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## G = C22×C10.D4order 320 = 26·5

### Direct product of C22 and C10.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22×C10.D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C22×C10.D4
 Lower central C5 — C10 — C22×C10.D4
 Upper central C1 — C24 — C23×C4

Generators and relations for C22×C10.D4
G = < a,b,c,d,e | a2=b2=c10=d4=1, e2=c5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=d-1 >

Subgroups: 1022 in 418 conjugacy classes, 247 normal (17 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C4⋊C4, C23×C4, C23×C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22×C4⋊C4, C10.D4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×C10, C2×C10.D4, C23×Dic5, C23×C20, C22×C10.D4
Quotients:

Smallest permutation representation of C22×C10.D4
Regular action on 320 points
Generators in S320
(1 110)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 235)(12 236)(13 237)(14 238)(15 239)(16 240)(17 231)(18 232)(19 233)(20 234)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 131)(62 132)(63 133)(64 134)(65 135)(66 136)(67 137)(68 138)(69 139)(70 140)(71 170)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 169)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)(171 270)(172 261)(173 262)(174 263)(175 264)(176 265)(177 266)(178 267)(179 268)(180 269)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)(187 258)(188 259)(189 260)(190 251)(191 290)(192 281)(193 282)(194 283)(195 284)(196 285)(197 286)(198 287)(199 288)(200 289)(201 272)(202 273)(203 274)(204 275)(205 276)(206 277)(207 278)(208 279)(209 280)(210 271)(211 310)(212 301)(213 302)(214 303)(215 304)(216 305)(217 306)(218 307)(219 308)(220 309)(221 292)(222 293)(223 294)(224 295)(225 296)(226 297)(227 298)(228 299)(229 300)(230 291)(241 312)(242 313)(243 314)(244 315)(245 316)(246 317)(247 318)(248 319)(249 320)(250 311)
(1 70)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 275)(12 276)(13 277)(14 278)(15 279)(16 280)(17 271)(18 272)(19 273)(20 274)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 170)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 169)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(171 230)(172 221)(173 222)(174 223)(175 224)(176 225)(177 226)(178 227)(179 228)(180 229)(181 212)(182 213)(183 214)(184 215)(185 216)(186 217)(187 218)(188 219)(189 220)(190 211)(191 250)(192 241)(193 242)(194 243)(195 244)(196 245)(197 246)(198 247)(199 248)(200 249)(201 232)(202 233)(203 234)(204 235)(205 236)(206 237)(207 238)(208 239)(209 240)(210 231)(251 310)(252 301)(253 302)(254 303)(255 304)(256 305)(257 306)(258 307)(259 308)(260 309)(261 292)(262 293)(263 294)(264 295)(265 296)(266 297)(267 298)(268 299)(269 300)(270 291)(281 312)(282 313)(283 314)(284 315)(285 316)(286 317)(287 318)(288 319)(289 320)(290 311)
(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)
(1 270 25 256)(2 269 26 255)(3 268 27 254)(4 267 28 253)(5 266 29 252)(6 265 30 251)(7 264 21 260)(8 263 22 259)(9 262 23 258)(10 261 24 257)(11 81 320 76)(12 90 311 75)(13 89 312 74)(14 88 313 73)(15 87 314 72)(16 86 315 71)(17 85 316 80)(18 84 317 79)(19 83 318 78)(20 82 319 77)(31 289 46 275)(32 288 47 274)(33 287 48 273)(34 286 49 272)(35 285 50 271)(36 284 41 280)(37 283 42 279)(38 282 43 278)(39 281 44 277)(40 290 45 276)(51 309 66 295)(52 308 67 294)(53 307 68 293)(54 306 69 292)(55 305 70 291)(56 304 61 300)(57 303 62 299)(58 302 63 298)(59 301 64 297)(60 310 65 296)(91 189 106 175)(92 188 107 174)(93 187 108 173)(94 186 109 172)(95 185 110 171)(96 184 101 180)(97 183 102 179)(98 182 103 178)(99 181 104 177)(100 190 105 176)(111 209 126 195)(112 208 127 194)(113 207 128 193)(114 206 129 192)(115 205 130 191)(116 204 121 200)(117 203 122 199)(118 202 123 198)(119 201 124 197)(120 210 125 196)(131 229 146 215)(132 228 147 214)(133 227 148 213)(134 226 149 212)(135 225 150 211)(136 224 141 220)(137 223 142 219)(138 222 143 218)(139 221 144 217)(140 230 145 216)(151 249 165 235)(152 248 166 234)(153 247 167 233)(154 246 168 232)(155 245 169 231)(156 244 170 240)(157 243 161 239)(158 242 162 238)(159 241 163 237)(160 250 164 236)
(1 231 6 236)(2 240 7 235)(3 239 8 234)(4 238 9 233)(5 237 10 232)(11 101 16 106)(12 110 17 105)(13 109 18 104)(14 108 19 103)(15 107 20 102)(21 249 26 244)(22 248 27 243)(23 247 28 242)(24 246 29 241)(25 245 30 250)(31 215 36 220)(32 214 37 219)(33 213 38 218)(34 212 39 217)(35 211 40 216)(41 224 46 229)(42 223 47 228)(43 222 48 227)(44 221 49 226)(45 230 50 225)(51 200 56 195)(52 199 57 194)(53 198 58 193)(54 197 59 192)(55 196 60 191)(61 209 66 204)(62 208 67 203)(63 207 68 202)(64 206 69 201)(65 205 70 210)(71 175 76 180)(72 174 77 179)(73 173 78 178)(74 172 79 177)(75 171 80 176)(81 184 86 189)(82 183 87 188)(83 182 88 187)(84 181 89 186)(85 190 90 185)(91 320 96 315)(92 319 97 314)(93 318 98 313)(94 317 99 312)(95 316 100 311)(111 295 116 300)(112 294 117 299)(113 293 118 298)(114 292 119 297)(115 291 120 296)(121 304 126 309)(122 303 127 308)(123 302 128 307)(124 301 129 306)(125 310 130 305)(131 280 136 275)(132 279 137 274)(133 278 138 273)(134 277 139 272)(135 276 140 271)(141 289 146 284)(142 288 147 283)(143 287 148 282)(144 286 149 281)(145 285 150 290)(151 255 156 260)(152 254 157 259)(153 253 158 258)(154 252 159 257)(155 251 160 256)(161 263 166 268)(162 262 167 267)(163 261 168 266)(164 270 169 265)(165 269 170 264)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2O 4A ··· 4H 4I ··· 4X 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + - + + + - image C1 C2 C2 C2 C4 D4 Q8 D5 D10 D10 Dic10 C4×D5 C5⋊D4 kernel C22×C10.D4 C2×C10.D4 C23×Dic5 C23×C20 C22×Dic5 C22×C10 C22×C10 C23×C4 C22×C4 C24 C23 C23 C23 # reps 1 12 2 1 16 4 4 2 12 2 16 16 16

Matrix representation of C22×C10.D4 in GL5(𝔽41)

 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 35 0 0 0 6 6
,
 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 6 23 0 0 0 18 35
,
 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 32 0 0 0 32 0

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,6,0,0,0,35,6],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,6,18,0,0,0,23,35],[9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,32,0] >;

C22×C10.D4 in GAP, Magma, Sage, TeX

C_2^2\times C_{10}.D_4
% in TeX

G:=Group("C2^2xC10.D4");
// GroupNames label

G:=SmallGroup(320,1455);
// by ID

G=gap.SmallGroup(320,1455);
# by ID

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

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

׿
×
𝔽