(267,286)(268,287)(269,288)(270,289)(271,290)(272,291)(273,292)(274,293)(275,294)(276,295)(277,296)(278,297)(279,298)(280,299), (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), (1,247,227,110)(2,276,228,99)(3,265,229,88)(4,254,230,117)(5,243,231,106)(6,272,232,95)(7,261,233,84)(8,250,234,113)(9,279,235,102)(10,268,236,91)(11,257,237,120)(12,246,238,109)(13,275,239,98)(14,264,240,87)(15,253,201,116)(16,242,202,105)(17,271,203,94)(18,260,204,83)(19,249,205,112)(20,278,206,101)(21,267,207,90)(22,256,208,119)(23,245,209,108)(24,274,210,97)(25,263,211,86)(26,252,212,115)(27,241,213,104)(28,270,214,93)(29,259,215,82)(30,248,216,111)(31,277,217,100)(32,266,218,89)(33,255,219,118)(34,244,220,107)(35,273,221,96)(36,262,222,85)(37,251,223,114)(38,280,224,103)(39,269,225,92)(40,258,226,81)(41,161,157,288)(42,190,158,317)(43,179,159,306)(44,168,160,295)(45,197,121,284)(46,186,122,313)(47,175,123,302)(48,164,124,291)(49,193,125,320)(50,182,126,309)(51,171,127,298)(52,200,128,287)(53,189,129,316)(54,178,130,305)(55,167,131,294)(56,196,132,283)(57,185,133,312)(58,174,134,301)(59,163,135,290)(60,192,136,319)(61,181,137,308)(62,170,138,297)(63,199,139,286)(64,188,140,315)(65,177,141,304)(66,166,142,293)(67,195,143,282)(68,184,144,311)(69,173,145,300)(70,162,146,289)(71,191,147,318)(72,180,148,307)(73,169,149,296)(74,198,150,285)(75,187,151,314)(76,176,152,303)(77,165,153,292)(78,194,154,281)(79,183,155,310)(80,172,156,299)>;

G:=Group( (1,159)(2,160)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,127)(10,128)(11,129)(12,130)(13,131)(14,132)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,141)(24,142)(25,143)(26,144)(27,145)(28,146)(29,147)(30,148)(31,149)(32,150)(33,151)(34,152)(35,153)(36,154)(37,155)(38,156)(39,157)(40,158)(41,225)(42,226)(43,227)(44,228)(45,229)(46,230)(47,231)(48,232)(49,233)(50,234)(51,235)(52,236)(53,237)(54,238)(55,239)(56,240)(57,201)(58,202)(59,203)(60,204)(61,205)(62,206)(63,207)(64,208)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,217)(74,218)(75,219)(76,220)(77,221)(78,222)(79,223)(80,224)(81,190)(82,191)(83,192)(84,193)(85,194)(86,195)(87,196)(88,197)(89,198)(90,199)(91,200)(92,161)(93,162)(94,163)(95,164)(96,165)(97,166)(98,167)(99,168)(100,169)(101,170)(102,171)(103,172)(104,173)(105,174)(106,175)(107,176)(108,177)(109,178)(110,179)(111,180)(112,181)(113,182)(114,183)(115,184)(116,185)(117,186)(118,187)(119,188)(120,189)(241,300)(242,301)(243,302)(244,303)(245,304)(246,305)(247,306)(248,307)(249,308)(250,309)(251,310)(252,311)(253,312)(254,313)(255,314)(256,315)(257,316)(258,317)(259,318)(260,319)(261,320)(262,281)(263,282)(264,283)(265,284)(266,285)(267,286)(268,287)(269,288)(270,289)(271,290)(272,291)(273,292)(274,293)(275,294)(276,295)(277,296)(278,297)(279,298)(280,299), (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), (1,247,227,110)(2,276,228,99)(3,265,229,88)(4,254,230,117)(5,243,231,106)(6,272,232,95)(7,261,233,84)(8,250,234,113)(9,279,235,102)(10,268,236,91)(11,257,237,120)(12,246,238,109)(13,275,239,98)(14,264,240,87)(15,253,201,116)(16,242,202,105)(17,271,203,94)(18,260,204,83)(19,249,205,112)(20,278,206,101)(21,267,207,90)(22,256,208,119)(23,245,209,108)(24,274,210,97)(25,263,211,86)(26,252,212,115)(27,241,213,104)(28,270,214,93)(29,259,215,82)(30,248,216,111)(31,277,217,100)(32,266,218,89)(33,255,219,118)(34,244,220,107)(35,273,221,96)(36,262,222,85)(37,251,223,114)(38,280,224,103)(39,269,225,92)(40,258,226,81)(41,161,157,288)(42,190,158,317)(43,179,159,306)(44,168,160,295)(45,197,121,284)(46,186,122,313)(47,175,123,302)(48,164,124,291)(49,193,125,320)(50,182,126,309)(51,171,127,298)(52,200,128,287)(53,189,129,316)(54,178,130,305)(55,167,131,294)(56,196,132,283)(57,185,133,312)(58,174,134,301)(59,163,135,290)(60,192,136,319)(61,181,137,308)(62,170,138,297)(63,199,139,286)(64,188,140,315)(65,177,141,304)(66,166,142,293)(67,195,143,282)(68,184,144,311)(69,173,145,300)(70,162,146,289)(71,191,147,318)(72,180,148,307)(73,169,149,296)(74,198,150,285)(75,187,151,314)(76,176,152,303)(77,165,153,292)(78,194,154,281)(79,183,155,310)(80,172,156,299) );

