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G = C22×C4×Dic5order 320 = 26·5

Direct product of C22×C4 and Dic5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22×C4×Dic5, C24.75D10, (C2×C10)⋊7C42, C103(C2×C42), C53(C22×C42), C2012(C22×C4), (C22×C20)⋊29C4, (C23×C4).15D5, C23.67(C4×D5), C10.57(C23×C4), (C23×C20).21C2, C2.2(C23×Dic5), (C2×C20).884C23, (C2×C10).280C24, (C22×C4).473D10, C23.44(C2×Dic5), C22.38(C23×D5), (C23×Dic5).16C2, C23.332(C22×D5), (C23×C10).102C22, (C22×C10).409C23, (C22×C20).569C22, (C2×Dic5).381C23, C22.29(C22×Dic5), (C22×Dic5).289C22, (C2×C20)⋊51(C2×C4), C2.3(D5×C22×C4), C22.77(C2×C4×D5), (C2×C4).828(C22×D5), (C2×C10).307(C22×C4), (C22×C10).209(C2×C4), SmallGroup(320,1454)

Series: Derived Chief Lower central Upper central

C1C5 — C22×C4×Dic5
C1C5C10C2×C10C2×Dic5C22×Dic5C23×Dic5 — C22×C4×Dic5
C5 — C22×C4×Dic5
C1C23×C4

Generators and relations for C22×C4×Dic5
 G = < a,b,c,d,e | a2=b2=c4=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 1022 in 498 conjugacy classes, 367 normal (11 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×16], C22, C22 [×34], C5, C2×C4 [×28], C2×C4 [×56], C23 [×15], C10, C10 [×14], C42 [×16], C22×C4 [×14], C22×C4 [×28], C24, Dic5 [×16], C20 [×8], C2×C10, C2×C10 [×34], C2×C42 [×12], C23×C4, C23×C4 [×2], C2×Dic5 [×56], C2×C20 [×28], C22×C10 [×15], C22×C42, C4×Dic5 [×16], C22×Dic5 [×28], C22×C20 [×14], C23×C10, C2×C4×Dic5 [×12], C23×Dic5 [×2], C23×C20, C22×C4×Dic5
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], D5, C42 [×16], C22×C4 [×42], C24, Dic5 [×8], D10 [×7], C2×C42 [×12], C23×C4 [×3], C4×D5 [×8], C2×Dic5 [×28], C22×D5 [×7], C22×C42, C4×Dic5 [×16], C2×C4×D5 [×12], C22×Dic5 [×14], C23×D5, C2×C4×Dic5 [×12], D5×C22×C4 [×2], C23×Dic5, C22×C4×Dic5

