Copied to
clipboard

## G = C22×C4×Dic5order 320 = 26·5

### Direct product of C22×C4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C22×C4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C22×C4×Dic5
 Lower central C5 — C22×C4×Dic5
 Upper central C1 — 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, ce=ec, ede-1=d-1 >

Subgroups: 1022 in 498 conjugacy classes, 367 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C42, C22×C4, C22×C4, C24, Dic5, C20, C2×C10, C2×C10, C2×C42, C23×C4, C23×C4, C2×Dic5, C2×C20, C22×C10, C22×C42, C4×Dic5, C22×Dic5, C22×C20, C23×C10, C2×C4×Dic5, C23×Dic5, C23×C20, C22×C4×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, C22×C4, C24, Dic5, D10, C2×C42, C23×C4, C4×D5, C2×Dic5, C22×D5, C22×C42, C4×Dic5, C2×C4×D5, C22×Dic5, C23×D5, C2×C4×Dic5, D5×C22×C4, 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)(11 234)(12 235)(13 236)(14 237)(15 238)(16 239)(17 240)(18 231)(19 232)(20 233)(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)(71 165)(72 166)(73 167)(74 168)(75 169)(76 170)(77 161)(78 162)(79 163)(80 164)(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 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(11 274)(12 275)(13 276)(14 277)(15 278)(16 279)(17 280)(18 271)(19 272)(20 273)(21 54)(22 55)(23 56)(24 57)(25 58)(26 59)(27 60)(28 51)(29 52)(30 53)(31 88)(32 89)(33 90)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 71)(49 72)(50 73)(91 148)(92 149)(93 150)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 131)(109 132)(110 133)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 161)(118 162)(119 163)(120 164)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 151)(129 152)(130 153)(171 229)(172 230)(173 221)(174 222)(175 223)(176 224)(177 225)(178 226)(179 227)(180 228)(181 213)(182 214)(183 215)(184 216)(185 217)(186 218)(187 219)(188 220)(189 211)(190 212)(191 249)(192 250)(193 241)(194 242)(195 243)(196 244)(197 245)(198 246)(199 247)(200 248)(201 233)(202 234)(203 235)(204 236)(205 237)(206 238)(207 239)(208 240)(209 231)(210 232)(251 309)(252 310)(253 301)(254 302)(255 303)(256 304)(257 305)(258 306)(259 307)(260 308)(261 293)(262 294)(263 295)(264 296)(265 297)(266 298)(267 299)(268 300)(269 291)(270 292)(281 313)(282 314)(283 315)(284 316)(285 317)(286 318)(287 319)(288 320)(289 311)(290 312)
(1 48 28 36)(2 49 29 37)(3 50 30 38)(4 41 21 39)(5 42 22 40)(6 43 23 31)(7 44 24 32)(8 45 25 33)(9 46 26 34)(10 47 27 35)(11 302 319 299)(12 303 320 300)(13 304 311 291)(14 305 312 292)(15 306 313 293)(16 307 314 294)(17 308 315 295)(18 309 316 296)(19 310 317 297)(20 301 318 298)(51 83 63 71)(52 84 64 72)(53 85 65 73)(54 86 66 74)(55 87 67 75)(56 88 68 76)(57 89 69 77)(58 90 70 78)(59 81 61 79)(60 82 62 80)(91 123 103 111)(92 124 104 112)(93 125 105 113)(94 126 106 114)(95 127 107 115)(96 128 108 116)(97 129 109 117)(98 130 110 118)(99 121 101 119)(100 122 102 120)(131 170 143 151)(132 161 144 152)(133 162 145 153)(134 163 146 154)(135 164 147 155)(136 165 148 156)(137 166 149 157)(138 167 150 158)(139 168 141 159)(140 169 142 160)(171 204 184 191)(172 205 185 192)(173 206 186 193)(174 207 187 194)(175 208 188 195)(176 209 189 196)(177 210 190 197)(178 201 181 198)(179 202 182 199)(180 203 183 200)(211 244 224 231)(212 245 225 232)(213 246 226 233)(214 247 227 234)(215 248 228 235)(216 249 229 236)(217 250 230 237)(218 241 221 238)(219 242 222 239)(220 243 223 240)(251 284 264 271)(252 285 265 272)(253 286 266 273)(254 287 267 274)(255 288 268 275)(256 289 269 276)(257 290 270 277)(258 281 261 278)(259 282 262 279)(260 283 263 280)
(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)
(1 299 6 294)(2 298 7 293)(3 297 8 292)(4 296 9 291)(5 295 10 300)(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)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,231)(19,232)(20,233)(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)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,161)(78,162)(79,163)(80,164)(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), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,274)(12,275)(13,276)(14,277)(15,278)(16,279)(17,280)(18,271)(19,272)(20,273)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,88)(32,89)(33,90)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,71)(49,72)(50,73)(91,148)(92,149)(93,150)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