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## G = C42⋊9Dic5order 320 = 26·5

### 6th semidirect product of C42 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊9Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C10.10C42 — C42⋊9Dic5
 Lower central C5 — C2×C10 — C42⋊9Dic5
 Upper central C1 — C23 — C2×C42

Generators and relations for C429Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 462 in 154 conjugacy classes, 87 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×C10, C428C4, C4⋊Dic5, C4×C20, C22×Dic5, C22×C20, C22×C20, C10.10C42, C2×C4⋊Dic5, C2×C4×C20, C429Dic5
Quotients:

Smallest permutation representation of C429Dic5
Regular action on 320 points
Generators in S320
(1 110 15 98)(2 101 16 99)(3 102 17 100)(4 103 18 91)(5 104 19 92)(6 105 20 93)(7 106 11 94)(8 107 12 95)(9 108 13 96)(10 109 14 97)(21 215 316 226)(22 216 317 227)(23 217 318 228)(24 218 319 229)(25 219 320 230)(26 220 311 221)(27 211 312 222)(28 212 313 223)(29 213 314 224)(30 214 315 225)(31 130 50 111)(32 121 41 112)(33 122 42 113)(34 123 43 114)(35 124 44 115)(36 125 45 116)(37 126 46 117)(38 127 47 118)(39 128 48 119)(40 129 49 120)(51 143 63 131)(52 144 64 132)(53 145 65 133)(54 146 66 134)(55 147 67 135)(56 148 68 136)(57 149 69 137)(58 150 70 138)(59 141 61 139)(60 142 62 140)(71 167 90 151)(72 168 81 152)(73 169 82 153)(74 170 83 154)(75 161 84 155)(76 162 85 156)(77 163 86 157)(78 164 87 158)(79 165 88 159)(80 166 89 160)(171 272 182 288)(172 273 183 289)(173 274 184 290)(174 275 185 281)(175 276 186 282)(176 277 187 283)(177 278 188 284)(178 279 189 285)(179 280 190 286)(180 271 181 287)(191 260 207 261)(192 251 208 262)(193 252 209 263)(194 253 210 264)(195 254 201 265)(196 255 202 266)(197 256 203 267)(198 257 204 268)(199 258 205 269)(200 259 206 270)(231 300 247 301)(232 291 248 302)(233 292 249 303)(234 293 250 304)(235 294 241 305)(236 295 242 306)(237 296 243 307)(238 297 244 308)(239 298 245 309)(240 299 246 310)
(1 79 39 58)(2 80 40 59)(3 71 31 60)(4 72 32 51)(5 73 33 52)(6 74 34 53)(7 75 35 54)(8 76 36 55)(9 77 37 56)(10 78 38 57)(11 84 44 66)(12 85 45 67)(13 86 46 68)(14 87 47 69)(15 88 48 70)(16 89 49 61)(17 90 50 62)(18 81 41 63)(19 82 42 64)(20 83 43 65)(21 276 295 266)(22 277 296 267)(23 278 297 268)(24 279 298 269)(25 280 299 270)(26 271 300 261)(27 272 291 262)(28 273 292 263)(29 274 293 264)(30 275 294 265)(91 152 112 131)(92 153 113 132)(93 154 114 133)(94 155 115 134)(95 156 116 135)(96 157 117 136)(97 158 118 137)(98 159 119 138)(99 160 120 139)(100 151 111 140)(101 166 129 141)(102 167 130 142)(103 168 121 143)(104 169 122 144)(105 170 123 145)(106 161 124 146)(107 162 125 147)(108 163 126 148)(109 164 127 149)(110 165 128 150)(171 232 208 222)(172 233 209 223)(173 234 210 224)(174 235 201 225)(175 236 202 226)(176 237 203 227)(177 238 204 228)(178 239 205 229)(179 240 206 230)(180 231 207 221)(181 247 191 220)(182 248 192 211)(183 249 193 212)(184 250 194 213)(185 241 195 214)(186 242 196 215)(187 243 197 216)(188 244 198 217)(189 245 199 218)(190 246 200 219)(251 312 288 302)(252 313 289 303)(253 314 290 304)(254 315 281 305)(255 316 282 306)(256 317 283 307)(257 318 284 308)(258 319 285 309)(259 320 286 310)(260 311 287 301)
(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)(11 179 16 174)(12 178 17 173)(13 177 18 172)(14 176 19 171)(15 175 20 180)(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)(51 223 56 228)(52 222 57 227)(53 221 58 226)(54 230 59 225)(55 229 60 224)(61 214 66 219)(62 213 67 218)(63 212 68 217)(64 211 69 216)(65 220 70 215)(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)(91 263 96 268)(92 262 97 267)(93 261 98 266)(94 270 99 265)(95 269 100 264)(101 254 106 259)(102 253 107 258)(103 252 108 257)(104 251 109 256)(105 260 110 255)(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)(131 303 136 308)(132 302 137 307)(133 301 138 306)(134 310 139 305)(135 309 140 304)(141 294 146 299)(142 293 147 298)(143 292 148 297)(144 291 149 296)(145 300 150 295)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

Matrix representation of C429Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 24 1 0 0 0 40 17 0 0 0 0 0 32 0 0 0 0 0 32
,
 1 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 17 40 0 0 0 1 24
,
 40 0 0 0 0 0 0 40 0 0 0 1 7 0 0 0 0 0 0 40 0 0 0 1 7
,
 32 0 0 0 0 0 9 22 0 0 0 0 32 0 0 0 0 0 17 40 0 0 0 3 24

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,24,40,0,0,0,1,17,0,0,0,0,0,32,0,0,0,0,0,32],[1,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,17,1,0,0,0,40,24],[40,0,0,0,0,0,0,1,0,0,0,40,7,0,0,0,0,0,0,1,0,0,0,40,7],[32,0,0,0,0,0,9,0,0,0,0,22,32,0,0,0,0,0,17,3,0,0,0,40,24] >;

C429Dic5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_9{\rm Dic}_5
% in TeX

G:=Group("C4^2:9Dic5");
// GroupNames label

G:=SmallGroup(320,563);
// by ID

G=gap.SmallGroup(320,563);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,422,184,12550]);
// 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*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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