direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊Dic5, (Q8×C10)⋊15C4, Q8⋊3(C2×Dic5), (C2×Q8)⋊3Dic5, (C2×C20).192D4, C20.207(C2×D4), C10.46(C2×Q16), (C2×C10).18Q16, C10⋊4(Q8⋊C4), (C22×Q8).1D5, (C2×Q8).165D10, (C2×C10).42SD16, C10.79(C2×SD16), C20.80(C22⋊C4), C20.140(C22×C4), (C2×C20).474C23, (C22×C4).354D10, (C22×C10).198D4, C22.12(Q8⋊D5), C4.10(C23.D5), C4.11(C22×Dic5), C22.9(C5⋊Q16), C23.101(C5⋊D4), C4⋊Dic5.352C22, (Q8×C10).200C22, (C22×C20).200C22, C22.35(C23.D5), C5⋊5(C2×Q8⋊C4), C2.4(C2×Q8⋊D5), (Q8×C2×C10).1C2, (C5×Q8)⋊24(C2×C4), C4.91(C2×C5⋊D4), C2.4(C2×C5⋊Q16), (C2×C20).292(C2×C4), (C2×C10).558(C2×D4), (C2×C4⋊Dic5).41C2, (C2×C4).51(C2×Dic5), C22.93(C2×C5⋊D4), C2.13(C2×C23.D5), (C2×C4).149(C5⋊D4), C10.118(C2×C22⋊C4), (C22×C5⋊2C8).13C2, (C2×C4).560(C22×D5), (C2×C5⋊2C8).289C22, (C2×C10).178(C22⋊C4), SmallGroup(320,851)
Series: Derived ►Chief ►Lower central ►Upper central
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=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >
Subgroups: 414 in 162 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic5, C20, C20, C20, C2×C10, C2×C10, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C5⋊2C8, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×C10, C2×Q8⋊C4, C2×C5⋊2C8, C2×C5⋊2C8, C4⋊Dic5, C4⋊Dic5, C22×Dic5, C22×C20, C22×C20, Q8×C10, Q8×C10, Q8⋊Dic5, C22×C5⋊2C8, C2×C4⋊Dic5, Q8×C2×C10, C2×Q8⋊Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, SD16, Q16, C22×C4, C2×D4, Dic5, D10, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C2×Dic5, C5⋊D4, C22×D5, C2×Q8⋊C4, Q8⋊D5, C5⋊Q16, C23.D5, C22×Dic5, C2×C5⋊D4, Q8⋊Dic5, C2×Q8⋊D5, C2×C5⋊Q16, C2×C23.D5, C2×Q8⋊Dic5
►Regular action on 320 points
Generators in S320
(1 100)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 243)(22 244)(23 245)(24 246)(25 247)(26 248)(27 249)(28 250)(29 241)(30 242)(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)(81 169)(82 170)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(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 47 31 19)(2 48 32 20)(3 49 33 11)(4 50 34 12)(5 41 35 13)(6 42 36 14)(7 43 37 15)(8 44 38 16)(9 45 39 17)(10 46 40 18)(21 319 306 300)(22 320 307 291)(23 311 308 292)(24 312 309 293)(25 313 310 294)(26 314 301 295)(27 315 302 296)(28 316 303 297)(29 317 304 298)(30 318 305 299)(51 70 77 83)(52 61 78 84)(53 62 79 85)(54 63 80 86)(55 64 71 87)(56 65 72 88)(57 66 73 89)(58 67 74 90)(59 68 75 81)(60 69 76 82)(91 123 117 110)(92 124 118 101)(93 125 119 102)(94 126 120 103)(95 127 111 104)(96 128 112 105)(97 129 113 106)(98 130 114 107)(99 121 115 108)(100 122 116 109)(131 150 157 166)(132 141 158 167)(133 142 159 168)(134 143 160 169)(135 144 151 170)(136 145 152 161)(137 146 153 162)(138 147 154 163)(139 148 155 164)(140 149 156 165)(171 187 200 209)(172 188 191 210)(173 189 192 201)(174 190 193 202)(175 181 194 203)(176 182 195 204)(177 183 196 205)(178 184 197 206)(179 185 198 207)(180 186 199 208)(211 249 240 227)(212 250 231 228)(213 241 232 229)(214 242 233 230)(215 243 234 221)(216 244 235 222)(217 245 236 223)(218 246 237 224)(219 247 238 225)(220 248 239 226)(251 267 280 289)(252 268 271 290)(253 269 272 281)(254 270 273 282)(255 261 274 283)(256 262 275 284)(257 263 276 285)(258 264 277 286)(259 265 278 287)(260 266 279 288)
