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G = C2×Q8×Dic5  order 320 = 26·5

Direct product of C2, Q8 and Dic5

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

Aliases: C2×Q8×Dic5, C10⋊4(C4×Q8), (Q8×C10)⋊21C4, C22.36(Q8×D5), C10.67(C23×C4), (C2×Q8).206D10, (C22×Q8).10D5, C10.52(C22×Q8), C2.8(C23×Dic5), C20.154(C22×C4), (C2×C10).303C24, (C2×C20).550C23, (C22×C4).383D10, C4.18(C22×Dic5), C22.46(C23×D5), C4⋊Dic5.388C22, (Q8×C10).232C22, C23.340(C22×D5), (C22×C20).283C22, (C22×C10).421C23, C22.38(Q8⋊2D5), (C2×Dic5).297C23, (C4×Dic5).291C22, C22.32(C22×Dic5), (C22×Dic5).290C22, C5⋊5(C2×C4×Q8), C2.4(C2×Q8×D5), (Q8×C2×C10).8C2, (C5×Q8)⋊27(C2×C4), C2.4(C2×Q8⋊2D5), (C2×C10).97(C2×Q8), (C2×C4×Dic5).17C2, (C2×C20).306(C2×C4), C10.126(C2×C4○D4), (C2×C4⋊Dic5).46C2, (C2×C4).55(C2×Dic5), (C2×C4).631(C22×D5), (C2×C10).199(C4○D4), (C2×C10).310(C22×C4), SmallGroup(320,1483)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×Q8×Dic5
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C4×Dic5 — C2×Q8×Dic5
Lower central
C5 — C10 — C2×Q8×Dic5
Upper central
C1 — C23 — C22×Q8

Generators and relations for C2×Q8×Dic5
 G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 654 in 298 conjugacy classes, 215 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C42, C2×C4⋊C4, C4×Q8, C22×Q8, C2×Dic5, C2×Dic5, C2×C20, C5×Q8, C22×C10, C2×C4×Q8, C4×Dic5, C4⋊Dic5, C22×Dic5, C22×Dic5, C22×C20, Q8×C10, C2×C4×Dic5, C2×C4⋊Dic5, Q8×Dic5, Q8×C2×C10, C2×Q8×Dic5
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, D5, C22×C4, C2×Q8, C4○D4, C24, Dic5, D10, C4×Q8, C23×C4, C22×Q8, C2×C4○D4, C2×Dic5, C22×D5, C2×C4×Q8, Q8×D5, Q8⋊2D5, C22×Dic5, C23×D5, Q8×Dic5, C2×Q8×D5, C2×Q8⋊2D5, C23×Dic5, C2×Q8×Dic5

Smallest permutation representation of C2×Q8×Dic5
►Regular action on 320 points
Generators in S320
(1 108)(2 109)(3 110)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 107)(11 235)(12 236)(13 237)(14 238)(15 239)(16 240)(17 231)(18 232)(19 233)(20 234)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 91)(30 92)(31 129)(32 130)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 111)(50 112)(51 149)(52 150)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 131)(70 132)(71 166)(72 167)(73 168)(74 169)(75 170)(76 161)(77 162)(78 163)(79 164)(80 165)(81 153)(82 154)(83 155)(84 156)(85 157)(86 158)(87 159)(88 160)(89 151)(90 152)(171 268)(172 269)(173 270)(174 261)(175 262)(176 263)(177 264)(178 265)(179 266)(180 267)(181 254)(182 255)(183 256)(184 257)(185 258)(186 259)(187 260)(188 251)(189 252)(190 253)(191 288)(192 289)(193 290)(194 281)(195 282)(196 283)(197 284)(198 285)(199 286)(200 287)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 271)(209 272)(210 273)(211 308)(212 309)(213 310)(214 301)(215 302)(216 303)(217 304)(218 305)(219 306)(220 307)(221 294)(222 295)(223 296)(224 297)(225 298)(226 299)(227 300)(228 291)(229 292)(230 293)(241 314)(242 315)(243 316)(244 317)(245 318)(246 319)(247 320)(248 311)(249 312)(250 313)

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

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

(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)

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

80 conjugacy classes

class 1 2A···2G4A···4L4M···4T4U···4AF5A5B10A···10N20A···20X
order12···24···44···44···45510···1020···20
size11···12···25···510···10222···24···4

80 irreducible representations

dim11111122222244
type+++++-++-+-+
imageC1C2C2C2C2C4Q8D5C4○D4D10Dic5D10Q8×D5Q8⋊2D5
kernelC2×Q8×Dic5C2×C4×Dic5C2×C4⋊Dic5Q8×Dic5Q8×C2×C10Q8×C10C2×Dic5C22×Q8C2×C10C22×C4C2×Q8C2×Q8C22C22
# reps1338116424616844

Matrix representation of C2×Q8×Dic5 ►in GL5(𝔽41)

400000
040000
004000
00010
00001
,
10000
00100
040000
00010
00001
,
400000
09000
003200
00010
00001
,
10000
01000
00100
000740
00010
,
400000
040000
004000
0002316
0001318

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,7,1,0,0,0,40,0],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,23,13,0,0,0,16,18] >;
 

C2×Q8×Dic5 in GAP, Magma, Sage, TeX 

C_2\times Q_8\times {\rm Dic}_5
 
% in TeX
 
G:=Group("C2xQ8xDic5");
 
// GroupNames label
 
G:=SmallGroup(320,1483);
 
// by ID
 
G=gap.SmallGroup(320,1483);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,184,297,136,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=d^10=1,c^2=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
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≀
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