,131)(109,132)(110,133)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,161)(118,162)(119,163)(120,164)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,151)(129,152)(130,153)(171,229)(172,230)(173,221)(174,222)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,213)(182,214)(183,215)(184,216)(185,217)(186,218)(187,219)(188,220)(189,211)(190,212)(191,249)(192,250)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)(208,240)(209,231)(210,232)(251,309)(252,310)(253,301)(254,302)(255,303)(256,304)(257,305)(258,306)(259,307)(260,308)(261,293)(262,294)(263,295)(264,296)(265,297)(266,298)(267,299)(268,300)(269,291)(270,292)(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)(288,320)(289,311)(290,312), (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,302,319,299)(12,303,320,300)(13,304,311,291)(14,305,312,292)(15,306,313,293)(16,307,314,294)(17,308,315,295)(18,309,316,296)(19,310,317,297)(20,301,318,298)(51,83,63,71)(52,84,64,72)(53,85,65,73)(54,86,66,74)(55,87,67,75)(56,88,68,76)(57,89,69,77)(58,90,70,78)(59,81,61,79)(60,82,62,80)(91,123,103,111)(92,124,104,112)(93,125,105,113)(94,126,106,114)(95,127,107,115)(96,128,108,116)(97,129,109,117)(98,130,110,118)(99,121,101,119)(100,122,102,120)(131,170,143,151)(132,161,144,152)(133,162,145,153)(134,163,146,154)(135,164,147,155)(136,165,148,156)(137,166,149,157)(138,167,150,158)(139,168,141,159)(140,169,142,160)(171,204,184,191)(172,205,185,192)(173,206,186,193)(174,207,187,194)(175,208,188,195)(176,209,189,196)(177,210,190,197)(178,201,181,198)(179,202,182,199)(180,203,183,200)(211,244,224,231)(212,245,225,232)(213,246,226,233)(214,247,227,234)(215,248,228,235)(216,249,229,236)(217,250,230,237)(218,241,221,238)(219,242,222,239)(220,243,223,240)(251,284,264,271)(252,285,265,272)(253,286,266,273)(254,287,267,274)(255,288,268,275)(256,289,269,276)(257,290,270,277)(258,281,261,278)(259,282,262,279)(260,283,263,280), (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), (1,299,6,294)(2,298,7,293)(3,297,8,292)(4,296,9,291)(5,295,10,300)(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)(11,234)(12,235)(13,236)(14,237)(15,238)(16,239)(17,240)(18,231)(19,232)(20,233)(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)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,161)(78,162)(79,163)(80,164)(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), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,274)(12,275)(13,276)(14,277)(15,278)(16,279)(17,280)(18,271)(19,272)(20,273)(21,54)(22,55)(23,56)(24,57)(25,58)(26,59)(27,60)(28,51)(29,52)(30,53)(31,88)(32,89)(33,90)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,71)(49,72)(50,73)(91,148)(92,149)(93,150)(94,141)(95,142)(96,143)(97,144)(98,145)(99,146)(100,147)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,131)(109,132)(110,133)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,161)(118,162)(119,163)(120,164)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,151)(129,152)(130,153)(171,229)(172,230)(173,221)(174,222)(175,223)(176,224)(177,225)(178,226)(179,227)(180,228)(181,213)(182,214)(183,215)(184,216)(185,217)(186,218)(187,219)(188,220)(189,211)(190,212)(191,249)(192,250)(193,241)(194,242)(195,243)(196,244)(197,245)(198,246)(199,247)(200,248)(201,233)(202,234)(203,235)(204,236)(205,237)(206,238)(207,239)(208,240)(209,231)(210,232)(251,309)(252,310)(253,301)(254,302)(255,303)(256,304)(257,305)(258,306)(259,307)(260,308)(261,293)(262,294)(263,295)(264,296)(265,297)(266,298)(267,299)(268,300)(269,291)(270,292)(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)(288,320)(289,311)(290,312), (1,48,28,36)(2,49,29,37)(3,50,30,38)(4,41,21,39)(5,42,22,40)(6,43,23,31)(7,44,24,32)(8,45,25,33)(9,46,26,34)(10,47,27,35)(11,302,319,299)(12,303,320,300)(13,304,311,291)(14,305,312,292)(15,306,313,293)(16,307,314,294)(17,308,315,295)(18,309,316,296)(19,310,317,297)(20,301,318,298)(51,83,63,71)(52,84,64,72)(53,85,65,73)(54,86,66,74)(55,87,67,75)(56,88,68,76)(57,89,69,77)(58,90,70,78)(59,81,61,79)(60,82,62,80)(91,123,103,111)(92,124,104,112)(93,125,105,113)(94,126,106,114)(95,127,107,115)(96,128,108,116)(97,129,109,117)(98,130,110,118)(99,121,101,119)(100,122,102,120)(131,170,143,151)(132,161,144,152)(133,162,145,153)(134,163,146,154)(135,164,147,155)(136,165,148,156)(137,166,149,157)(138,167,150,158)(139,168,141,159)(140,169,142,160)(171,204,184,191)