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## G = C4⋊Dic20order 320 = 26·5

### The semidirect product of C4 and Dic20 acting via Dic20/Dic10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4⋊Dic20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C4×Dic10 — C4⋊Dic20
 Lower central C5 — C10 — C2×C20 — C4⋊Dic20
 Upper central C1 — C22 — C42 — C4⋊C8

Generators and relations for C4⋊Dic20
G = < a,b,c | a4=b40=1, c2=b20, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 422 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C42, C4⋊C4, C2×C8, Q16, C2×Q8, Dic5, C20, C20, C20, C2×C10, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C40, Dic10, Dic10, C2×Dic5, C2×C20, C42Q16, Dic20, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C4×C20, C2×C40, C2×Dic10, C2×Dic10, C20.44D4, C5×C4⋊C8, C4×Dic10, C202Q8, C2×Dic20, C4⋊Dic20
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C4⋊D4, C2×Q16, C8.C22, D20, C22×D5, C42Q16, Dic20, C2×D20, D4×D5, Q82D5, C4⋊D20, C2×Dic20, C8.D10, C4⋊Dic20

Smallest permutation representation of C4⋊Dic20
Regular action on 320 points
Generators in S320
(1 194 134 315)(2 316 135 195)(3 196 136 317)(4 318 137 197)(5 198 138 319)(6 320 139 199)(7 200 140 281)(8 282 141 161)(9 162 142 283)(10 284 143 163)(11 164 144 285)(12 286 145 165)(13 166 146 287)(14 288 147 167)(15 168 148 289)(16 290 149 169)(17 170 150 291)(18 292 151 171)(19 172 152 293)(20 294 153 173)(21 174 154 295)(22 296 155 175)(23 176 156 297)(24 298 157 177)(25 178 158 299)(26 300 159 179)(27 180 160 301)(28 302 121 181)(29 182 122 303)(30 304 123 183)(31 184 124 305)(32 306 125 185)(33 186 126 307)(34 308 127 187)(35 188 128 309)(36 310 129 189)(37 190 130 311)(38 312 131 191)(39 192 132 313)(40 314 133 193)(41 280 96 236)(42 237 97 241)(43 242 98 238)(44 239 99 243)(45 244 100 240)(46 201 101 245)(47 246 102 202)(48 203 103 247)(49 248 104 204)(50 205 105 249)(51 250 106 206)(52 207 107 251)(53 252 108 208)(54 209 109 253)(55 254 110 210)(56 211 111 255)(57 256 112 212)(58 213 113 257)(59 258 114 214)(60 215 115 259)(61 260 116 216)(62 217 117 261)(63 262 118 218)(64 219 119 263)(65 264 120 220)(66 221 81 265)(67 266 82 222)(68 223 83 267)(69 268 84 224)(70 225 85 269)(71 270 86 226)(72 227 87 271)(73 272 88 228)(74 229 89 273)(75 274 90 230)(76 231 91 275)(77 276 92 232)(78 233 93 277)(79 278 94 234)(80 235 95 279)
(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)
(1 275 21 255)(2 274 22 254)(3 273 23 253)(4 272 24 252)(5 271 25 251)(6 270 26 250)(7 269 27 249)(8 268 28 248)(9 267 29 247)(10 266 30 246)(11 265 31 245)(12 264 32 244)(13 263 33 243)(14 262 34 242)(15 261 35 241)(16 260 36 280)(17 259 37 279)(18 258 38 278)(19 257 39 277)(20 256 40 276)(41 169 61 189)(42 168 62 188)(43 167 63 187)(44 166 64 186)(45 165 65 185)(46 164 66 184)(47 163 67 183)(48 162 68 182)(49 161 69 181)(50 200 70 180)(51 199 71 179)(52 198 72 178)(53 197 73 177)(54 196 74 176)(55 195 75 175)(56 194 76 174)(57 193 77 173)(58 192 78 172)(59 191 79 171)(60 190 80 170)(81 305 101 285)(82 304 102 284)(83 303 103 283)(84 302 104 282)(85 301 105 281)(86 300 106 320)(87 299 107 319)(88 298 108 318)(89 297 109 317)(90 296 110 316)(91 295 111 315)(92 294 112 314)(93 293 113 313)(94 292 114 312)(95 291 115 311)(96 290 116 310)(97 289 117 309)(98 288 118 308)(99 287 119 307)(100 286 120 306)(121 204 141 224)(122 203 142 223)(123 202 143 222)(124 201 144 221)(125 240 145 220)(126 239 146 219)(127 238 147 218)(128 237 148 217)(129 236 149 216)(130 235 150 215)(131 234 151 214)(132 233 152 213)(133 232 153 212)(134 231 154 211)(135 230 155 210)(136 229 156 209)(137 228 157 208)(138 227 158 207)(139 226 159 206)(140 225 160 205)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20P 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 2 2 4 20 20 20 20 40 40 2 2 4 4 4 4 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - - + + - image C1 C2 C2 C2 C2 C2 D4 D4 D5 Q16 C4○D4 D10 D10 D20 Dic20 C8.C22 D4×D5 Q8⋊2D5 C8.D10 kernel C4⋊Dic20 C20.44D4 C5×C4⋊C8 C4×Dic10 C20⋊2Q8 C2×Dic20 Dic10 C2×C20 C4⋊C8 C20 C20 C42 C2×C8 C2×C4 C4 C10 C4 C4 C2 # reps 1 2 1 1 1 2 2 2 2 4 2 2 4 8 16 1 2 2 4

Matrix representation of C4⋊Dic20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 3 25 0 0 0 0 16 38 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 1 0 0 0 0 33 7 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 29 12 0 0 0 0 29 29
,
 32 25 0 0 0 0 5 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 27 34 0 0 0 0 34 14

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,16,0,0,0,0,25,38,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,33,0,0,0,0,1,7,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,29,29,0,0,0,0,12,29],[32,5,0,0,0,0,25,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,34,0,0,0,0,34,14] >;

C4⋊Dic20 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_{20}
% in TeX

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

G:=SmallGroup(320,476);
// by ID

G=gap.SmallGroup(320,476);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,120,254,219,226,1123,136,12550]);
// Polycyclic

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

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