metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C40)⋊15C4, (C2×C8)⋊3Dic5, C10.20(C4×C8), (C2×Dic5)⋊3C8, C10.19(C4⋊C8), C20.74(C4⋊C4), (C2×C20).64Q8, C2.5(C8×Dic5), (C22×C8).2D5, (C2×C4).164D20, (C2×C20).491D4, (C22×C40).1C2, C2.3(C40⋊8C4), C22.12(C8×D5), C23.59(C4×D5), (C2×C10).41C42, C4.21(C4⋊Dic5), (C2×C4).54Dic10, C10.14(C8⋊C4), C2.2(D10⋊1C8), C10.26(C22⋊C8), (C22×C4).452D10, C22.8(C8⋊D5), (C2×C10).28M4(2), C4.25(C23.D5), C2.2(C20.8Q8), C4.48(D10⋊C4), C22.17(C4×Dic5), C20.109(C22⋊C4), C4.31(C10.D4), C5⋊4(C22.7C42), (C22×Dic5).12C4, (C22×C20).549C22, C22.39(D10⋊C4), C2.2(C10.10C42), C10.29(C2.C42), C22.21(C10.D4), (C2×C5⋊2C8)⋊13C4, (C2×C10).41(C2×C8), (C2×C4).171(C4×D5), (C2×C4×Dic5).18C2, (C2×C10).65(C4⋊C4), (C2×C20).486(C2×C4), (C2×C4).94(C2×Dic5), (C2×C4).269(C5⋊D4), (C22×C5⋊2C8).16C2, (C22×C10).155(C2×C4), (C2×C10).115(C22⋊C4), SmallGroup(320,108)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C40)⋊15C4
G = < a,b,c | a2=b40=c4=1, ab=ba, ac=ca, cbc-1=ab9 >
Subgroups: 310 in 118 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×C42, C22×C8, C22×C8, C5⋊2C8, C40, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22.7C42, C2×C5⋊2C8, C2×C5⋊2C8, C4×Dic5, C2×C40, C2×C40, C22×Dic5, C22×C20, C22×C5⋊2C8, C2×C4×Dic5, C22×C40, (C2×C40)⋊15C4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), Dic5, D10, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C22.7C42, C8×D5, C8⋊D5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C8×Dic5, C20.8Q8, C40⋊8C4, D10⋊1C8, C10.10C42, (C2×C40)⋊15C4
►Regular action on 320 points
Generators in S320
(1 312)(2 313)(3 314)(4 315)(5 316)(6 317)(7 318)(8 319)(9 320)(10 281)(11 282)(12 283)(13 284)(14 285)(15 286)(16 287)(17 288)(18 289)(19 290)(20 291)(21 292)(22 293)(23 294)(24 295)(25 296)(26 297)(27 298)(28 299)(29 300)(30 301)(31 302)(32 303)(33 304)(34 305)(35 306)(36 307)(37 308)(38 309)(39 310)(40 311)(41 247)(42 248)(43 249)(44 250)(45 251)(46 252)(47 253)(48 254)(49 255)(50 256)(51 257)(52 258)(53 259)(54 260)(55 261)(56 262)(57 263)(58 264)(59 265)(60 266)(61 267)(62 268)(63 269)(64 270)(65 271)(66 272)(67 273)(68 274)(69 275)(70 276)(71 277)(72 278)(73 279)(74 280)(75 241)(76 242)(77 243)(78 244)(79 245)(80 246)(81 127)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 145)(100 146)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)(161 230)(162 231)(163 232)(164 233)(165 234)(166 235)(167 236)(168 237)(169 238)(170 239)(171 240)(172 201)(173 202)(174 203)(175 204)(176 205)(177 206)(178 207)(179 208)(180 209)(181 210)(182 211)(183 212)(184 213)(185 214)(186 215)(187 216)(188 217)(189 218)(190 219)(191 220)(192 221)(193 222)(194 223)(195 224)(196 225)(197 226)(198 227)(199 228)(200 229)
(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)
