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## G = C2×C8×Dic5order 320 = 26·5

### Direct product of C2×C8 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C8×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — C2×C4×Dic5 — C2×C8×Dic5
 Lower central C5 — C2×C8×Dic5
 Upper central C1 — C22×C8

Generators and relations for C2×C8×Dic5
G = < a,b,c,d | a2=b8=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 334 in 162 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, C2×C42, C22×C8, C22×C8, C52C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C2×C4×C8, C2×C52C8, C4×Dic5, C2×C40, C22×Dic5, C22×C20, C8×Dic5, C22×C52C8, C2×C4×Dic5, C22×C40, C2×C8×Dic5
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D5, C42, C2×C8, C22×C4, Dic5, D10, C4×C8, C2×C42, C22×C8, C4×D5, C2×Dic5, C22×D5, C2×C4×C8, C8×D5, C4×Dic5, C2×C4×D5, C22×Dic5, C8×Dic5, D5×C2×C8, C2×C4×Dic5, C2×C8×Dic5

Smallest permutation representation of C2×C8×Dic5
Regular action on 320 points
Generators in S320
(1 106)(2 107)(3 108)(4 109)(5 110)(6 101)(7 102)(8 103)(9 104)(10 105)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 247)(22 248)(23 249)(24 250)(25 241)(26 242)(27 243)(28 244)(29 245)(30 246)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 169)(88 170)(89 161)(90 162)(171 256)(172 257)(173 258)(174 259)(175 260)(176 251)(177 252)(178 253)(179 254)(180 255)(181 266)(182 267)(183 268)(184 269)(185 270)(186 261)(187 262)(188 263)(189 264)(190 265)(191 276)(192 277)(193 278)(194 279)(195 280)(196 271)(197 272)(198 273)(199 274)(200 275)(201 286)(202 287)(203 288)(204 289)(205 290)(206 281)(207 282)(208 283)(209 284)(210 285)(211 296)(212 297)(213 298)(214 299)(215 300)(216 291)(217 292)(218 293)(219 294)(220 295)(221 306)(222 307)(223 308)(224 309)(225 310)(226 301)(227 302)(228 303)(229 304)(230 305)(231 316)(232 317)(233 318)(234 319)(235 320)(236 311)(237 312)(238 313)(239 314)(240 315)
(1 83 48 74 39 58 13 66)(2 84 49 75 40 59 14 67)(3 85 50 76 31 60 15 68)(4 86 41 77 32 51 16 69)(5 87 42 78 33 52 17 70)(6 88 43 79 34 53 18 61)(7 89 44 80 35 54 19 62)(8 90 45 71 36 55 20 63)(9 81 46 72 37 56 11 64)(10 82 47 73 38 57 12 65)(21 287 316 271 299 254 310 265)(22 288 317 272 300 255 301 266)(23 289 318 273 291 256 302 267)(24 290 319 274 292 257 303 268)(25 281 320 275 293 258 304 269)(26 282 311 276 294 259 305 270)(27 283 312 277 295 260 306 261)(28 284 313 278 296 251 307 262)(29 285 314 279 297 252 308 263)(30 286 315 280 298 253 309 264)(91 144 109 168 126 152 117 136)(92 145 110 169 127 153 118 137)(93 146 101 170 128 154 119 138)(94 147 102 161 129 155 120 139)(95 148 103 162 130 156 111 140)(96 149 104 163 121 157 112 131)(97 150 105 164 122 158 113 132)(98 141 106 165 123 159 114 133)(99 142 107 166 124 160 115 134)(100 143 108 167 125 151 116 135)(171 227 182 249 204 233 198 216)(172 228 183 250 205 234 199 217)(173 229 184 241 206 235 200 218)(174 230 185 242 207 236 191 219)(175 221 186 243 208 237 192 220)(176 222 187 244 209 238 193 211)(177 223 188 245 210 239 194 212)(178 224 189 246 201 240 195 213)(179 225 190 247 202 231 196 214)(180 226 181 248 203 232 197 215)
(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)
(1 190 6 185)(2 189 7 184)(3 188 8 183)(4 187 9 182)(5 186 10 181)(11 171 16 176)(12 180 17 175)(13 179 18 174)(14 178 19 173)(15 177 20 172)(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)(51 211 56 216)(52 220 57 215)(53 219 58 214)(54 218 59 213)(55 217 60 212)(61 230 66 225)(62 229 67 224)(63 228 68 223)(64 227 69 222)(65 226 70 221)(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)(91 251 96 256)(92 260 97 255)(93 259 98 254)(94 258 99 253)(95 257 100 252)(101 270 106 265)(102 269 107 264)(103 268 108 263)(104 267 109 262)(105 266 110 261)(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)(131 291 136 296)(132 300 137 295)(133 299 138 294)(134 298 139 293)(135 297 140 292)(141 310 146 305)(142 309 147 304)(143 308 148 303)(144 307 149 302)(145 306 150 301)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4X 5A 5B 8A ··· 8P 8Q ··· 8AF 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 ··· 1 1 ··· 1 5 ··· 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 C8 D5 Dic5 D10 D10 C4×D5 C4×D5 C8×D5 kernel C2×C8×Dic5 C8×Dic5 C22×C5⋊2C8 C2×C4×Dic5 C22×C40 C2×C5⋊2C8 C4×Dic5 C2×C40 C22×Dic5 C2×Dic5 C22×C8 C2×C8 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 4 1 1 1 8 4 8 4 32 2 8 4 2 12 4 32

Matrix representation of C2×C8×Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 1
,
 38 0 0 0 0 1 0 0 0 0 38 0 0 0 0 38
,
 1 0 0 0 0 1 0 0 0 0 1 40 0 0 36 6
,
 40 0 0 0 0 40 0 0 0 0 37 16 0 0 22 4
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,0,1,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,37,22,0,0,16,4] >;

C2×C8×Dic5 in GAP, Magma, Sage, TeX

C_2\times C_8\times {\rm Dic}_5
% in TeX

G:=Group("C2xC8xDic5");
// GroupNames label

G:=SmallGroup(320,725);
// by ID

G=gap.SmallGroup(320,725);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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