Copied to
clipboard

?

G = C23×Dic10  order 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, C5⋊1(Q8×C23), C10⋊1(C22×Q8), (C22×C10)⋊8Q8, C2.3(D5×C24), (C23×C4).13D5, C4.61(C23×D5), (C23×C20).14C2, (C2×C10).323C24, (C2×C20).790C23, (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

Derived series
C1 — C10 — C23×Dic10
Chief series
C1 — C5 — C10 — Dic5 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C23×Dic10
Lower central
C5 — C10 — C23×Dic10

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

Generators and relations
 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 >

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

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

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

(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)

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

Matrix representation ►G ⊆ 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] >;
 

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

In GAP, Magma, Sage, TeX 

C_2^3\times 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
 

׿
×
⋊
ℤ
𝔽
○
≀