(1 143 52 185)(2 106 53 223)(3 121 54 163)(4 84 55 201)(5 139 56 181)(6 102 57 219)(7 157 58 199)(8 120 59 237)(9 135 60 177)(10 98 61 215)(11 153 62 195)(12 116 63 233)(13 131 64 173)(14 94 65 211)(15 149 66 191)(16 112 67 229)(17 127 68 169)(18 90 69 207)(19 145 70 187)(20 108 71 225)(21 123 72 165)(22 86 73 203)(23 141 74 183)(24 104 75 221)(25 159 76 161)(26 82 77 239)(27 137 78 179)(28 100 79 217)(29 155 80 197)(30 118 41 235)(31 133 42 175)(32 96 43 213)(33 151 44 193)(34 114 45 231)(35 129 46 171)(36 92 47 209)(37 147 48 189)(38 110 49 227)(39 125 50 167)(40 88 51 205)(81 274 238 288)(83 252 240 306)(85 270 202 284)(87 248 204 302)(89 266 206 320)(91 244 208 298)(93 262 210 316)(95 280 212 294)(97 258 214 312)(99 276 216 290)(101 254 218 308)(103 272 220 286)(105 250 222 304)(107 268 224 282)(109 246 226 300)(111 264 228 318)(113 242 230 296)(115 260 232 314)(117 278 234 292)(119 256 236 310)(122 269 164 283)(124 247 166 301)(126 265 168 319)(128 243 170 297)(130 261 172 315)(132 279 174 293)(134 257 176 311)(136 275 178 289)(138 253 180 307)(140 271 182 285)(142 249 184 303)(144 267 186 281)(146 245 188 299)(148 263 190 317)(150 241 192 295)(152 259 194 313)(154 277 196 291)(156 255 198 309)(158 273 200 287)(160 251 162 305)
G:=sub<Sym(320)| (1,312)(2,313)(3,314)(4,315)(5,316)(6,317)(7,318)(8,319)(9,320)(10,281)(11,282)(12,283)(13,284)(14,285)(15,286)(16,287)(17,288)(18,289)(19,290)(20,291)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,301)(31,302)(32,303)(33,304)(34,305)(35,306)(36,307)(37,308)(38,309)(39,310)(40,311)(41,247)(42,248)(43,249)(44,250)(45,251)(46,252)(47,253)(48,254)(49,255)(50,256)(51,257)(52,258)(53,259)(54,260)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(61,267)(62,268)(63,269)(64,270)(65,271)(66,272)(67,273)(68,274)(69,275)(70,276)(71,277)(72,278)(73,279)(74,280)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(161,230)(162,231)(163,232)(164,233)(165,234)(166,235)(167,236)(168,237)(169,238)(170,239)(171,240)(172,201)(173,202)(174,203)(175,204)(176,205)(177,206)(178,207)(179,208)(180,209)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229), (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), (1,143,52,185)(2,106,53,223)(3,121,54,163)(4,84,55,201)(5,139,56,181)(6,102,57,219)(7,157,58,199)(8,120,59,237)(9,135,60,177)(10,98,61,215)(11,153,62,195)(12,116,63,233)(13,131,64,173)(14,94,65,211)(15,149,66,191)(16,112,67,229)(17,127,68,169)(18,90,69,207)(19,145,70,187)(20,108,71,225)(21,123,72,165)(22,86,73,203)(23,141,74,183)(24,104,75,221)(25,159,76,161)(26,82,77,239)(27,137,78,179)(28,100,79,217)(29,155,80,197)(30,118,41,235)(31,133,42,175)(32,96,43,213)(33,151,44,193)(34,114,45,231)(35,129,46,171)(36,92,47,209)(37,147,48,189)(38,110,49,227)(39,125,50,167)(40,88,51,205)(81,274,238,288)(83,252,240,306)(85,270,202,284)(87,248,204,302)(89,266,206,320)(91,244,208,298)(93,262,210,316)(95,280,212,294)(97,258,214,312)(99,276,216,290)(101,254,218,308)(103,272,220,286)(105,250,222,304)(107,268,224,282)(109,246,226,300)(111,264,228,318)(113,242,230,296)(115,260,232,314)(117,278,234,292)(119,256,236,310)(122,269,164,283)(124,247,166,301)(126,265,168,319)(128,243,170,297)(130,261,172,315)(132,279,174,293)(134,257,176,311)(136,275,178,289)(138,253,180,307)(140,271,182,285)(142,249,184,303)(144,267,186,281)(146,245,188,299)(148,263,190,317)(150,241,192,295)(152,259,194,313)(154,277,196,291)(156,255,198,309)(158,273,200,287)(160,251,162,305)>;
