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G = C22×C4⋊Dic5  order 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, C20⋊11(C22×C4), (C22×C20)⋊26C4, C4⋊2(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, C10⋊4(C2×C4⋊C4), C5⋊4(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

Derived series
C1 — C10 — C22×C4⋊Dic5
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C22×C4⋊Dic5
Lower central
C5 — C10 — C22×C4⋊Dic5
Upper central
C1 — C24 — C23×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, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C4⋊C4, C23×C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C22×C4⋊C4, C4⋊Dic5, C22×Dic5, C22×Dic5, C22×C20, C23×C10, C2×C4⋊Dic5, C23×Dic5, C23×C20, C22×C4⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, D10, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, Dic10, D20, C2×Dic5, C22×D5, C22×C4⋊C4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C23×D5, C2×C4⋊Dic5, 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)(11 236)(12 237)(13 238)(14 239)(15 240)(16 231)(17 232)(18 233)(19 234)(20 235)(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)(71 167)(72 168)(73 169)(74 170)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(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 59)(2 60)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(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 27 154)(2 161 28 155)(3 162 29 156)(4 163 30 157)(5 164 21 158)(6 165 22 159)(7 166 23 160)(8 167 24 151)(9 168 25 152)(10 169 26 153)(11 176 316 188)(12 177 317 189)(13 178 318 190)(14 179 319 181)(15 180 320 182)(16 171 311 183)(17 172 312 184)(18 173 313 185)(19 174 314 186)(20 175 315 187)(31 131 44 144)(32 132 45 145)(33 133 46 146)(34 134 47 147)(35 135 48 148)(36 136 49 149)(37 137 50 150)(38 138 41 141)(39 139 42 142)(40 140 43 143)(51 124 64 111)(52 125 65 112)(53 126 66 113)(54 127 67 114)(55 128 68 115)(56 129 69 116)(57 130 70 117)(58 121 61 118)(59 122 62 119)(60 123 63 120)(71 91 84 104)(72 92 85 105)(73 93 86 106)(74 94 87 107)(75 95 88 108)(76 96 89 109)(77 97 90 110)(78 98 81 101)(79 99 82 102)(80 100 83 103)(191 303 203 291)(192 304 204 292)(193 305 205 293)(194 306 206 294)(195 307 207 295)(196 308 208 296)(197 309 209 297)(198 310 210 298)(199 301 201 299)(200 302 202 300)(211 271 223 283)(212 272 224 284)(213 273 225 285)(214 274 226 286)(215 275 227 287)(216 276 228 288)(217 277 229 289)(218 278 230 290)(219 279 221 281)(220 280 222 282)(231 263 243 251)(232 264 244 252)(233 265 245 253)(234 266 246 254)(235 267 247 255)(236 268 248 256)(237 269 249 257)(238 270 250 258)(239 261 241 259)(240 262 242 260)

(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)

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

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
 

׿
×
⋊
ℤ
𝔽
○
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