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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 [×3], C4 [×2], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], Q8 [×7], C10 [×3], C42, C42, C4⋊C4 [×4], C2×C8 [×2], Q16 [×4], C2×Q8 [×3], Dic5 [×5], C20 [×2], C20 [×2], C20, C2×C10, Q8⋊C4 [×2], C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16 [×2], C40 [×2], Dic10 [×2], Dic10 [×5], C2×Dic5 [×4], C2×C20 [×3], C42Q16, Dic20 [×4], C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5 [×2], C4×C20, C2×C40 [×2], C2×Dic10, C2×Dic10 [×2], C20.44D4 [×2], C5×C4⋊C8, C4×Dic10, C202Q8, C2×Dic20 [×2], C4⋊Dic20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, Q16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×Q16, C8.C22, D20 [×2], C22×D5, C42Q16, Dic20 [×2], 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
(1 258 198 236)(2 237 199 259)(3 260 200 238)(4 239 161 261)(5 262 162 240)(6 201 163 263)(7 264 164 202)(8 203 165 265)(9 266 166 204)(10 205 167 267)(11 268 168 206)(12 207 169 269)(13 270 170 208)(14 209 171 271)(15 272 172 210)(16 211 173 273)(17 274 174 212)(18 213 175 275)(19 276 176 214)(20 215 177 277)(21 278 178 216)(22 217 179 279)(23 280 180 218)(24 219 181 241)(25 242 182 220)(26 221 183 243)(27 244 184 222)(28 223 185 245)(29 246 186 224)(30 225 187 247)(31 248 188 226)(32 227 189 249)(33 250 190 228)(34 229 191 251)(35 252 192 230)(36 231 193 253)(37 254 194 232)(38 233 195 255)(39 256 196 234)(40 235 197 257)(41 297 98 126)(42 127 99 298)(43 299 100 128)(44 129 101 300)(45 301 102 130)(46 131 103 302)(47 303 104 132)(48 133 105 304)(49 305 106 134)(50 135 107 306)(51 307 108 136)(52 137 109 308)(53 309 110 138)(54 139 111 310)(55 311 112 140)(56 141 113 312)(57 313 114 142)(58 143 115 314)(59 315 116 144)(60 145 117 316)(61 317 118 146)(62 147 119 318)(63 319 120 148)(64 149 81 320)(65 281 82 150)(66 151 83 282)(67 283 84 152)(68 153 85 284)(69 285 86 154)(70 155 87 286)(71 287 88 156)(72 157 89 288)(73 289 90 158)(74 159 91 290)(75 291 92 160)(76 121 93 292)(77 293 94 122)(78 123 95 294)(79 295 96 124)(80 125 97 296)
(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)
(1 127 21 147)(2 126 22 146)(3 125 23 145)(4 124 24 144)(5 123 25 143)(6 122 26 142)(7 121 27 141)(8 160 28 140)(9 159 29 139)(10 158 30 138)(11 157 31 137)(12 156 32 136)(13 155 33 135)(14 154 34 134)(15 153 35 133)(16 152 36 132)(17 151 37 131)(18 150 38 130)(19 149 39 129)(20 148 40 128)(41 217 61 237)(42 216 62 236)(43 215 63 235)(44 214 64 234)(45 213 65 233)(46 212 66 232)(47 211 67 231)(48 210 68 230)(49 209 69 229)(50 208 70 228)(51 207 71 227)(52 206 72 226)(53 205 73 225)(54 204 74 224)(55 203 75 223)(56 202 76 222)(57 201 77 221)(58 240 78 220)(59 239 79 219)(60 238 80 218)(81 256 101 276)(82 255 102 275)(83 254 103 274)(84 253 104 273)(85 252 105 272)(86 251 106 271)(87 250 107 270)(88 249 108 269)(89 248 109 268)(90 247 110 267)(91 246 111 266)(92 245 112 265)(93 244 113 264)(94 243 114 263)(95 242 115 262)(96 241 116 261)(97 280 117 260)(98 279 118 259)(99 278 119 258)(100 277 120 257)(161 295 181 315)(162 294 182 314)(163 293 183 313)(164 292 184 312)(165 291 185 311)(166 290 186 310)(167 289 187 309)(168 288 188 308)(169 287 189 307)(170 286 190 306)(171 285 191 305)(172 284 192 304)(173 283 193 303)(174 282 194 302)(175 281 195 301)(176 320 196 300)(177 319 197 299)(178 318 198 298)(179 317 199 297)(180 316 200 296)

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

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

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

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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