G:=Group( (1,312)(2,313)(3,314)(4,315)(5,316)(6,317)(7,318)(8,319)(9,320)(10,281)(11,282)(12,283)(13,284)(14,285)(15,286)(16,287)(17,288)(18,289)(19,290)(20,291)(21,292)(22,293)(23,294)(24,295)(25,296)(26,297)(27,298)(28,299)(29,300)(30,301)(31,302)(32,303)(33,304)(34,305)(35,306)(36,307)(37,308)(38,309)(39,310)(40,311)(41,247)(42,248)(43,249)(44,250)(45,251)(46,252)(47,253)(48,254)(49,255)(50,256)(51,257)(52,258)(53,259)(54,260)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(61,267)(62,268)(63,269)(64,270)(65,271)(66,272)(67,273)(68,274)(69,275)(70,276)(71,277)(72,278)(73,279)(74,280)(75,241)(76,242)(77,243)(78,244)(79,245)(80,246)(81,127)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(96,142)(97,143)(98,144)(99,145)(100,146)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)(161,230)(162,231)(163,232)(164,233)(165,234)(166,235)(167,236)(168,237)(169,238)(170,239)(171,240)(172,201)(173,202)(174,203)(175,204)(176,205)(177,206)(178,207)(179,208)(180,209)(181,210)(182,211)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)(193,222)(194,223)(195,224)(196,225)(197,226)(198,227)(199,228)(200,229), (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), (1,143,52,185)(2,106,53,223)(3,121,54,163)(4,84,55,201)(5,139,56,181)(6,102,57,219)(7,157,58,199)(8,120,59,237)(9,135,60,177)(10,98,61,215)(11,153,62,195)(12,116,63,233)(13,131,64,173)(14,94,65,211)(15,149,66,191)(16,112,67,229)(17,127,68,169)(18,90,69,207)(19,145,70,187)(20,108,71,225)(21,123,72,165)(22,86,73,203)(23,141,74,183)(24,104,75,221)(25,159,76,161)(26,82,77,239)(27,137,78,179)(28,100,79,217)(29,155,80,197)(30,118,41,235)(31,133,42,175)(32,96,43,213)(33,151,44,193)(34,114,45,231)(35,129,46,171)(36,92,47,209)(37,147,48,189)(38,110,49,227)(39,125,50,167)(40,88,51,205)(81,274,238,288)(83,252,240,306)(85,270,202,284)(87,248,204,302)(89,266,206,320)(91,244,208,298)(93,262,210,316)(95,280,212,294)(97,258,214,312)(99,276,216,290)(101,254,218,308)(103,272,220,286)(105,250,222,304)(107,268,224,282)(109,246,226,300)(111,264,228,318)(113,242,230,296)(115,260,232,314)(117,278,234,292)(119,256,236,310)(122,269,164,283)(124,247,166,301)(126,265,168,319)(128,243,170,297)(130,261,172,315)(132,279,174,293)(134,257,176,311)(136,275,178,289)(138,253,180,307)(140,271,182,285)(142,249,184,303)(144,267,186,281)(146,245,188,299)(148,263,190,317)(150,241,192,295)(152,259,194,313)(154,277,196,291)(156,255,198,309)(158,273,200,287)(160,251,162,305) );
G=PermutationGroup([[(1,312),(2,313),(3,314),(4,315),(5,316),(6,317),(7,318),(8,319),(9,320),(10,281),(11,282),(12,283),(13,284),(14,285),(15,286),(16,287),(17,288),(18,289),(19,290),(20,291),(21,292),(22,293),(23,294),(24,295),(25,296),(26,297),(27,298),(28,299),(29,300),(30,301),(31,302),(32,303),(33,304),(34,305),(35,306),(36,307),(37,308),(38,309),(39,310),(40,311),(41,247),(42,248),(43,249),(44,250),(45,251),(46,252),(47,253),(48,254),(49,255),(50,256),(51,257),(52,258),(53,259),(54,260),(55,261),(56,262),(57,263),(58,264),(59,265),(60,266),(61,267),(62,268),(63,269),(64,270),(65,271),(66,272),(67,273),(68,274),(69,275),(70,276),(71,277),(72,278),(73,279),(74,280),(75,241),(76,242),(77,243),(78,244),(79,245),