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G = C23×Dic10order 320 = 26·5

Direct product of C23 and Dic10

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

Aliases: C23×Dic10, C10.1C25, C20.64C24, C24.80D10, Dic5.1C24, C51(Q8×C23), C101(C22×Q8), (C22×C10)⋊8Q8, C2.3(D5×C24), C4.61(C23×D5), (C23×C4).13D5, (C23×C20).14C2, (C2×C20).790C23, (C2×C10).323C24, (C22×C4).450D10, C22.51(C23×D5), (C23×Dic5).11C2, C23.344(C22×D5), (C23×C10).113C22, (C22×C10).430C23, (C22×C20).531C22, (C2×Dic5).305C23, (C22×Dic5).261C22, (C2×C10)⋊7(C2×Q8), (C2×C4).741(C22×D5), SmallGroup(320,1608)

Series: Derived Chief Lower central Upper central

C1C10 — C23×Dic10
C1C5C10Dic5C2×Dic5C22×Dic5C23×Dic5 — C23×Dic10
C5C10 — C23×Dic10
C1C24C23×C4

Generators and relations for C23×Dic10
 G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=d10, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 2078 in 850 conjugacy classes, 543 normal (9 characteristic)
C1, C2, C2 [×14], C4 [×8], C4 [×16], C22 [×35], C5, C2×C4 [×28], C2×C4 [×56], Q8 [×64], C23 [×15], C10, C10 [×14], C22×C4 [×14], C22×C4 [×28], C2×Q8 [×112], C24, Dic5 [×16], C20 [×8], C2×C10 [×35], C23×C4, C23×C4 [×2], C22×Q8 [×28], Dic10 [×64], C2×Dic5 [×56], C2×C20 [×28], C22×C10 [×15], Q8×C23, C2×Dic10 [×112], C22×Dic5 [×28], C22×C20 [×14], C23×C10, C22×Dic10 [×28], C23×Dic5 [×2], C23×C20, C23×Dic10
Quotients: C1, C2 [×31], C22 [×155], Q8 [×8], C23 [×155], D5, C2×Q8 [×28], C24 [×31], D10 [×15], C22×Q8 [×14], C25, Dic10 [×8], C22×D5 [×35], Q8×C23, C2×Dic10 [×28], C23×D5 [×15], C22×Dic10 [×14], D5×C24, C23×Dic10

