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## G = C2×C40⋊8C4order 320 = 26·5

### Direct product of C2 and C40⋊8C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C40⋊8C4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — C2×C40⋊8C4
 Lower central C5 — C10 — C2×C40⋊8C4
 Upper central C1 — C22×C4 — C22×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, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C8⋊C4, C2×C42, C22×C8, C22×C8, C52C8, C40, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C8⋊C4, C2×C52C8, C4×Dic5, C2×C40, C22×Dic5, C22×C20, C408C4, C22×C52C8, C2×C4×Dic5, C22×C40, C2×C408C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, Dic5, D10, C8⋊C4, C2×C42, C2×M4(2), C4×D5, C2×Dic5, C22×D5, C2×C8⋊C4, C8⋊D5, C4×Dic5, C2×C4×D5, C22×Dic5, C408C4, C2×C8⋊D5, C2×C4×Dic5, C2×C408C4

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

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 1 ··· 1 10 ··· 10 2 2 2 ··· 2 10 ··· 10 2 ··· 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 D5 M4(2) Dic5 D10 D10 C4×D5 C4×D5 C8⋊D5 kernel C2×C40⋊8C4 C40⋊8C4 C22×C5⋊2C8 C2×C4×Dic5 C22×C40 C2×C5⋊2C8 C4×Dic5 C2×C40 C22×Dic5 C22×C8 C2×C10 C2×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 4 1 1 1 8 4 8 4 2 8 8 4 2 12 4 32

Matrix representation of C2×C408C4 in GL5(𝔽41)

 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 40 0 0 0 9 0 0 0 0 0 0 0 34 0 0 0 6 35
,
 9 0 0 0 0 0 6 28 0 0 0 6 35 0 0 0 0 0 35 40 0 0 0 35 6

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

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