G=PermutationGroup([(1,159),(2,160),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,127),(10,128),(11,129),(12,130),(13,131),(14,132),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,141),(24,142),(25,143),(26,144),(27,145),(28,146),(29,147),(30,148),(31,149),(32,150),(33,151),(34,152),(35,153),(36,154),(37,155),(38,156),(39,157),(40,158),(41,225),(42,226),(43,227),(44,228),(45,229),(46,230),(47,231),(48,232),(49,233),(50,234),(51,235),(52,236),(53,237),(54,238),(55,239),(56,240),(57,201),(58,202),(59,203),(60,204),(61,205),(62,206),(63,207),(64,208),(65,209),(66,210),(67,211),(68,212),(69,213),(70,214),(71,215),(72,216),(73,217),(74,218),(75,219),(76,220),(77,221),(78,222),(79,223),(80,224),(81,190),(82,191),(83,192),(84,193),(85,194),(86,195),(87,196),(88,197),(89,198),(90,199),(91,200),(92,161),(93,162),(94,163),(95,164),(96,165),(97,166),(98,167),(99,168),(100,169),(101,170),(102,171),(103,172),(104,173),(105,174),(106,175),(107,176),(108,177),(109,178),(110,179),(111,180),(112,181),(113,182),(114,183),(115,184),(116,185),(117,186),(118,187),(119,188),(120,189),(241,300),(242,301),(243,302),(244,303),(245,304),(246,305),(247,306),(248,307),(249,308),(250,309),(251,310),(252,311),(253,312),(254,313),(255,314),(256,315),(257,316),(258,317),(259,318),(260,319),(261,320),(262,281),(263,282),(264,283),(265,284),(266,285),(267,286),(268,287),(269,288),(270,289),(271,290),(272,291),(273,292),(274,293),(275,294),(276,295),(277,296),(278,297),(279,298),(280,299)], [(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)], [(1,247,227,110),(2,276,228,99),(3,265,229,88),(4,254,230,117),(5,243,231,106),(6,272,232,95),(7,261,233,84),(8,250,234,113),(9,279,235,102),(10,268,236,91),(11,257,237,120),(12,246,238,109),(13,275,239,98),(14,264,240,87),(15,253,201,116),(16,242,202,105),(17,271,203,94),(18,260,204,83),(19,249,205,112),(20,278,206,101),(21,267,207,90),(22,256,208,119),(23,245,209,108),(24,274,210,97),(25,263,211,86),(26,252,212,115),(27,241,213,104),(28,270,214,93),(29,259,215,82),(30,248,216,111),(31,277,217,100),(32,266,218,89),(33,255,219,118),(34,244,220,107),(35,273,221,96),(36,262,222,85),(37,251,223,114),(38,280,224,103),(39,269,225,92),(40,258,226,81),(41,161,157,288),(42,190,158,317),(43,179,159,306),(44,168,160,295),(45,197,121,284),(46,186,122,313),(47,175,123,302),(48,164,124,291),(49,193,125,320),(50,182,126,309),(51,171,127,298),(52,200,128,287),(53,189,129,316),(54,178,130,305),(55,167,131,294),(56,196,132,283),(57,185,133,312),(58,174,134,301),(59,163,135,290),(60,192,136,319),(61,181,137,308),(62,170,138,297),(63,199,139,286),(64,188,140,315),(65,177,141,304),(66,166,142,293),(67,195,143,282),(68,184,144,311),(69,173,145,300),(70,162,146,289),(71,191,147,318),(72,180,148,307),(73,169,149,296),(74,198,150,285),(75,187,151,314),(76,176,152,303),(77,165,153,292),(78,194,154,281),(79,183,155,310),(80,172,156,299)])

104 conjugacy classes

class 1 2A···2G4A···4H4I···4P5A5B8A···8H8I···8P10A···10N20A···20P40A···40AF
order12···24···44···4558···88···810···1020···2040···40
size11···11···110···10222···210···102···22···22···2

104 irreducible representations

dim11111111122222222
type++++++-++
imageC1C2C2C2C2C4C4C4C4D5M4(2)Dic5D10D10C4×D5C4×D5C8⋊D5
kernelC2×C408C4C408C4C22×C52C8C2×C4×Dic5C22×C40C2×C52C8C4×Dic5C2×C40C22×Dic5C22×C8C2×C10C2×C8C2×C8C22×C4C2×C4C23C22
# reps1411184842884212432

Matrix representation of C2×C408C4 in GL5(𝔽41)

10000
01000
00100
000400
000040
,
90000
004000
09000
000034
000635
,
90000
062800
063500
0003540
000356

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,40,0,0,0,0,0,0,0,6,0,0,0,34,35],[9,0,0,0,0,0,6,6,0,0,0,28,35,0,0,0,0,0,35,35,0,0,0,40,6] >;

C2×C408C4 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes_8C_4
% in TeX

G:=Group("C2xC40:8C4");
// GroupNames label

G:=SmallGroup(320,727);
// by ID

G=gap.SmallGroup(320,727);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,100,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations

׿
×
𝔽