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## G = C2×Dic5⋊3Q8order 320 = 26·5

### Direct product of C2 and Dic5⋊3Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×Dic5⋊3Q8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C2×Dic5⋊3Q8
 Lower central C5 — C10 — C2×Dic5⋊3Q8
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×Dic53Q8
G = < a,b,c,d,e | a2=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=b-1, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 750 in 298 conjugacy classes, 175 normal (21 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×18], C22, C22 [×6], C5, C2×C4 [×10], C2×C4 [×26], Q8 [×16], C23, C10 [×3], C10 [×4], C42 [×12], C4⋊C4 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×12], Dic5 [×12], Dic5 [×2], C20 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C4×Q8 [×8], C22×Q8, Dic10 [×16], C2×Dic5 [×20], C2×Dic5 [×2], C2×C20 [×10], C2×C20 [×4], C22×C10, C2×C4×Q8, C4×Dic5 [×12], C10.D4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×12], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], Dic53Q8 [×8], C2×C4×Dic5, C2×C4×Dic5 [×2], C2×C10.D4 [×2], C10×C4⋊C4, C22×Dic10, C2×Dic53Q8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], D5, C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C4×D5 [×4], C22×D5 [×7], C2×C4×Q8, C2×C4×D5 [×6], D42D5 [×2], Q8×D5 [×2], C23×D5, Dic53Q8 [×4], D5×C22×C4, C2×D42D5, C2×Q8×D5, C2×Dic53Q8