(1 76 31 60)(2 77 32 51)(3 78 33 52)(4 79 34 53)(5 80 35 54)(6 71 36 55)(7 72 37 56)(8 73 38 57)(9 74 39 58)(10 75 40 59)(11 84 49 61)(12 85 50 62)(13 86 41 63)(14 87 42 64)(15 88 43 65)(16 89 44 66)(17 90 45 67)(18 81 46 68)(19 82 47 69)(20 83 48 70)(21 274 306 255)(22 275 307 256)(23 276 308 257)(24 277 309 258)(25 278 310 259)(26 279 301 260)(27 280 302 251)(28 271 303 252)(29 272 304 253)(30 273 305 254)(91 152 117 136)(92 153 118 137)(93 154 119 138)(94 155 120 139)(95 156 111 140)(96 157 112 131)(97 158 113 132)(98 159 114 133)(99 160 115 134)(100 151 116 135)(101 162 124 146)(102 163 125 147)(103 164 126 148)(104 165 127 149)(105 166 128 150)(106 167 129 141)(107 168 130 142)(108 169 121 143)(109 170 122 144)(110 161 123 145)(171 244 200 222)(172 245 191 223)(173 246 192 224)(174 247 193 225)(175 248 194 226)(176 249 195 227)(177 250 196 228)(178 241 197 229)(179 242 198 230)(180 243 199 221)(181 220 203 239)(182 211 204 240)(183 212 205 231)(184 213 206 232)(185 214 207 233)(186 215 208 234)(187 216 209 235)(188 217 210 236)(189 218 201 237)(190 219 202 238)(261 300 283 319)(262 291 284 320)(263 292 285 311)(264 293 286 312)(265 294 287 313)(266 295 288 314)(267 296 289 315)(268 297 290 316)(269 298 281 317)(270 299 282 318)
(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)(11 269 16 264)(12 268 17 263)(13 267 18 262)(14 266 19 261)(15 265 20 270)(21 87 26 82)(22 86 27 81)(23 85 28 90)(24 84 29 89)(25 83 30 88)(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)(51 299 56 294)(52 298 57 293)(53 297 58 292)(54 296 59 291)(55 295 60 300)(61 304 66 309)(62 303 67 308)(63 302 68 307)(64 301 69 306)(65 310 70 305)(71 314 76 319)(72 313 77 318)(73 312 78 317)(74 311 79 316)(75 320 80 315)(91 179 96 174)(92 178 97 173)(93 177 98 172)(94 176 99 171)(95 175 100 180)(101 184 106 189)(102 183 107 188)(103 182 108 187)(104 181 109 186)(105 190 110 185)(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)(131 219 136 214)(132 218 137 213)(133 217 138 212)(134 216 139 211)(135 215 140 220)(141 224 146 229)(142 223 147 228)(143 222 148 227)(144 221 149 226)(145 230 150 225)(151 234 156 239)(152 233 157 238)(153 232 158 237)(154 231 159 236)(155 240 160 235)(161 242 166 247)(162 241 167 246)(163 250 168 245)(164 249 169 244)(165 248 170 243)
G:=sub<Sym(320)| (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,243)(22,244)(23,245)(24,246)(25,247)(26,248)(27,249)(28,250)(29,241)(30,242)(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)(81,169)(82,170)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(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), (1,47,31,19)(2,48,32,20)(3,49,33,11)(4,50,34,12)(5,41,35,13)(6,42,36,14)(7,43,37,15)(8,44,38,16)(9,45,39,17)(10,46,40,18)(21,319,306,300)(22,320,307,291)(23,311,308,292)(24,312,309,293)(25,313,310,294)(26,314,301,295)(27,315,302,296)(28,316,303,297)(29,317,304,298)(30,318,305,299)(51,70,77,83)(52,61,78,84)(53,62,79,85)(54,63,80,86)(55,64,71,87)(56,65,72,88)(57,66,73,89)(58,67,74,90)(59,68,75,81)(60,69,76,82)(91,123,117,110)(92,124,118,101)(93,125,119,102)(94,126,120,103)(95,127,111,104)(96,128,112,105)(97,129,113,106)(98,130,114,107)(99,121,115,108)(100,122,116,109)(131,150,157,166)(132,141,158,167)(133,142,159,168)(134,143,160,169)(135,144,151,170)(136,145,152,161)(137,146,153,162)(138,147,154,163)(139,148,155,164)(140,149,156,165)(171,187,200,209)(172,188,191,210)(173,189,192,201)(174,190,193,202)(175,181,194,203)(176,182,195,204)(177,183,196,205)(178,184,197,206)(179,185,198,207)(180,186,199,208)(211