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G = (C2×C40)⋊15C4order 320 = 26·5

1st semidirect product of C2×C40 and C4 acting via C4/C2=C2

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(C408C4), C22.12(C8×D5), C23.59(C4×D5), (C2×C10).41C42, C4.21(C4⋊Dic5), (C2×C4).54Dic10, C10.14(C8⋊C4), C2.2(D101C8), 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), C54(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×C52C8)⋊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×C52C8).16C2, (C22×C10).155(C2×C4), (C2×C10).115(C22⋊C4), SmallGroup(320,108)

Series: Derived Chief Lower central Upper central

C1C10 — (C2×C40)⋊15C4
C1C5C10C2×C10C2×C20C22×C20C2×C4×Dic5 — (C2×C40)⋊15C4
C5C10 — (C2×C40)⋊15C4
C1C22×C4C22×C8

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 [×3], C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×4], C5, C8 [×4], C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], C42 [×2], C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2×C42, C22×C8, C22×C8, C52C8 [×2], C40 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C22.7C42, C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5 [×2], C2×C40 [×2], C2×C40 [×2], C22×Dic5 [×2], C22×C20, C22×C52C8, C2×C4×Dic5, C22×C40, (C2×C40)⋊15C4
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], Dic5 [×2], D10, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C22.7C42, C8×D5 [×2], C8⋊D5 [×2], C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C23.D5, C8×Dic5, C20.8Q8 [×2], C408C4, D101C8 [×2], C10.10C42, (C2×C40)⋊15C4

Smallest permutation representation of (C2×C40)⋊15C4
Regular action on 320 points
Generators in S320
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 195)(28 196)(29 197)(30 198)(31 199)(32 200)(33 161)(34 162)(35 163)(36 164)(37 165)(38 166)(39 167)(40 168)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(121 299)(122 300)(123 301)(124 302)(125 303)(126 304)(127 305)(128 306)(129 307)(130 308)(131 309)(132 310)(133 311)(134 312)(135 313)(136 314)(137 315)(138 316)(139 317)(140 318)(141 319)(142 320)(143 281)(144 282)(145 283)(146 284)(147 285)(148 286)(149 287)(150 288)(151 289)(152 290)(153 291)(154 292)(155 293)(156 294)(157 295)(158 296)(159 297)(160 298)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 241)(209 242)(210 243)(211 244)(212 245)(213 246)(214 247)(215 248)(216 249)(217 250)(218 251)(219 252)(220 253)(221 254)(222 255)(223 256)(224 257)(225 258)(226 259)(227 260)(228 261)(229 262)(230 263)(231 264)(232 265)(233 266)(234 267)(235 268)(236 269)(237 270)(238 271)(239 272)(240 273)
(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)
(1 320 233 64)(2 151 234 86)(3 298 235 42)(4 129 236 104)(5 316 237 60)(6 147 238 82)(7 294 239 78)(8 125 240 100)(9 312 201 56)(10 143 202 118)(11 290 203 74)(12 121 204 96)(13 308 205 52)(14 139 206 114)(15 286 207 70)(16 157 208 92)(17 304 209 48)(18 135 210 110)(19 282 211 66)(20 153 212 88)(21 300 213 44)(22 131 214 106)(23 318 215 62)(24 149 216 84)(25 296 217 80)(26 127 218 102)(27 314 219 58)(28 145 220 120)(29 292 221 76)(30 123 222 98)(31 310 223 54)(32 141 224 116)(33 288 225 72)(34 159 226 94)(35 306 227 50)(36 137 228 112)(37 284 229 68)(38 155 230 90)(39 302 231 46)(40 133 232 108)(41 162 297 259)(43 180 299 277)(45 198 301 255)(47 176 303 273)(49 194 305 251)(51 172 307 269)(53 190 309 247)(55 168 311 265)(57 186 313 243)(59 164 315 261)(61 182 317 279)(63 200 319 257)(65 178 281 275)(67 196 283 253)(69 174 285 271)(71 192 287 249)(73 170 289 267)(75 188 291 245)(77 166 293 263)(79 184 295 241)(81 165 146 262)(83 183 148 280)(85 161 150 258)(87 179 152 276)(89 197 154 254)(91 175 156 272)(93 193 158 250)(95 171 160 268)(97 189 122 246)(99 167 124 264)(101 185 126 242)(103 163 128 260)(105 181 130 278)(107 199 132 256)(109 177 134 274)(111 195 136 252)(113 173 138 270)(115 191 140 248)(117 169 142 266)(119 187 144 244)

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

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

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

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

dim111111112222222222222
type+++++-+-+-+
imageC1C2C2C2C4C4C4C8D4Q8D5M4(2)Dic5D10Dic10C4×D5D20C5⋊D4C4×D5C8×D5C8⋊D5
kernel(C2×C40)⋊15C4C22×C52C8C2×C4×Dic5C22×C40C2×C52C8C2×C40C22×Dic5C2×Dic5C2×C20C2×C20C22×C8C2×C10C2×C8C22×C4C2×C4C2×C4C2×C4C2×C4C23C22C22
# reps111144416312442444841616

Matrix representation of (C2×C40)⋊15C4 in GL4(𝔽41) generated by

1000
0100
00400
00040
,
27000
03200
00243
00383
,
32000
03200
00315
003538
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

׿
×
𝔽