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G = C22×C4⋊Dic5order 320 = 26·5

Direct product of C22 and C4⋊Dic5

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

Aliases: C22×C4⋊Dic5, C23.63D20, C24.77D10, C23.19Dic10, C2011(C22×C4), (C22×C20)⋊26C4, C42(C22×Dic5), (C22×C4)⋊9Dic5, (C23×C4).11D5, C2.2(C22×D20), (C23×C20).13C2, C10.63(C23×C4), C10.31(C22×D4), C22.75(C2×D20), (C22×C10).28Q8, C10.20(C22×Q8), C2.4(C23×Dic5), (C2×C10).283C24, (C2×C20).789C23, (C22×C10).148D4, (C22×C4).447D10, C23.45(C2×Dic5), C2.3(C22×Dic10), C22.40(C23×D5), (C23×Dic5).10C2, C22.39(C2×Dic10), C23.334(C22×D5), (C22×C20).529C22, (C23×C10).105C22, (C22×C10).412C23, (C2×Dic5).287C23, C22.30(C22×Dic5), (C22×Dic5).251C22, C104(C2×C4⋊C4), C54(C22×C4⋊C4), (C2×C20)⋊49(C2×C4), (C2×C10)⋊12(C4⋊C4), (C2×C4)⋊10(C2×Dic5), (C2×C10).54(C2×Q8), (C2×C10).181(C2×D4), (C2×C4).739(C22×D5), (C22×C10).210(C2×C4), (C2×C10).308(C22×C4), SmallGroup(320,1457)

Series: Derived Chief Lower central Upper central

C1C10 — C22×C4⋊Dic5
C1C5C10C2×C10C2×Dic5C22×Dic5C23×Dic5 — C22×C4⋊Dic5
C5C10 — C22×C4⋊Dic5
C1C24C23×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, ece-1=c-1, ede-1=d-1 >

Subgroups: 1022 in 418 conjugacy classes, 287 normal (15 characteristic)
C1, C2 [×3], C2 [×12], C4 [×8], C4 [×8], C22, C22 [×34], C5, C2×C4 [×28], C2×C4 [×32], C23 [×15], C10 [×3], C10 [×12], C4⋊C4 [×16], C22×C4 [×14], C22×C4 [×20], C24, Dic5 [×8], C20 [×8], C2×C10, C2×C10 [×34], C2×C4⋊C4 [×12], C23×C4, C23×C4 [×2], C2×Dic5 [×8], C2×Dic5 [×24], C2×C20 [×28], C22×C10 [×15], C22×C4⋊C4, C4⋊Dic5 [×16], C22×Dic5 [×12], C22×Dic5 [×8], C22×C20 [×14], C23×C10, C2×C4⋊Dic5 [×12], C23×Dic5 [×2], C23×C20, C22×C4⋊Dic5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], Q8 [×4], C23 [×15], D5, C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×6], C2×Q8 [×6], C24, Dic5 [×8], D10 [×7], C2×C4⋊C4 [×12], C23×C4, C22×D4, C22×Q8, Dic10 [×4], D20 [×4], C2×Dic5 [×28], C22×D5 [×7], C22×C4⋊C4, C4⋊Dic5 [×16], C2×Dic10 [×6], C2×D20 [×6], C22×Dic5 [×14], C23×D5, C2×C4⋊Dic5 [×12], C22×Dic10, C22×D20, C23×Dic5, C22×C4⋊Dic5

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

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

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

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

104 conjugacy classes

class 1 2A···2O4A···4H4I···4X5A5B10A···10AD20A···20AF
order12···24···44···45510···1020···20
size11···12···210···10222···22···2

104 irreducible representations

dim1111122222222
type+++++-+-++-+
imageC1C2C2C2C4D4Q8D5Dic5D10D10Dic10D20
kernelC22×C4⋊Dic5C2×C4⋊Dic5C23×Dic5C23×C20C22×C20C22×C10C22×C10C23×C4C22×C4C22×C4C24C23C23
# reps1122116442161221616

Matrix representation of C22×C4⋊Dic5 in GL6(𝔽41)

4000000
010000
001000
000100
000010
000001
,
4000000
0400000
001000
000100
000010
000001
,
4000000
010000
0004000
001000
000090
0000632
,
100000
010000
001000
000100
000040
00003231
,
100000
0400000
00122900
00292900
00001440
00003327

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,9,6,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,32,0,0,0,0,0,31],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,12,29,0,0,0,0,29,29,0,0,0,0,0,0,14,33,0,0,0,0,40,27] >;

C22×C4⋊Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("C2^2xC4:Dic5");
// GroupNames label

G:=SmallGroup(320,1457);
// by ID

G=gap.SmallGroup(320,1457);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,136,12550]);
// 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,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