(172,205,185,192)(173,206,186,193)(174,207,187,194)(175,208,188,195)(176,209,189,196)(177,210,190,197)(178,201,181,198)(179,202,182,199)(180,203,183,200)(211,244,224,231)(212,245,225,232)(213,246,226,233)(214,247,227,234)(215,248,228,235)(216,249,229,236)(217,250,230,237)(218,241,221,238)(219,242,222,239)(220,243,223,240)(251,284,264,271)(252,285,265,272)(253,286,266,273)(254,287,267,274)(255,288,268,275)(256,289,269,276)(257,290,270,277)(258,281,261,278)(259,282,262,279)(260,283,263,280), (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), (1,299,6,294)(2,298,7,293)(3,297,8,292)(4,296,9,291)(5,295,10,300)(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),(11,234),(12,235),(13,236),(14,237),(15,238),(16,239),(17,240),(18,231),(19,232),(20,233),(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),(71,165),(72,166),(73,167),(74,168),(75,169),(76,170),(77,161),(78,162),(79,163),(80,164),(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)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,61),(10,62),(11,274),(12,275),(13,276),(14,277),(15,278),(16,279),(17,280),(18,271),(19,272),(20,273),(21,54),(22,55),(23,56),(24,57),(25,58),(26,59),(27,60),(28,51),(29,52),(30,53),(31,88),(32,89),(33,90),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,71),(49,72),(50,73),(91,148),(92,149),(93,150),(94,141),(95,142),(96,143),(97,144),(98,145),(99,146),(100,147),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,131),(109,132),(110,133),(111,165),(112,166),(113,167),(114,168),(115,169),(116,170),(117,161),(118,162),(119,163),(120,164),(121,154),(122,155),(123,156),(124,157),(125,158),(126,159),(127,160),(128,151),(129,152),(130,153),(171,229),(172,230),(173,221),(174,222),(175,223),(176,224),(177,225),(178,226),(179,227),(180,228),(181,213),(182,214),(183,215),(184,216),(185,217),(186,218),(187,219),(188,220),(189,211),(190,212),(191,249),(192,250),(193,241),(194,242),(195,243),(196,244),(197,245),(198,246),(199,247),(200,248),(201,233),(202,234),(203,235),(204,236),(205,237),(206,238),(207,239),(208,240),(209,231),(210,232),(251,309),(252,310),(253,301),(254,302),(255,303),(256,304),(257,305),(258,306),(259,307),(260,308),(261,293),(262,294),(263,295),(264,296),(265,297),(266,298),(267,299),(268,300),(269,291),(270,292),(281,313),(282,314),(283,315),(284,316),(285,317),(286,318),(287,319),(288,320),(289,311),(290,312)], [(1,48,28,36),(2,49,29,37),(3,50,30,38),(4,41,21,39),(5,42,22,40),(6,43,23,31),(7,44,24,32),(8,45,25,33),(9,46,26,34),(10,47,27,35),(11,302,319,299),(12,303,320,300),(13,304,311,291),(14,305,312,292),(15,306,313,293),(16,307,314,294),(17,308,315,295),(18,309,316,296),(19,310,317,297),(20,301,318,298),(51,83,63,71),(52,84,64,72),(53,85,65,73),(54,86,66,74),(55,87,67,75),(56,88,68,76),(57,89,69,77),(58,90,70,78),(59,81,61,79),(60,82,62,80),(91,123,103,111),(92,124,104,112),(93,125,105,113),(94,126,106,114),(95,127,107,115),(96,128,108,116),(97,129,109,117),(98,130,110,118),(99,121,101,119),(100,122,102,120),(131,170,143,151),(132,161,144,152),(133,162,145,153),(134,163,146,154),(135,164,147,155),(136,165,148,156),(137,166,149,157),(138,167,150,158),(139,168,141,159),(140,169,142,160),(171,204,184,191),(172,205,185,192),(173,206,186,193),(174,207,187,194),(175,208,188,195),(176,209,189,196),(177,210,190,197),(178,201,181,198),(179,202,182,199),(180,203,183,200),(211,244,224,231),(212,245,225,232),(213,246,226,233),(214,247,227,234),(215,248,228,235),(216,249,229,236),(217,250,230,237),(218,241,221,238),(219,242,222,239),(220,243,223,240),(251,284,264,271),(252,285,265,272),(253,286,266,273),(254,287,267,274),(255,288,268,275),(256,289,269,276),(257,290,270,277),(258,281,261,278),(259,282,262,279),(260,283,263,280)], [(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)], [(1,299,6,294),(2,298,7,293),(3,297,8,292),(4,296,9,291),(5,295,10,300),(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 ··· 2O 4A ··· 4P 4Q ··· 4AV 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 D5 Dic5 D10 D10 C4×D5 kernel C22×C4×Dic5 C2×C4×Dic5 C23×Dic5 C23×C20 C22×Dic5 C22×C20 C23×C4 C22×C4 C22×C4 C24 C23 # reps 1 12 2 1 32 16 2 16 12 2 32

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

 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 40 0 0 0 36 6
,
 9 0 0 0 0 0 9 0 0 0 0 0 1 0 0 0 0 0 0 22 0 0 0 13 0

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