(80,246),(81,127),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(96,142),(97,143),(98,144),(99,145),(100,146),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126),(161,230),(162,231),(163,232),(164,233),(165,234),(166,235),(167,236),(168,237),(169,238),(170,239),(171,240),(172,201),(173,202),(174,203),(175,204),(176,205),(177,206),(178,207),(179,208),(180,209),(181,210),(182,211),(183,212),(184,213),(185,214),(186,215),(187,216),(188,217),(189,218),(190,219),(191,220),(192,221),(193,222),(194,223),(195,224),(196,225),(197,226),(198,227),(199,228),(200,229)], [(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)], [(1,143,52,185),(2,106,53,223),(3,121,54,163),(4,84,55,201),(5,139,56,181),(6,102,57,219),(7,157,58,199),(8,120,59,237),(9,135,60,177),(10,98,61,215),(11,153,62,195),(12,116,63,233),(13,131,64,173),(14,94,65,211),(15,149,66,191),(16,112,67,229),(17,127,68,169),(18,90,69,207),(19,145,70,187),(20,108,71,225),(21,123,72,165),(22,86,73,203),(23,141,74,183),(24,104,75,221),(25,159,76,161),(26,82,77,239),(27,137,78,179),(28,100,79,217),(29,155,80,197),(30,118,41,235),(31,133,42,175),(32,96,43,213),(33,151,44,193),(34,114,45,231),(35,129,46,171),(36,92,47,209),(37,147,48,189),(38,110,49,227),(39,125,50,167),(40,88,51,205),(81,274,238,288),(83,252,240,306),(85,270,202,284),(87,248,204,302),(89,266,206,320),(91,244,208,298),(93,262,210,316),(95,280,212,294),(97,258,214,312),(99,276,216,290),(101,254,218,308),(103,272,220,286),(105,250,222,304),(107,268,224,282),(109,246,226,300),(111,264,228,318),(113,242,230,296),(115,260,232,314),(117,278,234,292),(119,256,236,310),(122,269,164,283),(124,247,166,301),(126,265,168,319),(128,243,170,297),(130,261,172,315),(132,279,174,293),(134,257,176,311),(136,275,178,289),(138,253,180,307),(140,271,182,285),(142,249,184,303),(144,267,186,281),(146,245,188,299),(148,263,190,317),(150,241,192,295),(152,259,194,313),(154,277,196,291),(156,255,198,309),(158,273,200,287),(160,251,162,305)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | D4 | Q8 | D5 | M4(2) | Dic5 | D10 | Dic10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C8×D5 | C8⋊D5 |
| kernel | (C2×C40)⋊15C4 | C22×C5⋊2C8 | C2×C4×Dic5 | C22×C40 | C2×C5⋊2C8 | C2×C40 | C22×Dic5 | C2×Dic5 | C2×C20 | C2×C20 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 16 | 3 | 1 | 2 | 4 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 16 | 16 |
Matrix representation of (C2×C40)⋊15C4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 27 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 24 | 3 |
| 0 | 0 | 38 | 3 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 3 | 15 |
| 0 | 0 | 35 | 38 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[27,0,0,0,0,32,0,0,0,0,24,38,0,0,3,3],[32,0,0,0,0,32,0,0,0,0,3,35,0,0,15,38] >;
(C2×C40)⋊15C4 in GAP, Magma, Sage, TeX
(C_2\times C_{40})\rtimes_{15}C_4 % in TeX
G:=Group("(C2xC40):15C4"); // GroupNames label
G:=SmallGroup(320,108);
// by ID
G=gap.SmallGroup(320,108);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,12550]);
// 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=a*b^9>;
// generators/relations