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## G = C42⋊4Dic5order 320 = 26·5

### 1st semidirect product of C42 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C42⋊4Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4×Dic5 — C42⋊4Dic5
 Lower central C5 — C10 — C42⋊4Dic5
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C424Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 462 in 178 conjugacy classes, 103 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C42, C22×C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C42, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C424C4, C4×Dic5, C4×C20, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4×Dic5, C2×C4×C20, C424Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, C4○D4, Dic5, D10, C2×C42, C42⋊C2, C4×D5, C2×Dic5, C22×D5, C424C4, C4×Dic5, C2×C4×D5, C4○D20, C22×Dic5, C42⋊D5, C2×C4×Dic5, C23.21D10, C424Dic5

Smallest permutation representation of C424Dic5
Regular action on 320 points
Generators in S320
(1 119 39 103)(2 120 40 104)(3 111 31 105)(4 112 32 106)(5 113 33 107)(6 114 34 108)(7 115 35 109)(8 116 36 110)(9 117 37 101)(10 118 38 102)(11 128 48 91)(12 129 49 92)(13 130 50 93)(14 121 41 94)(15 122 42 95)(16 123 43 96)(17 124 44 97)(18 125 45 98)(19 126 46 99)(20 127 47 100)(21 242 295 215)(22 243 296 216)(23 244 297 217)(24 245 298 218)(25 246 299 219)(26 247 300 220)(27 248 291 211)(28 249 292 212)(29 250 293 213)(30 241 294 214)(51 165 88 131)(52 166 89 132)(53 167 90 133)(54 168 81 134)(55 169 82 135)(56 170 83 136)(57 161 84 137)(58 162 85 138)(59 163 86 139)(60 164 87 140)(61 157 77 141)(62 158 78 142)(63 159 79 143)(64 160 80 144)(65 151 71 145)(66 152 72 146)(67 153 73 147)(68 154 74 148)(69 155 75 149)(70 156 76 150)(171 288 208 251)(172 289 209 252)(173 290 210 253)(174 281 201 254)(175 282 202 255)(176 283 203 256)(177 284 204 257)(178 285 205 258)(179 286 206 259)(180 287 207 260)(181 271 191 261)(182 272 192 262)(183 273 193 263)(184 274 194 264)(185 275 195 265)(186 276 196 266)(187 277 197 267)(188 278 198 268)(189 279 199 269)(190 280 200 270)(221 311 231 301)(222 312 232 302)(223 313 233 303)(224 314 234 304)(225 315 235 305)(226 316 236 306)(227 317 237 307)(228 318 238 308)(229 319 239 309)(230 320 240 310)
(1 51 11 63)(2 52 12 64)(3 53 13 65)(4 54 14 66)(5 55 15 67)(6 56 16 68)(7 57 17 69)(8 58 18 70)(9 59 19 61)(10 60 20 62)(21 255 316 266)(22 256 317 267)(23 257 318 268)(24 258 319 269)(25 259 320 270)(26 260 311 261)(27 251 312 262)(28 252 313 263)(29 253 314 264)(30 254 315 265)(31 90 50 71)(32 81 41 72)(33 82 42 73)(34 83 43 74)(35 84 44 75)(36 85 45 76)(37 86 46 77)(38 87 47 78)(39 88 48 79)(40 89 49 80)(91 143 103 131)(92 144 104 132)(93 145 105 133)(94 146 106 134)(95 147 107 135)(96 148 108 136)(97 149 109 137)(98 150 110 138)(99 141 101 139)(100 142 102 140)(111 167 130 151)(112 168 121 152)(113 169 122 153)(114 170 123 154)(115 161 124 155)(116 162 125 156)(117 163 126 157)(118 164 127 158)(119 165 128 159)(120 166 129 160)(171 232 182 248)(172 233 183 249)(173 234 184 250)(174 235 185 241)(175 236 186 242)(176 237 187 243)(177 238 188 244)(178 239 189 245)(179 240 190 246)(180 231 181 247)(191 220 207 221)(192 211 208 222)(193 212 209 223)(194 213 210 224)(195 214 201 225)(196 215 202 226)(197 216 203 227)(198 217 204 228)(199 218 205 229)(200 219 206 230)(271 300 287 301)(272 291 288 302)(273 292 289 303)(274 293 290 304)(275 294 281 305)(276 295 282 306)(277 296 283 307)(278 297 284 308)(279 298 285 309)(280 299 286 310)
(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)
(1 186 6 181)(2 185 7 190)(3 184 8 189)(4 183 9 188)(5 182 10 187)(11 175 16 180)(12 174 17 179)(13 173 18 178)(14 172 19 177)(15 171 20 176)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 215 56 220)(52 214 57 219)(53 213 58 218)(54 212 59 217)(55 211 60 216)(61 228 66 223)(62 227 67 222)(63 226 68 221)(64 225 69 230)(65 224 70 229)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 255 96 260)(92 254 97 259)(93 253 98 258)(94 252 99 257)(95 251 100 256)(101 268 106 263)(102 267 107 262)(103 266 108 261)(104 265 109 270)(105 264 110 269)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 295 136 300)(132 294 137 299)(133 293 138 298)(134 292 139 297)(135 291 140 296)(141 308 146 303)(142 307 147 302)(143 306 148 301)(144 305 149 310)(145 304 150 309)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4P 4Q ··· 4AF 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 D5 C4○D4 Dic5 D10 C4×D5 C4○D20 kernel C42⋊4Dic5 C10.10C42 C2×C4×Dic5 C2×C4×C20 C4×Dic5 C4×C20 C2×C42 C2×C10 C42 C22×C4 C2×C4 C22 # reps 1 4 2 1 16 8 2 8 8 6 16 32

Matrix representation of C424Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 30 13 0 0 0 19 11
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 1 0 0 0 33 7
,
 32 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 11 28 0 0 0 25 30

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,30,19,0,0,0,13,11],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,33,0,0,0,1,7],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,11,25,0,0,0,28,30] >;

C424Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_4{\rm Dic}_5
% in TeX

G:=Group("C4^2:4Dic5");
// GroupNames label

G:=SmallGroup(320,559);
// by ID

G=gap.SmallGroup(320,559);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,100,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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