Smallest permutation representation of C23×Dic10
Regular action on 320 points
Generators in S320
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 61)(20 62)(21 182)(22 183)(23 184)(24 185)(25 186)(26 187)(27 188)(28 189)(29 190)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 197)(37 198)(38 199)(39 200)(40 181)(41 232)(42 233)(43 234)(44 235)(45 236)(46 237)(47 238)(48 239)(49 240)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 229)(59 230)(60 231)(81 270)(82 271)(83 272)(84 273)(85 274)(86 275)(87 276)(88 277)(89 278)(90 279)(91 280)(92 261)(93 262)(94 263)(95 264)(96 265)(97 266)(98 267)(99 268)(100 269)(101 299)(102 300)(103 281)(104 282)(105 283)(106 284)(107 285)(108 286)(109 287)(110 288)(111 289)(112 290)(113 291)(114 292)(115 293)(116 294)(117 295)(118 296)(119 297)(120 298)(121 206)(122 207)(123 208)(124 209)(125 210)(126 211)(127 212)(128 213)(129 214)(130 215)(131 216)(132 217)(133 218)(134 219)(135 220)(136 201)(137 202)(138 203)(139 204)(140 205)(141 243)(142 244)(143 245)(144 246)(145 247)(146 248)(147 249)(148 250)(149 251)(150 252)(151 253)(152 254)(153 255)(154 256)(155 257)(156 258)(157 259)(158 260)(159 241)(160 242)(161 317)(162 318)(163 319)(164 320)(165 301)(166 302)(167 303)(168 304)(169 305)(170 306)(171 307)(172 308)(173 309)(174 310)(175 311)(176 312)(177 313)(178 314)(179 315)(180 316)
(1 167)(2 168)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 177)(12 178)(13 179)(14 180)(15 161)(16 162)(17 163)(18 164)(19 165)(20 166)(21 289)(22 290)(23 291)(24 292)(25 293)(26 294)(27 295)(28 296)(29 297)(30 298)(31 299)(32 300)(33 281)(34 282)(35 283)(36 284)(37 285)(38 286)(39 287)(40 288)(41 253)(42 254)(43 255)(44 256)(45 257)(46 258)(47 259)(48 260)(49 241)(50 242)(51 243)(52 244)(53 245)(54 246)(55 247)(56 248)(57 249)(58 250)(59 251)(60 252)(61 301)(62 302)(63 303)(64 304)(65 305)(66 306)(67 307)(68 308)(69 309)(70 310)(71 311)(72 312)(73 313)(74 314)(75 315)(76 316)(77 317)(78 318)(79 319)(80 320)(81 139)(82 140)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)(101 192)(102 193)(103 194)(104 195)(105 196)(106 197)(107 198)(108 199)(109 200)(110 181)(111 182)(112 183)(113 184)(114 185)(115 186)(116 187)(117 188)(118 189)(119 190)(120 191)(141 222)(142 223)(143 224)(144 225)(145 226)(146 227)(147 228)(148 229)(149 230)(150 231)(151 232)(152 233)(153 234)(154 235)(155 236)(156 237)(157 238)(158 239)(159 240)(160 221)(201 267)(202 268)(203 269)(204 270)(205 271)(206 272)(207 273)(208 274)(209 275)(210 276)(211 277)(212 278)(213 279)(214 280)(215 261)(216 262)(217 263)(218 264)(219 265)(220 266)
(1 259)(2 260)(3 241)(4 242)(5 243)(6 244)(7 245)(8 246)(9 247)(10 248)(11 249)(12 250)(13 251)(14 252)(15 253)(16 254)(17 255)(18 256)(19 257)(20 258)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 97)(30 98)(31 99)(32 100)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 169)(50 170)(51 171)(52 172)(53 173)(54 174)(55 175)(56 176)(57 177)(58 178)(59 179)(60 180)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(101 202)(102 203)(103 204)(104 205)(105 206)(106 207)(107 208)(108 209)(109 210)(110 211)(111 212)(112 213)(113 214)(114 215)(115 216)(116 217)(117 218)(118 219)(119 220)(120 201)(121 283)(122 284)(123 285)(124 286)(125 287)(126 288)(127 289)(128 290)(129 291)(130 292)(131 293)(132 294)(133 295)(134 296)(135 297)(136 298)(137 299)(138 300)(139 281)(140 282)(181 277)(182 278)(183 279)(184 280)(185 261)(186 262)(187 263)(188 264)(189 265)(190 266)(191 267)(192 268)(193 269)(194 270)(195 271)(196 272)(197 273)(198 274)(199 275)(200 276)(221 306)(222 307)(223 308)(224 309)(225 310)(226 311)(227 312)(228 313)(229 314)(230 315)(231 316)(232 317)(233 318)(234 319)(235 320)(236 301)(237 302)(238 303)(239 304)(240 305)
(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)
(1 269 11 279)(2 268 12 278)(3 267 13 277)(4 266 14 276)(5 265 15 275)(6 264 16 274)(7 263 17 273)(8 262 18 272)(9 261 19 271)(10 280 20 270)(21 158 31 148)(22 157 32 147)(23 156 33 146)(24 155 34 145)(25 154 35 144)(26 153 36 143)(27 152 37 142)(28 151 38 141)(29 150 39 160)(30 149 40 159)(41 108 51 118)(42 107 52 117)(43 106 53 116)(44 105 54 115)(45 104 55 114)(46 103 56 113)(47 102 57 112)(48 101 58 111)(49 120 59 110)(50 119 60 109)(61 82 71 92)(62 81 72 91)(63 100 73 90)(64 99 74 89)(65 98 75 88)(66 97 76 87)(67 96 77 86)(68 95 78 85)(69 94 79 84)(70 93 80 83)(121 310 131 320)(122 309 132 319)(123 308 133 318)(124 307 134 317)(125 306 135 316)(126 305 136 315)(127 304 137 314)(128 303 138 313)(129 302 139 312)(130 301 140 311)(161 209 171 219)(162 208 172 218)(163 207 173 217)(164 206 174 216)(165 205 175 215)(166 204 176 214)(167 203 177 213)(168 202 178 212)(169 201 179 211)(170 220 180 210)(181 241 191 251)(182 260 192 250)(183 259 193 249)(184 258 194 248)(185 257 195 247)(186 256 196 246)(187 255 197 245)(188 254 198 244)(189 253 199 243)(190 252 200 242)(221 297 231 287)(222 296 232 286)(223 295 233 285)(224 294 234 284)(225 293 235 283)(226 292 236 282)(227 291 237 281)(228 290 238 300)(229 289 239 299)(230 288 240 298)

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

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

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

104 conjugacy classes

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

104 irreducible representations

dim111122222
type++++-+++-
imageC1C2C2C2Q8D5D10D10Dic10
kernelC23×Dic10C22×Dic10C23×Dic5C23×C20C22×C10C23×C4C22×C4C24C23
# reps128218228232

Matrix representation of C23×Dic10 in GL6(𝔽41)

4000000
010000
0040000
0004000
000010
000001
,
4000000
010000
001000
000100
0000400
0000040
,
100000
0400000
001000
000100
000010
000001
,
100000
010000
000100
0040000
0000140
0000366
,
100000
0400000
000900
009000
000007
000060

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,36,0,0,0,0,40,6],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,6,0,0,0,0,7,0] >;

C23×Dic10 in GAP, Magma, Sage, TeX

C_2^3\times {\rm Dic}_{10}
% in TeX

G:=Group("C2^3xDic10");
// GroupNames label

G:=SmallGroup(320,1608);
// by ID

G=gap.SmallGroup(320,1608);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=d^10,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