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G = C2×C8×Dic5order 320 = 26·5

Direct product of C2×C8 and Dic5

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

Aliases: C2×C8×Dic5, C20.55C42, C104(C4×C8), C4041(C2×C4), (C2×C40)⋊22C4, (C2×C8).344D10, C4.20(C4×Dic5), C22.15(C8×D5), C23.60(C4×D5), (C22×C8).18D5, (C22×C40).23C2, (C2×C10).46C42, C10.41(C22×C8), C10.42(C2×C42), (C4×Dic5).51C4, C20.196(C22×C4), (C2×C40).413C22, (C2×C20).853C23, (C22×C4).460D10, C22.20(C4×Dic5), C4.31(C22×Dic5), (C22×Dic5).40C4, (C22×C20).556C22, (C4×Dic5).360C22, C56(C2×C4×C8), C2.3(D5×C2×C8), C4.111(C2×C4×D5), (C2×C52C8)⋊24C4, C52C839(C2×C4), C2.3(C2×C4×Dic5), C22.56(C2×C4×D5), (C2×C10).45(C2×C8), (C2×C4).181(C4×D5), (C2×C4×Dic5).54C2, (C2×C20).492(C2×C4), (C22×C52C8).25C2, (C2×C4).102(C2×Dic5), (C2×C4).795(C22×D5), (C2×C10).224(C22×C4), (C22×C10).156(C2×C4), (C2×C52C8).356C22, (C2×Dic5).206(C2×C4), SmallGroup(320,725)

Series: Derived Chief Lower central Upper central

C1C5 — C2×C8×Dic5
C1C5C10C2×C10C2×C20C4×Dic5C2×C4×Dic5 — C2×C8×Dic5
C5 — C2×C8×Dic5
C1C22×C8

Generators and relations for C2×C8×Dic5
 G = < a,b,c,d | a2=b8=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 334 in 162 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], C23, C10, C10 [×6], C42 [×4], C2×C8 [×6], C2×C8 [×6], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C4×C8 [×4], C2×C42, C22×C8, C22×C8, C52C8 [×4], C40 [×4], C2×Dic5 [×12], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×C4×C8, C2×C52C8 [×6], C4×Dic5 [×4], C2×C40 [×6], C22×Dic5 [×2], C22×C20, C8×Dic5 [×4], C22×C52C8, C2×C4×Dic5, C22×C40, C2×C8×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, D5, C42 [×4], C2×C8 [×12], C22×C4 [×3], Dic5 [×4], D10 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C2×C4×C8, C8×D5 [×4], C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C8×Dic5 [×4], D5×C2×C8 [×2], C2×C4×Dic5, C2×C8×Dic5

Smallest permutation representation of C2×C8×Dic5
Regular action on 320 points
Generators in S320
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 91)(8 92)(9 93)(10 94)(11 108)(12 109)(13 110)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 247)(22 248)(23 249)(24 250)(25 241)(26 242)(27 243)(28 244)(29 245)(30 246)(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)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 169)(88 170)(89 161)(90 162)(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 83 48 74 39 61 14 55)(2 84 49 75 40 62 15 56)(3 85 50 76 31 63 16 57)(4 86 41 77 32 64 17 58)(5 87 42 78 33 65 18 59)(6 88 43 79 34 66 19 60)(7 89 44 80 35 67 20 51)(8 90 45 71 36 68 11 52)(9 81 46 72 37 69 12 53)(10 82 47 73 38 70 13 54)(21 287 316 271 309 264 300 255)(22 288 317 272 310 265 291 256)(23 289 318 273 301 266 292 257)(24 290 319 274 302 267 293 258)(25 281 320 275 303 268 294 259)(26 282 311 276 304 269 295 260)(27 283 312 277 305 270 296 251)(28 284 313 278 306 261 297 252)(29 285 314 279 307 262 298 253)(30 286 315 280 308 263 299 254)(91 161 129 155 120 142 107 136)(92 162 130 156 111 143 108 137)(93 163 121 157 112 144 109 138)(94 164 122 158 113 145 110 139)(95 165 123 159 114 146 101 140)(96 166 124 160 115 147 102 131)(97 167 125 151 116 148 103 132)(98 168 126 152 117 149 104 133)(99 169 127 153 118 150 105 134)(100 170 128 154 119 141 106 135)(171 248 203 232 197 225 190 216)(172 249 204 233 198 226 181 217)(173 250 205 234 199 227 182 218)(174 241 206 235 200 228 183 219)(175 242 207 236 191 229 184 220)(176 243 208 237 192 230 185 211)(177 244 209 238 193 221 186 212)(178 245 210 239 194 222 187 213)(179 246 201 240 195 223 188 214)(180 247 202 231 196 224 189 215)
(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)
(1 180 6 175)(2 179 7 174)(3 178 8 173)(4 177 9 172)(5 176 10 171)(11 182 16 187)(12 181 17 186)(13 190 18 185)(14 189 19 184)(15 188 20 183)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

128 conjugacy classes

class 1 2A···2G4A···4H4I···4X5A5B8A···8P8Q···8AF10A···10N20A···20P40A···40AF
order12···24···44···4558···88···810···1020···2040···40
size11···11···15···5221···15···52···22···22···2

128 irreducible representations

dim11111111112222222
type++++++-++
imageC1C2C2C2C2C4C4C4C4C8D5Dic5D10D10C4×D5C4×D5C8×D5
kernelC2×C8×Dic5C8×Dic5C22×C52C8C2×C4×Dic5C22×C40C2×C52C8C4×Dic5C2×C40C22×Dic5C2×Dic5C22×C8C2×C8C2×C8C22×C4C2×C4C23C22
# reps14111848432284212432

Matrix representation of C2×C8×Dic5 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
38000
0100
00380
00038
,
1000
0100
00140
00366
,
40000
04000
003716
00224
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[38,0,0,0,0,1,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,0,0,0,0,1,36,0,0,40,6],[40,0,0,0,0,40,0,0,0,0,37,22,0,0,16,4] >;

C2×C8×Dic5 in GAP, Magma, Sage, TeX

C_2\times C_8\times {\rm Dic}_5
% in TeX

G:=Group("C2xC8xDic5");
// GroupNames label

G:=SmallGroup(320,725);
// by ID

G=gap.SmallGroup(320,725);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