Smallest permutation representation of C22×C4×Dic5
Regular action on 320 points
Generators in S320
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 109)(8 110)(9 101)(10 102)(11 240)(12 231)(13 232)(14 233)(15 234)(16 235)(17 236)(18 237)(19 238)(20 239)(21 98)(22 99)(23 100)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 121)(39 122)(40 123)(41 118)(42 119)(43 120)(44 111)(45 112)(46 113)(47 114)(48 115)(49 116)(50 117)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 141)(59 142)(60 143)(61 138)(62 139)(63 140)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 161)(72 162)(73 163)(74 164)(75 165)(76 166)(77 167)(78 168)(79 169)(80 170)(81 158)(82 159)(83 160)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(171 263)(172 264)(173 265)(174 266)(175 267)(176 268)(177 269)(178 270)(179 261)(180 262)(181 259)(182 260)(183 251)(184 252)(185 253)(186 254)(187 255)(188 256)(189 257)(190 258)(191 283)(192 284)(193 285)(194 286)(195 287)(196 288)(197 289)(198 290)(199 281)(200 282)(201 279)(202 280)(203 271)(204 272)(205 273)(206 274)(207 275)(208 276)(209 277)(210 278)(211 303)(212 304)(213 305)(214 306)(215 307)(216 308)(217 309)(218 310)(219 301)(220 302)(221 299)(222 300)(223 291)(224 292)(225 293)(226 294)(227 295)(228 296)(229 297)(230 298)(241 319)(242 320)(243 311)(244 312)(245 313)(246 314)(247 315)(248 316)(249 317)(250 318)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(11 280)(12 271)(13 272)(14 273)(15 274)(16 275)(17 276)(18 277)(19 278)(20 279)(21 58)(22 59)(23 60)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 81)(39 82)(40 83)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(111 161)(112 162)(113 163)(114 164)(115 165)(116 166)(117 167)(118 168)(119 169)(120 170)(121 158)(122 159)(123 160)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(171 223)(172 224)(173 225)(174 226)(175 227)(176 228)(177 229)(178 230)(179 221)(180 222)(181 219)(182 220)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 243)(192 244)(193 245)(194 246)(195 247)(196 248)(197 249)(198 250)(199 241)(200 242)(201 239)(202 240)(203 231)(204 232)(205 233)(206 234)(207 235)(208 236)(209 237)(210 238)(251 303)(252 304)(253 305)(254 306)(255 307)(256 308)(257 309)(258 310)(259 301)(260 302)(261 299)(262 300)(263 291)(264 292)(265 293)(266 294)(267 295)(268 296)(269 297)(270 298)(281 319)(282 320)(283 311)(284 312)(285 313)(286 314)(287 315)(288 316)(289 317)(290 318)
(1 48 28 40)(2 49 29 31)(3 50 30 32)(4 41 21 33)(5 42 22 34)(6 43 23 35)(7 44 24 36)(8 45 25 37)(9 46 26 38)(10 47 27 39)(11 302 315 295)(12 303 316 296)(13 304 317 297)(14 305 318 298)(15 306 319 299)(16 307 320 300)(17 308 311 291)(18 309 312 292)(19 310 313 293)(20 301 314 294)(51 89 69 71)(52 90 70 72)(53 81 61 73)(54 82 62 74)(55 83 63 75)(56 84 64 76)(57 85 65 77)(58 86 66 78)(59 87 67 79)(60 88 68 80)(91 129 109 111)(92 130 110 112)(93 121 101 113)(94 122 102 114)(95 123 103 115)(96 124 104 116)(97 125 105 117)(98 126 106 118)(99 127 107 119)(100 128 108 120)(131 166 149 151)(132 167 150 152)(133 168 141 153)(134 169 142 154)(135 170 143 155)(136 161 144 156)(137 162 145 157)(138 163 146 158)(139 164 147 159)(140 165 148 160)(171 208 188 191)(172 209 189 192)(173 210 190 193)(174 201 181 194)(175 202 182 195)(176 203 183 196)(177 204 184 197)(178 205 185 198)(179 206 186 199)(180 207 187 200)(211 248 228 231)(212 249 229 232)(213 250 230 233)(214 241 221 234)(215 242 222 235)(216 243 223 236)(217 244 224 237)(218 245 225 238)(219 246 226 239)(220 247 227 240)(251 288 268 271)(252 289 269 272)(253 290 270 273)(254 281 261 274)(255 282 262 275)(256 283 263 276)(257 284 264 277)(258 285 265 278)(259 286 266 279)(260 287 267 280)
(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)
(1 295 6 300)(2 294 7 299)(3 293 8 298)(4 292 9 297)(5 291 10 296)(11 43 16 48)(12 42 17 47)(13 41 18 46)(14 50 19 45)(15 49 20 44)(21 309 26 304)(22 308 27 303)(23 307 28 302)(24 306 29 301)(25 305 30 310)(31 314 36 319)(32 313 37 318)(33 312 38 317)(34 311 39 316)(35 320 40 315)(51 254 56 259)(52 253 57 258)(53 252 58 257)(54 251 59 256)(55 260 60 255)(61 269 66 264)(62 268 67 263)(63 267 68 262)(64 266 69 261)(65 265 70 270)(71 274 76 279)(72 273 77 278)(73 272 78 277)(74 271 79 276)(75 280 80 275)(81 289 86 284)(82 288 87 283)(83 287 88 282)(84 286 89 281)(85 285 90 290)(91 214 96 219)(92 213 97 218)(93 212 98 217)(94 211 99 216)(95 220 100 215)(101 229 106 224)(102 228 107 223)(103 227 108 222)(104 226 109 221)(105 225 110 230)(111 234 116 239)(112 233 117 238)(113 232 118 237)(114 231 119 236)(115 240 120 235)(121 249 126 244)(122 248 127 243)(123 247 128 242)(124 246 129 241)(125 245 130 250)(131 174 136 179)(132 173 137 178)(133 172 138 177)(134 171 139 176)(135 180 140 175)(141 189 146 184)(142 188 147 183)(143 187 148 182)(144 186 149 181)(145 185 150 190)(151 194 156 199)(152 193 157 198)(153 192 158 197)(154 191 159 196)(155 200 160 195)(161 206 166 201)(162 205 167 210)(163 204 168 209)(164 203 169 208)(165 202 170 207)

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

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

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

128 conjugacy classes

class 1 2A···2O4A···4P4Q···4AV5A5B10A···10AD20A···20AF
order12···24···44···45510···1020···20
size11···11···15···5222···22···2

128 irreducible representations

dim11111122222
type+++++-++
imageC1C2C2C2C4C4D5Dic5D10D10C4×D5
kernelC22×C4×Dic5C2×C4×Dic5C23×Dic5C23×C20C22×Dic5C22×C20C23×C4C22×C4C22×C4C24C23
# reps11221321621612232

Matrix representation of C22×C4×Dic5 in GL5(𝔽41)

10000
040000
004000
00010
00001
,
10000
040000
004000
000400
000040
,
400000
032000
004000
000400
000040
,
400000
040000
00100
000140
000366
,
90000
09000
00100
000022
000130

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,32,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,36,0,0,0,40,6],[9,0,0,0,0,0,9,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,22,0] >;

C22×C4×Dic5 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times {\rm Dic}_5
% in TeX

G:=Group("C2^2xC4xDic5");
// GroupNames label

G:=SmallGroup(320,1454);
// by ID

G=gap.SmallGroup(320,1454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,136,12550]);
// Polycyclic

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

׿
×
𝔽