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G = Dic209C4order 320 = 26·5

3rd semidirect product of Dic20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic209C4, C42.22D10, C8.7(C4×D5), C40.45(C2×C4), C2.16(C4×D20), (C2×C8).57D10, C10.43(C4×D4), C8⋊C4.2D5, C406C4.3C2, C52(Q16⋊C4), (C2×C4).116D20, (C2×C20).238D4, (C4×C20).16C22, (C2×C40).58C22, (C4×Dic10).5C2, (C2×Dic20).7C2, C22.32(C2×D20), C20.226(C4○D4), C4.110(C4○D20), (C2×C20).737C23, C20.166(C22×C4), Dic10.29(C2×C4), C2.2(C8.D10), C10.6(C8.C22), C20.44D4.16C2, C4⋊Dic5.267C22, (C2×Dic10).215C22, C4.65(C2×C4×D5), (C5×C8⋊C4).2C2, (C2×C10).120(C2×D4), (C2×C4).681(C22×D5), SmallGroup(320,343)

Series: Derived Chief Lower central Upper central

C1C20 — Dic209C4
C1C5C10C2×C10C2×C20C2×Dic10C2×Dic20 — Dic209C4
C5C10C20 — Dic209C4
C1C22C42C8⋊C4

Generators and relations for Dic209C4
 G = < a,b,c | a40=c4=1, b2=a20, bab-1=a-1, cac-1=a21, bc=cb >

Subgroups: 374 in 108 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8, C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×6], C10, C10 [×2], C42, C42 [×2], C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×2], C2×C10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8 [×2], C2×Q16, C40 [×2], C40, Dic10 [×4], Dic10 [×2], C2×Dic5 [×4], C2×C20, C2×C20 [×2], Q16⋊C4, Dic20 [×4], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×Dic10 [×2], C20.44D4 [×2], C406C4, C5×C8⋊C4, C4×Dic10 [×2], C2×Dic20, Dic209C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8.C22 [×2], C4×D5 [×2], D20 [×2], C22×D5, Q16⋊C4, C2×C4×D5, C2×D20, C4○D20, C4×D20, C8.D10 [×2], Dic209C4

Smallest permutation representation of Dic209C4
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)
(1 176 21 196)(2 175 22 195)(3 174 23 194)(4 173 24 193)(5 172 25 192)(6 171 26 191)(7 170 27 190)(8 169 28 189)(9 168 29 188)(10 167 30 187)(11 166 31 186)(12 165 32 185)(13 164 33 184)(14 163 34 183)(15 162 35 182)(16 161 36 181)(17 200 37 180)(18 199 38 179)(19 198 39 178)(20 197 40 177)(41 297 61 317)(42 296 62 316)(43 295 63 315)(44 294 64 314)(45 293 65 313)(46 292 66 312)(47 291 67 311)(48 290 68 310)(49 289 69 309)(50 288 70 308)(51 287 71 307)(52 286 72 306)(53 285 73 305)(54 284 74 304)(55 283 75 303)(56 282 76 302)(57 281 77 301)(58 320 78 300)(59 319 79 299)(60 318 80 298)(81 138 101 158)(82 137 102 157)(83 136 103 156)(84 135 104 155)(85 134 105 154)(86 133 106 153)(87 132 107 152)(88 131 108 151)(89 130 109 150)(90 129 110 149)(91 128 111 148)(92 127 112 147)(93 126 113 146)(94 125 114 145)(95 124 115 144)(96 123 116 143)(97 122 117 142)(98 121 118 141)(99 160 119 140)(100 159 120 139)(201 260 221 280)(202 259 222 279)(203 258 223 278)(204 257 224 277)(205 256 225 276)(206 255 226 275)(207 254 227 274)(208 253 228 273)(209 252 229 272)(210 251 230 271)(211 250 231 270)(212 249 232 269)(213 248 233 268)(214 247 234 267)(215 246 235 266)(216 245 236 265)(217 244 237 264)(218 243 238 263)(219 242 239 262)(220 241 240 261)
(1 133 221 43)(2 154 222 64)(3 135 223 45)(4 156 224 66)(5 137 225 47)(6 158 226 68)(7 139 227 49)(8 160 228 70)(9 141 229 51)(10 122 230 72)(11 143 231 53)(12 124 232 74)(13 145 233 55)(14 126 234 76)(15 147 235 57)(16 128 236 78)(17 149 237 59)(18 130 238 80)(19 151 239 61)(20 132 240 42)(21 153 201 63)(22 134 202 44)(23 155 203 65)(24 136 204 46)(25 157 205 67)(26 138 206 48)(27 159 207 69)(28 140 208 50)(29 121 209 71)(30 142 210 52)(31 123 211 73)(32 144 212 54)(33 125 213 75)(34 146 214 56)(35 127 215 77)(36 148 216 58)(37 129 217 79)(38 150 218 60)(39 131 219 41)(40 152 220 62)(81 275 310 171)(82 256 311 192)(83 277 312 173)(84 258 313 194)(85 279 314 175)(86 260 315 196)(87 241 316 177)(88 262 317 198)(89 243 318 179)(90 264 319 200)(91 245 320 181)(92 266 281 162)(93 247 282 183)(94 268 283 164)(95 249 284 185)(96 270 285 166)(97 251 286 187)(98 272 287 168)(99 253 288 189)(100 274 289 170)(101 255 290 191)(102 276 291 172)(103 257 292 193)(104 278 293 174)(105 259 294 195)(106 280 295 176)(107 261 296 197)(108 242 297 178)(109 263 298 199)(110 244 299 180)(111 265 300 161)(112 246 301 182)(113 267 302 163)(114 248 303 184)(115 269 304 165)(116 250 305 186)(117 271 306 167)(118 252 307 188)(119 273 308 169)(120 254 309 190)

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

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

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

62 conjugacy classes

class 1 2A2B2C4A···4F4G···4N5A5B8A8B8C8D10A···10F20A···20H20I···20P40A···40P
order12224···44···455888810···1020···2020···2040···40
size11112···220···202244442···22···24···44···4

62 irreducible representations

dim11111112222222244
type+++++++++++--
imageC1C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5D20C4○D20C8.C22C8.D10
kernelDic209C4C20.44D4C406C4C5×C8⋊C4C4×Dic10C2×Dic20Dic20C2×C20C8⋊C4C20C42C2×C8C8C2×C4C4C10C2
# reps12112182222488828

Matrix representation of Dic209C4 in GL6(𝔽41)

1230000
32400000
003219342
0022223920
00342922
0039201919
,
16270000
27250000
002723512
001814136
00629272
0028351814
,
3200000
0320000
000010
000001
0040000
0004000

G:=sub<GL(6,GF(41))| [1,32,0,0,0,0,23,40,0,0,0,0,0,0,32,22,34,39,0,0,19,22,2,20,0,0,34,39,9,19,0,0,2,20,22,19],[16,27,0,0,0,0,27,25,0,0,0,0,0,0,27,18,6,28,0,0,2,14,29,35,0,0,35,13,27,18,0,0,12,6,2,14],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;

Dic209C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{20}\rtimes_9C_4
% in TeX

G:=Group("Dic20:9C4");
// GroupNames label

G:=SmallGroup(320,343);
// by ID

G=gap.SmallGroup(320,343);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,387,58,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=c^4=1,b^2=a^20,b*a*b^-1=a^-1,c*a*c^-1=a^21,b*c=c*b>;
// generators/relations

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