C2^3xDic10 - GroupNames

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), 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

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

Upper central

C₁ — C₂⁴ — C₂³×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, C₄, C₄, C₂², C₅, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₀, C₁₀, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₅, C₂₀, C₂×C₁₀, C₂³×C₄, C₂³×C₄, C₂²×Q₈, Dic₁₀, C₂×Dic₅, C₂×C₂₀, C₂²×C₁₀, Q₈×C₂³, C₂×Dic₁₀, C₂²×Dic₅, C₂²×C₂₀, C₂³×C₁₀, C₂²×Dic₁₀, C₂³×Dic₅, C₂³×C₂₀, C₂³×Dic₁₀
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, C₂⁴, D₁₀, C₂²×Q₈, C₂⁵, Dic₁₀, C₂²×D₅, Q₈×C₂³, C₂×Dic₁₀, C₂³×D₅, C₂²×Dic₁₀, D₅×C₂⁴, C₂³×Dic₁₀

Smallest permutation representation of C₂³×Dic₁₀
►Regular action on 320 points

Generators in S₃₂₀

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

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

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

(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)

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

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

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

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

104 conjugacy classes

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

104 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+	+	-	+	+	+	-
image	C₁	C₂	C₂	C₂	Q₈	D₅	D₁₀	D₁₀	Dic₁₀
kernel	C₂³×Dic₁₀	C₂²×Dic₁₀	C₂³×Dic₅	C₂³×C₂₀	C₂²×C₁₀	C₂³×C₄	C₂²×C₄	C₂⁴	C₂³
# reps	1	28	2	1	8	2	28	2	32

Matrix representation of C₂³×Dic₁₀ ►in GL₆(𝔽₄₁)

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	1	0	0
0	0	40	0	0	0
0	0	0	0	1	40
0	0	0	0	36	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	7
0	0	0	0	6	0

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

G = C23×Dic10 order 320 = 26·5

Direct product of C23 and Dic10

G = C₂³×Dic₁₀ order 320 = 2⁶·5

Direct product of C₂³ and Dic₁₀