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## G = C24×Dic5order 320 = 26·5

### Direct product of C24 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C24×Dic5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C24×Dic5
 Lower central C5 — C24×Dic5
 Upper central C1 — C25

Generators and relations for C24×Dic5
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=e5, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 2590 in 1362 conjugacy classes, 1055 normal (7 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, C10, C10, C22×C4, C24, Dic5, C2×C10, C23×C4, C25, C2×Dic5, C22×C10, C24×C4, C22×Dic5, C23×C10, C23×Dic5, C24×C10, C24×Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, C25, C2×Dic5, C22×D5, C24×C4, C22×Dic5, C23×D5, C23×Dic5, D5×C24, C24×Dic5

Smallest permutation representation of C24×Dic5
Regular action on 320 points
Generators in S320
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 81)(9 82)(10 83)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)(161 246)(162 247)(163 248)(164 249)(165 250)(166 241)(167 242)(168 243)(169 244)(170 245)(171 256)(172 257)(173 258)(174 259)(175 260)(176 251)(177 252)(178 253)(179 254)(180 255)(181 266)(182 267)(183 268)(184 269)(185 270)(186 261)(187 262)(188 263)(189 264)(190 265)(191 276)(192 277)(193 278)(194 279)(195 280)(196 271)(197 272)(198 273)(199 274)(200 275)(201 286)(202 287)(203 288)(204 289)(205 290)(206 281)(207 282)(208 283)(209 284)(210 285)(211 296)(212 297)(213 298)(214 299)(215 300)(216 291)(217 292)(218 293)(219 294)(220 295)(221 306)(222 307)(223 308)(224 309)(225 310)(226 301)(227 302)(228 303)(229 304)(230 305)(231 316)(232 317)(233 318)(234 319)(235 320)(236 311)(237 312)(238 313)(239 314)(240 315)
(1 79)(2 80)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 51)(22 52)(23 53)(24 54)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(81 151)(82 152)(83 153)(84 154)(85 155)(86 156)(87 157)(88 158)(89 159)(90 160)(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)(111 121)(112 122)(113 123)(114 124)(115 125)(116 126)(117 127)(118 128)(119 129)(120 130)(161 231)(162 232)(163 233)(164 234)(165 235)(166 236)(167 237)(168 238)(169 239)(170 240)(171 221)(172 222)(173 223)(174 224)(175 225)(176 226)(177 227)(178 228)(179 229)(180 230)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 201)(192 202)(193 203)(194 204)(195 205)(196 206)(197 207)(198 208)(199 209)(200 210)(241 311)(242 312)(243 313)(244 314)(245 315)(246 316)(247 317)(248 318)(249 319)(250 320)(251 301)(252 302)(253 303)(254 304)(255 305)(256 306)(257 307)(258 308)(259 309)(260 310)(261 291)(262 292)(263 293)(264 294)(265 295)(266 296)(267 297)(268 298)(269 299)(270 300)(271 281)(272 282)(273 283)(274 284)(275 285)(276 286)(277 287)(278 288)(279 289)(280 290)
(1 39)(2 40)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(99 109)(100 110)(121 151)(122 152)(123 153)(124 154)(125 155)(126 156)(127 157)(128 158)(129 159)(130 160)(131 141)(132 142)(133 143)(134 144)(135 145)(136 146)(137 147)(138 148)(139 149)(140 150)(161 191)(162 192)(163 193)(164 194)(165 195)(166 196)(167 197)(168 198)(169 199)(170 200)(171 181)(172 182)(173 183)(174 184)(175 185)(176 186)(177 187)(178 188)(179 189)(180 190)(201 231)(202 232)(203 233)(204 234)(205 235)(206 236)(207 237)(208 238)(209 239)(210 240)(211 221)(212 222)(213 223)(214 224)(215 225)(216 226)(217 227)(218 228)(219 229)(220 230)(241 271)(242 272)(243 273)(244 274)(245 275)(246 276)(247 277)(248 278)(249 279)(250 280)(251 261)(252 262)(253 263)(254 264)(255 265)(256 266)(257 267)(258 268)(259 269)(260 270)(281 311)(282 312)(283 313)(284 314)(285 315)(286 316)(287 317)(288 318)(289 319)(290 320)(291 301)(292 302)(293 303)(294 304)(295 305)(296 306)(297 307)(298 308)(299 