Smallest permutation representation of C2×Dic53Q8
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)
(1 259 6 254)(2 258 7 253)(3 257 8 252)(4 256 9 251)(5 255 10 260)(11 249 16 244)(12 248 17 243)(13 247 18 242)(14 246 19 241)(15 245 20 250)(21 279 26 274)(22 278 27 273)(23 277 28 272)(24 276 29 271)(25 275 30 280)(31 269 36 264)(32 268 37 263)(33 267 38 262)(34 266 39 261)(35 265 40 270)(41 299 46 294)(42 298 47 293)(43 297 48 292)(44 296 49 291)(45 295 50 300)(51 289 56 284)(52 288 57 283)(53 287 58 282)(54 286 59 281)(55 285 60 290)(61 319 66 314)(62 318 67 313)(63 317 68 312)(64 316 69 311)(65 315 70 320)(71 309 76 304)(72 308 77 303)(73 307 78 302)(74 306 79 301)(75 305 80 310)(81 179 86 174)(82 178 87 173)(83 177 88 172)(84 176 89 171)(85 175 90 180)(91 169 96 164)(92 168 97 163)(93 167 98 162)(94 166 99 161)(95 165 100 170)(101 199 106 194)(102 198 107 193)(103 197 108 192)(104 196 109 191)(105 195 110 200)(111 189 116 184)(112 188 117 183)(113 187 118 182)(114 186 119 181)(115 185 120 190)(121 219 126 214)(122 218 127 213)(123 217 128 212)(124 216 129 211)(125 215 130 220)(131 209 136 204)(132 208 137 203)(133 207 138 202)(134 206 139 201)(135 205 140 210)(141 239 146 234)(142 238 147 233)(143 237 148 232)(144 236 149 231)(145 235 150 240)(151 229 156 224)(152 228 157 223)(153 227 158 222)(154 226 159 221)(155 225 160 230)
(1 189 11 199)(2 188 12 198)(3 187 13 197)(4 186 14 196)(5 185 15 195)(6 184 16 194)(7 183 17 193)(8 182 18 192)(9 181 19 191)(10 190 20 200)(21 174 31 164)(22 173 32 163)(23 172 33 162)(24 171 34 161)(25 180 35 170)(26 179 36 169)(27 178 37 168)(28 177 38 167)(29 176 39 166)(30 175 40 165)(41 239 51 229)(42 238 52 228)(43 237 53 227)(44 236 54 226)(45 235 55 225)(46 234 56 224)(47 233 57 223)(48 232 58 222)(49 231 59 221)(50 240 60 230)(61 204 71 214)(62 203 72 213)(63 202 73 212)(64 201 74 211)(65 210 75 220)(66 209 76 219)(67 208 77 218)(68 207 78 217)(69 206 79 216)(70 205 80 215)(81 269 91 279)(82 268 92 278)(83 267 93 277)(84 266 94 276)(85 265 95 275)(86 264 96 274)(87 263 97 273)(88 262 98 272)(89 261 99 271)(90 270 100 280)(101 254 111 244)(102 253 112 243)(103 252 113 242)(104 251 114 241)(105 260 115 250)(106 259 116 249)(107 258 117 248)(108 257 118 247)(109 256 119 246)(110 255 120 245)(121 319 131 309)(122 318 132 308)(123 317 133 307)(124 316 134 306)(125 315 135 305)(126 314 136 304)(127 313 137 303)(128 312 138 302)(129 311 139 301)(130 320 140 310)(141 284 151 294)(142 283 152 293)(143 282 153 292)(144 281 154 291)(145 290 155 300)(146 289 156 299)(147 288 157 298)(148 287 158 297)(149 286 159 296)(150 285 160 295)
(1 56 11 46)(2 57 12 47)(3 58 13 48)(4 59 14 49)(5 60 15 50)(6 51 16 41)(7 52 17 42)(8 53 18 43)(9 54 19 44)(10 55 20 45)(21 76 31 66)(22 77 32 67)(23 78 33 68)(24 79 34 69)(25 80 35 70)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 40 65)(81 136 91 126)(82 137 92 127)(83 138 93 128)(84 139 94 129)(85 140 95 130)(86 131 96 121)(87 132 97 122)(88 133 98 123)(89 134 99 124)(90 135 100 125)(101 156 111 146)(102 157 112 147)(103 158 113 148)(104 159 114 149)(105 160 115 150)(106 151 116 141)(107 152 117 142)(108 153 118 143)(109 154 119 144)(110 155 120 145)(161 216 171 206)(162 217 172 207)(163 218 173 208)(164 219 174 209)(165 220 175 210)(166 211 176 201)(167 212 177 202)(168 213 178 203)(169 214 179 204)(170 215 180 205)(181 236 191 226)(182 237 192 227)(183 238 193 228)(184 239 194 229)(185 240 195 230)(186 231 196 221)(187 232 197 222)(188 233 198 223)(189 234 199 224)(190 235 200 225)(241 296 251 286)(242 297 252 287)(243 298 253 288)(244 299 254 289)(245 300 255 290)(246 291 256 281)(247 292 257 282)(248 293 258 283)(249 294 259 284)(250 295 260 285)(261 316 271 306)(262 317 272 307)(263 318 273 308)(264 319 274 309)(265 320 275 310)(266 311 276 301)(267 312 277 302)(268 313 278 303)(269 314 279 304)(270 315 280 305)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 4M ··· 4T 4U ··· 4AF 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C4 Q8 D5 C4○D4 D10 D10 C4×D5 D4⋊2D5 Q8×D5 kernel C2×Dic5⋊3Q8 Dic5⋊3Q8 C2×C4×Dic5 C2×C10.D4 C10×C4⋊C4 C22×Dic10 C2×Dic10 C2×Dic5 C2×C4⋊C4 C2×C10 C4⋊C4 C22×C4 C2×C4 C22 C22 # reps 1 8 3 2 1 1 16 4 2 4 8 6 16 4 4

Matrix representation of C2×Dic53Q8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 1 0 0 0 0 5 6 0 0 0 0 0 0 40 40 0 0 0 0 8 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 27 14 0 0 0 0 24 14 0 0 0 0 0 0 35 12 0 0 0 0 21 6 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 38 3 0 0 0 0 11 3 0 0 0 0 0 0 35 12 0 0 0 0 21 6 0 0 0 0 0 0 1 18 0 0 0 0 9 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 37 15 0 0 0 0 18 4

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,5,0,0,0,0,1,6,0,0,0,0,0,0,40,8,0,0,0,0,40,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[27,24,0,0,0,0,14,14,0,0,0,0,0,0,35,21,0,0,0,0,12,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[38,11,0,0,0,0,3,3,0,0,0,0,0,0,35,21,0,0,0,0,12,6,0,0,0,0,0,0,1,9,0,0,0,0,18,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,37,18,0,0,0,0,15,4] >;

C2×Dic53Q8 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5\rtimes_3Q_8
% in TeX

G:=Group("C2xDic5:3Q8");
// GroupNames label

G:=SmallGroup(320,1168);
// by ID

G=gap.SmallGroup(320,1168);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,185,192,12550]);
// 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=d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