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## G = C2×Q8⋊Dic5order 320 = 26·5

### Direct product of C2 and Q8⋊Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×Q8⋊Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — C2×Q8⋊Dic5
 Lower central C5 — C10 — C20 — C2×Q8⋊Dic5
 Upper central C1 — C23 — C22×C4 — C22×Q8

Generators and relations for C2×Q8⋊Dic5
G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >

Subgroups: 414 in 162 conjugacy classes, 87 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], Q8 [×6], C23, C10 [×3], C10 [×4], C4⋊C4 [×3], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×3], Dic5 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C52C8 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C2×Q8⋊C4, C2×C52C8 [×2], C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C22×Dic5, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], Q8⋊Dic5 [×4], C22×C52C8, C2×C4⋊Dic5, Q8×C2×C10, C2×Q8⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×Q8⋊C4, Q8⋊D5 [×2], C5⋊Q16 [×2], C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], Q8⋊Dic5 [×4], C2×Q8⋊D5, C2×C5⋊Q16, C2×C23.D5, C2×Q8⋊Dic5

Smallest permutation representation of C2×Q8⋊Dic5
Regular action on 320 points
Generators in S320
(1 96)(2 97)(3 98)(4 99)(5 100)(6 91)(7 92)(8 93)(9 94)(10 95)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 101)(18 102)(19 103)(20 104)(21 243)(22 244)(23 245)(24 246)(25 247)(26 248)(27 249)(28 250)(29 241)(30 242)(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)(81 169)(82 170)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(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 47 31 11)(2 48 32 12)(3 49 33 13)(4 50 34 14)(5 41 35 15)(6 42 36 16)(7 43 37 17)(8 44 38 18)(9 45 39 19)(10 46 40 20)(21 319 310 294)(22 320 301 295)(23 311 302 296)(24 312 303 297)(25 313 304 298)(26 314 305 299)(27 315 306 300)(28 316 307 291)(29 317 308 292)(30 318 309 293)(51 70 71 87)(52 61 72 88)(53 62 73 89)(54 63 74 90)(55 64 75 81)(56 65 76 82)(57 66 77 83)(58 67 78 84)(59 68 79 85)(60 69 80 86)(91 127 111 110)(92 128 112 101)(93 129 113 102)(94 130 114 103)(95 121 115 104)(96 122 116 105)(97 123 117 106)(98 124 118 107)(99 125 119 108)(100 126 120 109)(131 150 151 170)(132 141 152 161)(133 142 153 162)(134 143 154 163)(135 144 155 164)(136 145 156 165)(137 146 157 166)(138 147 158 167)(139 148 159 168)(140 149 160 169)(171 187 196 205)(172 188 197 206)(173 189 198 207)(174 190 199 208)(175 181 200 209)(176 182 191 210)(177 183 192 201)(178 184 193 202)(179 185 194 203)(180 186 195 204)(211 245 236 227)(212 246 237 228)(213 247 238 229)(214 248 239 230)(215 249 240 221)(216 250 231 222)(217 241 232 223)(218 242 233 224)(219 243 234 225)(220 244 235 226)(251 267 276 285)(252 268 277 286)(253 269 278 287)(254 270 279 288)(255 261 280 289)(256 262 271 290)(257 263 272 281)(258 264 273 282)(259 265 274 283)(260 266 275 284)