309)(300 310)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 11)(9 12)(10 13)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 96)(82 97)(83 98)(84 99)(85 100)(86 91)(87 92)(88 93)(89 94)(90 95)(101 116)(102 117)(103 118)(104 119)(105 120)(106 111)(107 112)(108 113)(109 114)(110 115)(121 136)(122 137)(123 138)(124 139)(125 140)(126 131)(127 132)(128 133)(129 134)(130 135)(141 156)(142 157)(143 158)(144 159)(145 160)(146 151)(147 152)(148 153)(149 154)(150 155)(161 176)(162 177)(163 178)(164 179)(165 180)(166 171)(167 172)(168 173)(169 174)(170 175)(181 196)(182 197)(183 198)(184 199)(185 200)(186 191)(187 192)(188 193)(189 194)(190 195)(201 216)(202 217)(203 218)(204 219)(205 220)(206 211)(207 212)(208 213)(209 214)(210 215)(221 236)(222 237)(223 238)(224 239)(225 240)(226 231)(227 232)(228 233)(229 234)(230 235)(241 256)(242 257)(243 258)(244 259)(245 260)(246 251)(247 252)(248 253)(249 254)(250 255)(261 276)(262 277)(263 278)(264 279)(265 280)(266 271)(267 272)(268 273)(269 274)(270 275)(281 296)(282 297)(283 298)(284 299)(285 300)(286 291)(287 292)(288 293)(289 294)(290 295)(301 316)(302 317)(303 318)(304 319)(305 320)(306 311)(307 312)(308 313)(309 314)(310 315)
(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)
(1 206 6 201)(2 205 7 210)(3 204 8 209)(4 203 9 208)(5 202 10 207)(11 214 16 219)(12 213 17 218)(13 212 18 217)(14 211 19 216)(15 220 20 215)(21 224 26 229)(22 223 27 228)(23 222 28 227)(24 221 29 226)(25 230 30 225)(31 234 36 239)(32 233 37 238)(33 232 38 237)(34 231 39 236)(35 240 40 235)(41 164 46 169)(42 163 47 168)(43 162 48 167)(44 161 49 166)(45 170 50 165)(51 174 56 179)(52 173 57 178)(53 172 58 177)(54 171 59 176)(55 180 60 175)(61 184 66 189)(62 183 67 188)(63 182 68 187)(64 181 69 186)(65 190 70 185)(71 194 76 199)(72 193 77 198)(73 192 78 197)(74 191 79 196)(75 200 80 195)(81 284 86 289)(82 283 87 288)(83 282 88 287)(84 281 89 286)(85 290 90 285)(91 294 96 299)(92 293 97 298)(93 292 98 297)(94 291 99 296)(95 300 100 295)(101 304 106 309)(102 303 107 308)(103 302 108 307)(104 301 109 306)(105 310 110 305)(111 314 116 319)(112 313 117 318)(113 312 118 317)(114 311 119 316)(115 320 120 315)(121 244 126 249)(122 243 127 248)(123 242 128 247)(124 241 129 246)(125 250 130 245)(131 254 136 259)(132 253 137 258)(133 252 138 257)(134 251 139 256)(135 260 140 255)(141 264 146 269)(142 263 147 268)(143 262 148 267)(144 261 149 266)(145 270 150 265)(151 274 156 279)(152 273 157 278)(153 272 158 277)(154 271 159 276)(155 280 160 275)

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

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

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

128 conjugacy classes

 class 1 2A ··· 2AE 4A ··· 4AF 5A 5B 10A ··· 10BJ order 1 2 ··· 2 4 ··· 4 5 5 10 ··· 10 size 1 1 ··· 1 5 ··· 5 2 2 2 ··· 2

128 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D5 Dic5 D10 kernel C24×Dic5 C23×Dic5 C24×C10 C23×C10 C25 C24 C24 # reps 1 30 1 32 2 32 30

Matrix representation of C24×Dic5 in GL6(𝔽41)

 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 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
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 0 40 0 0 0 0 1 35
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 37 14 0 0 0 0 31 4

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,35],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,37,31,0,0,0,0,14,4] >;

C24×Dic5 in GAP, Magma, Sage, TeX

C_2^4\times {\rm Dic}_5
% in TeX

G:=Group("C2^4xDic5");
// GroupNames label

G:=SmallGroup(320,1626);
// by ID

G=gap.SmallGroup(320,1626);
# by ID

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

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

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