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## G = C2×Dic5⋊C8order 320 = 26·5

### Direct product of C2 and Dic5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×Dic5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C2×Dic5⋊C8
 Lower central C5 — C10 — C2×Dic5⋊C8
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×Dic5⋊C8
G = < a,b,c,d | a2=b10=d8=1, c2=b5, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b3, dcd-1=b5c >

Subgroups: 378 in 138 conjugacy classes, 76 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×10], C22, C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×14], C23, C10 [×3], C10 [×4], C42 [×4], C2×C8 [×8], C22×C4, C22×C4 [×2], Dic5 [×2], Dic5 [×6], C20 [×2], C2×C10, C2×C10 [×6], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×2], C22×C10, C2×C4⋊C8, C4×Dic5 [×4], C2×C5⋊C8 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×C20, Dic5⋊C8 [×4], C2×C4×Dic5, C22×C5⋊C8 [×2], C2×Dic5⋊C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, F5, C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), C2×F5 [×3], C2×C4⋊C8, D5⋊C8 [×2], C4⋊F5 [×2], C22.F5 [×2], C22×F5, Dic5⋊C8 [×4], C2×D5⋊C8, C2×C4⋊F5, C2×C22.F5, C2×Dic5⋊C8

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 5 8A ··· 8P 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 4 4 4 4 4 ··· 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 4 10 ··· 10 4 ··· 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 type + + + + + - + + + - image C1 C2 C2 C2 C4 C4 C4 C8 D4 Q8 M4(2) F5 C2×F5 C2×F5 D5⋊C8 C4⋊F5 C22.F5 kernel C2×Dic5⋊C8 Dic5⋊C8 C2×C4×Dic5 C22×C5⋊C8 C4×Dic5 C22×Dic5 C22×C20 C2×Dic5 C2×Dic5 C2×Dic5 C2×C10 C22×C4 C2×C4 C23 C22 C22 C22 # reps 1 4 1 2 4 2 2 16 2 2 4 1 2 1 4 4 4

Matrix representation of C2×Dic5⋊C8 in GL8(𝔽41)

 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 40 0 0 0 0 0 1 0 40 0 0 0 0 0 0 1 40
,
 1 23 0 0 0 0 0 0 32 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 20 25 9 35 0 0 0 0 4 34 3 28 0 0 0 0 13 28 37 7 0 0 0 0 7 21 16 32
,
 27 0 0 0 0 0 0 0 3 14 0 0 0 0 0 0 0 0 29 12 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 34 0 11 8 0 0 0 0 4 8 40 1 0 0 0 0 33 1 40 12 0 0 0 0 33 12 7 0

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[1,32,0,0,0,0,0,0,23,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,20,4,13,7,0,0,0,0,25,34,28,21,0,0,0,0,9,3,37,16,0,0,0,0,35,28,7,32],[27,3,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,0,29,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,34,4,33,33,0,0,0,0,0,8,1,12,0,0,0,0,11,40,40,7,0,0,0,0,8,1,12,0] >;

C2×Dic5⋊C8 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic5:C8");
// GroupNames label

G:=SmallGroup(320,1090);
// by ID

G=gap.SmallGroup(320,1090);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=d^8=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^5*c>;
// generators/relations

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