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## G = C2×Dic5⋊Q8order 320 = 26·5

### Direct product of C2 and Dic5⋊Q8

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×Dic5⋊Q8
G = < a,b,c,d,e | a2=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >

Subgroups: 798 in 290 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2 [×6], C4 [×4], C4 [×16], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×24], Q8 [×16], C23, C10, C10 [×6], C42 [×4], C4⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×12], Dic5 [×8], Dic5 [×4], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42, C2×C4⋊C4 [×4], C4⋊Q8 [×8], C22×Q8, C22×Q8, Dic10 [×8], C2×Dic5 [×16], C2×Dic5 [×4], C2×C20 [×10], C2×C20 [×4], C5×Q8 [×8], C22×C10, C2×C4⋊Q8, C4×Dic5 [×4], C10.D4 [×16], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C2×C4×Dic5, C2×C10.D4 [×4], Dic5⋊Q8 [×8], C22×Dic10, Q8×C2×C10, C2×Dic5⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×12], C24, D10 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C4⋊Q8, Q8×D5 [×4], C2×C5⋊D4 [×6], C23×D5, Dic5⋊Q8 [×4], C2×Q8×D5 [×2], C22×C5⋊D4, C2×Dic5⋊Q8

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

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 Q8 D4 D5 D10 D10 C5⋊D4 Q8×D5 kernel C2×Dic5⋊Q8 C2×C4×Dic5 C2×C10.D4 Dic5⋊Q8 C22×Dic10 Q8×C2×C10 C2×Dic5 C2×C20 C22×Q8 C22×C4 C2×Q8 C2×C4 C22 # reps 1 1 4 8 1 1 8 4 2 6 8 16 8

Matrix representation of C2×Dic5⋊Q8 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 34 1 0 0 0 0 33 1 0 0 0 0 0 0 34 1 0 0 0 0 33 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 20 0 0 0 0 4 40 0 0 0 0 0 0 40 21 0 0 0 0 37 1 0 0 0 0 0 0 2 23 0 0 0 0 39 39
,
 17 6 0 0 0 0 34 24 0 0 0 0 0 0 24 35 0 0 0 0 7 17 0 0 0 0 0 0 18 2 0 0 0 0 22 23
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 2 23 0 0 0 0 39 39

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[34,33,0,0,0,0,1,1,0,0,0,0,0,0,34,33,0,0,0,0,1,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,4,0,0,0,0,20,40,0,0,0,0,0,0,40,37,0,0,0,0,21,1,0,0,0,0,0,0,2,39,0,0,0,0,23,39],[17,34,0,0,0,0,6,24,0,0,0,0,0,0,24,7,0,0,0,0,35,17,0,0,0,0,0,0,18,22,0,0,0,0,2,23],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,2,39,0,0,0,0,23,39] >;

C2×Dic5⋊Q8 in GAP, Magma, Sage, TeX

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

G:=Group("C2xDic5:Q8");
// GroupNames label

G:=SmallGroup(320,1482);
// by ID

G=gap.SmallGroup(320,1482);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1123,185,80,12550]);
// Polycyclic

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

׿
×
𝔽