(1 76 31 56)(2 77 32 57)(3 78 33 58)(4 79 34 59)(5 80 35 60)(6 71 36 51)(7 72 37 52)(8 73 38 53)(9 74 39 54)(10 75 40 55)(11 82 47 65)(12 83 48 66)(13 84 49 67)(14 85 50 68)(15 86 41 69)(16 87 42 70)(17 88 43 61)(18 89 44 62)(19 90 45 63)(20 81 46 64)(21 274 310 259)(22 275 301 260)(23 276 302 251)(24 277 303 252)(25 278 304 253)(26 279 305 254)(27 280 306 255)(28 271 307 256)(29 272 308 257)(30 273 309 258)(91 156 111 136)(92 157 112 137)(93 158 113 138)(94 159 114 139)(95 160 115 140)(96 151 116 131)(97 152 117 132)(98 153 118 133)(99 154 119 134)(100 155 120 135)(101 166 128 146)(102 167 129 147)(103 168 130 148)(104 169 121 149)(105 170 122 150)(106 161 123 141)(107 162 124 142)(108 163 125 143)(109 164 126 144)(110 165 127 145)(171 250 196 222)(172 241 197 223)(173 242 198 224)(174 243 199 225)(175 244 200 226)(176 245 191 227)(177 246 192 228)(178 247 193 229)(179 248 194 230)(180 249 195 221)(181 220 209 235)(182 211 210 236)(183 212 201 237)(184 213 202 238)(185 214 203 239)(186 215 204 240)(187 216 205 231)(188 217 206 232)(189 218 207 233)(190 219 208 234)(261 300 289 315)(262 291 290 316)(263 292 281 317)(264 293 282 318)(265 294 283 319)(266 295 284 320)(267 296 285 311)(268 297 286 312)(269 298 287 313)(270 299 288 314)
(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)
(1 259 6 254)(2 258 7 253)(3 257 8 252)(4 256 9 251)(5 255 10 260)(11 265 16 270)(12 264 17 269)(13 263 18 268)(14 262 19 267)(15 261 20 266)(21 87 26 82)(22 86 27 81)(23 85 28 90)(24 84 29 89)(25 83 30 88)(31 274 36 279)(32 273 37 278)(33 272 38 277)(34 271 39 276)(35 280 40 275)(41 289 46 284)(42 288 47 283)(43 287 48 282)(44 286 49 281)(45 285 50 290)(51 299 56 294)(52 298 57 293)(53 297 58 292)(54 296 59 291)(55 295 60 300)(61 304 66 309)(62 303 67 308)(63 302 68 307)(64 301 69 306)(65 310 70 305)(71 314 76 319)(72 313 77 318)(73 312 78 317)(74 311 79 316)(75 320 80 315)(91 179 96 174)(92 178 97 173)(93 177 98 172)(94 176 99 171)(95 175 100 180)(101 184 106 189)(102 183 107 188)(103 182 108 187)(104 181 109 186)(105 190 110 185)(111 194 116 199)(112 193 117 198)(113 192 118 197)(114 191 119 196)(115 200 120 195)(121 209 126 204)(122 208 127 203)(123 207 128 202)(124 206 129 201)(125 205 130 210)(131 219 136 214)(132 218 137 213)(133 217 138 212)(134 216 139 211)(135 215 140 220)(141 224 146 229)(142 223 147 228)(143 222 148 227)(144 221 149 226)(145 230 150 225)(151 234 156 239)(152 233 157 238)(153 232 158 237)(154 231 159 236)(155 240 160 235)(161 242 166 247)(162 241 167 246)(163 250 168 245)(164 249 169 244)(165 248 170 243)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 20 20 20 20 2 2 10 ··· 10 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + - + + - image C1 C2 C2 C2 C2 C4 D4 D4 D5 SD16 Q16 D10 Dic5 D10 C5⋊D4 C5⋊D4 Q8⋊D5 C5⋊Q16 kernel C2×Q8⋊Dic5 Q8⋊Dic5 C22×C5⋊2C8 C2×C4⋊Dic5 Q8×C2×C10 Q8×C10 C2×C20 C22×C10 C22×Q8 C2×C10 C2×C10 C22×C4 C2×Q8 C2×Q8 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 3 1 2 4 4 2 8 4 12 4 4 4

Matrix representation of C2×Q8⋊Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40 9 0 0 0 18 1
,
 1 0 0 0 0 0 17 40 0 0 0 1 24 0 0 0 0 0 31 32 0 0 0 34 10
,
 40 0 0 0 0 0 34 40 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 0 0 0 0 0 7 1 0 0 0 34 34 0 0 0 0 0 39 38 0 0 0 1 2

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,18,0,0,0,9,1],[1,0,0,0,0,0,17,1,0,0,0,40,24,0,0,0,0,0,31,34,0,0,0,32,10],[40,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,7,34,0,0,0,1,34,0,0,0,0,0,39,1,0,0,0,38,2] >;

C2×Q8⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xQ8:Dic5");
// GroupNames label

G:=SmallGroup(320,851);
// by ID

G=gap.SmallGroup(320,851);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,1684,438,102,12550]);
// Polycyclic

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

׿
×
𝔽