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## G = C42⋊8Dic5order 320 = 26·5

### 5th semidirect product of C42 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊8Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — C42⋊8Dic5
 Lower central C5 — C2×C10 — C42⋊8Dic5
 Upper central C1 — C23 — C2×C42

Generators and relations for C428Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 558 in 186 conjugacy classes, 119 normal (11 characteristic)
C1, C2, C2 [×6], C4 [×12], C4 [×4], C22, C22 [×6], C5, C2×C4 [×18], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], C20 [×12], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×6], C2×Dic5 [×12], C2×C20 [×18], C22×C10, C429C4, C4⋊Dic5 [×12], C4×C20 [×4], C22×Dic5 [×4], C22×C20 [×3], C2×C4⋊Dic5 [×6], C2×C4×C20, C428Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, D5, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], Dic5 [×4], D10 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], Dic10 [×6], D20 [×6], C2×Dic5 [×6], C22×D5, C429C4, C4⋊Dic5 [×12], C2×Dic10 [×3], C2×D20 [×3], C22×Dic5, C202Q8 [×3], C204D4, C2×C4⋊Dic5 [×3], C428Dic5

Smallest permutation representation of C428Dic5
Regular action on 320 points
Generators in S320
(1 119 39 98)(2 120 40 99)(3 111 31 100)(4 112 32 91)(5 113 33 92)(6 114 34 93)(7 115 35 94)(8 116 36 95)(9 117 37 96)(10 118 38 97)(11 127 47 103)(12 128 48 104)(13 129 49 105)(14 130 50 106)(15 121 41 107)(16 122 42 108)(17 123 43 109)(18 124 44 110)(19 125 45 101)(20 126 46 102)(21 221 301 242)(22 222 302 243)(23 223 303 244)(24 224 304 245)(25 225 305 246)(26 226 306 247)(27 227 307 248)(28 228 308 249)(29 229 309 250)(30 230 310 241)(51 152 72 131)(52 153 73 132)(53 154 74 133)(54 155 75 134)(55 156 76 135)(56 157 77 136)(57 158 78 137)(58 159 79 138)(59 160 80 139)(60 151 71 140)(61 162 85 141)(62 163 86 142)(63 164 87 143)(64 165 88 144)(65 166 89 145)(66 167 90 146)(67 168 81 147)(68 169 82 148)(69 170 83 149)(70 161 84 150)(171 251 195 275)(172 252 196 276)(173 253 197 277)(174 254 198 278)(175 255 199 279)(176 256 200 280)(177 257 191 271)(178 258 192 272)(179 259 193 273)(180 260 194 274)(181 261 202 282)(182 262 203 283)(183 263 204 284)(184 264 205 285)(185 265 206 286)(186 266 207 287)(187 267 208 288)(188 268 209 289)(189 269 210 290)(190 270 201 281)(211 291 235 315)(212 292 236 316)(213 293 237 317)(214 294 238 318)(215 295 239 319)(216 296 240 320)(217 297 231 311)(218 298 232 312)(219 299 233 313)(220 300 234 314)
(1 69 12 53)(2 70 13 54)(3 61 14 55)(4 62 15 56)(5 63 16 57)(6 64 17 58)(7 65 18 59)(8 66 19 60)(9 67 20 51)(10 68 11 52)(21 271 316 287)(22 272 317 288)(23 273 318 289)(24 274 319 290)(25 275 320 281)(26 276 311 282)(27 277 312 283)(28 278 313 284)(29 279 314 285)(30 280 315 286)(31 85 50 76)(32 86 41 77)(33 87 42 78)(34 88 43 79)(35 89 44 80)(36 90 45 71)(37 81 46 72)(38 82 47 73)(39 83 48 74)(40 84 49 75)(91 142 107 136)(92 143 108 137)(93 144 109 138)(94 145 110 139)(95 146 101 140)(96 147 102 131)(97 148 103 132)(98 149 104 133)(99 150 105 134)(100 141 106 135)(111 162 130 156)(112 163 121 157)(113 164 122 158)(114 165 123 159)(115 166 124 160)(116 167 125 151)(117 168 126 152)(118 169 127 153)(119 170 128 154)(120 161 129 155)(171 216 190 225)(172 217 181 226)(173 218 182 227)(174 219 183 228)(175 220 184 229)(176 211 185 230)(177 212 186 221)(178 213 187 222)(179 214 188 223)(180 215 189 224)(191 236 207 242)(192 237 208 243)(193 238 209 244)(194 239 210 245)(195 240 201 246)(196 231 202 247)(197 232 203 248)(198 233 204 249)(199 234 205 250)(200 235 206 241)(251 296 270 305)(252 297 261 306)(253 298 262 307)(254 299 263 308)(255 300 264 309)(256 291 265 310)(257 292 266 301)(258 293 267 302)(259 294 268 303)(260 295 269 304)
(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)
(1 172 6 177)(2 171 7 176)(3 180 8 175)(4 179 9 174)(5 178 10 173)(11 182 16 187)(12 181 17 186)(13 190 18 185)(14 189 19 184)(15 188 20 183)(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)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(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)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(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)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

92 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 5A 5B 10A ··· 10N 20A ··· 20AV order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 20 ··· 20 2 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 type + + + + - + - + - + image C1 C2 C2 C4 D4 Q8 D5 Dic5 D10 Dic10 D20 kernel C42⋊8Dic5 C2×C4⋊Dic5 C2×C4×C20 C4×C20 C2×C20 C2×C20 C2×C42 C42 C22×C4 C2×C4 C2×C4 # reps 1 6 1 8 6 6 2 8 6 24 24

Matrix representation of C428Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 2 9 0 0 0 4 39 0 0 0 0 0 1 2 0 0 0 40 40
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 2 0 0 0 40 40
,
 40 0 0 0 0 0 1 40 0 0 0 36 6 0 0 0 0 0 40 0 0 0 0 0 40
,
 9 0 0 0 0 0 37 16 0 0 0 22 4 0 0 0 0 0 37 29 0 0 0 39 4

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,2,4,0,0,0,9,39,0,0,0,0,0,1,40,0,0,0,2,40],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,40,0,0,0,2,40],[40,0,0,0,0,0,1,36,0,0,0,40,6,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,37,22,0,0,0,16,4,0,0,0,0,0,37,39,0,0,0,29,4] >;

C428Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_8{\rm Dic}_5
% in TeX

G:=Group("C4^2:8Dic5");
// GroupNames label

G:=SmallGroup(320,562);
// by ID

G=gap.SmallGroup(320,562);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,100,12550]);
// 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,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
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