,249,240,227)(212,250,231,228)(213,241,232,229)(214,242,233,230)(215,243,234,221)(216,244,235,222)(217,245,236,223)(218,246,237,224)(219,247,238,225)(220,248,239,226)(251,267,280,289)(252,268,271,290)(253,269,272,281)(254,270,273,282)(255,261,274,283)(256,262,275,284)(257,263,276,285)(258,264,277,286)(259,265,278,287)(260,266,279,288), (1,76,31,60)(2,77,32,51)(3,78,33,52)(4,79,34,53)(5,80,35,54)(6,71,36,55)(7,72,37,56)(8,73,38,57)(9,74,39,58)(10,75,40,59)(11,84,49,61)(12,85,50,62)(13,86,41,63)(14,87,42,64)(15,88,43,65)(16,89,44,66)(17,90,45,67)(18,81,46,68)(19,82,47,69)(20,83,48,70)(21,274,306,255)(22,275,307,256)(23,276,308,257)(24,277,309,258)(25,278,310,259)(26,279,301,260)(27,280,302,251)(28,271,303,252)(29,272,304,253)(30,273,305,254)(91,152,117,136)(92,153,118,137)(93,154,119,138)(94,155,120,139)(95,156,111,140)(96,157,112,131)(97,158,113,132)(98,159,114,133)(99,160,115,134)(100,151,116,135)(101,162,124,146)(102,163,125,147)(103,164,126,148)(104,165,127,149)(105,166,128,150)(106,167,129,141)(107,168,130,142)(108,169,121,143)(109,170,122,144)(110,161,123,145)(171,244,200,222)(172,245,191,223)(173,246,192,224)(174,247,193,225)(175,248,194,226)(176,249,195,227)(177,250,196,228)(178,241,197,229)(179,242,198,230)(180,243,199,221)(181,220,203,239)(182,211,204,240)(183,212,205,231)(184,213,206,232)(185,214,207,233)(186,215,208,234)(187,216,209,235)(188,217,210,236)(189,218,201,237)(190,219,202,238)(261,300,283,319)(262,291,284,320)(263,292,285,311)(264,293,286,312)(265,294,287,313)(266,295,288,314)(267,296,289,315)(268,297,290,316)(269,298,281,317)(270,299,282,318), (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)(11,269,16,264)(12,268,17,263)(13,267,18,262)(14,266,19,261)(15,265,20,270)(21,87,26,82)(22,86,27,81)(23,85,28,90)(24,84,29,89)(25,83,30,88)(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)(51,299,56,294)(52,298,57,293)(53,297,58,292)(54,296,59,291)(55,295,60,300)(61,304,66,309)(62,303,67,308)(63,302,68,307)(64,301,69,306)(65,310,70,305)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,179,96,174)(92,178,97,173)(93,177,98,172)(94,176,99,171)(95,175,100,180)(101,184,106,189)(102,183,107,188)(103,182,108,187)(104,181,109,186)(105,190,110,185)(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)(131,219,136,214)(132,218,137,213)(133,217,138,212)(134,216,139,211)(135,215,140,220)(141,224,146,229)(142,223,147,228)(143,222,148,227)(144,221,149,226)(145,230,150,225)(151,234,156,239)(152,233,157,238)(153,232,158,237)(154,231,159,236)(155,240,160,235)(161,242,166,247)(162,241,167,246)(163,250,168,245)(164,249,169,244)(165,248,170,243)>;
G:=Group( (1,100)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,243)(22,244)(23,245)(24,246)(25,247)(26,248)(27,249)(28,250)(29,241)(30,242)(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)(81,169)(82,170)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(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), (1,47,31,19)(2,48,32,20)(3,49,33,11)(4,50,34,12)(5,41,35,13)(6,42,36,14)(7,43,37,15)(8,44,38,16)(9,45,39,17)(10,46,40,18)(21,319,306,300)(22,320,307,291)(23,311,308,292)(24,312,309,293)(25,313,310,294)(26,314,301,295)(27,315,302,296)(28,316,303,297)(29,317,304,298)(30,318,305,299)(51,70,77,83)(52,61,78,84)(53,62,79,85)(54,63,80,86)(55,64,71,87)(56,65,72,88)(57,66,73,89)(58,67,74,90)(59,68,75,81)(60,69,76,82)(91,123,117,110)(92,124,118,101)(93,125,119,102)(94,126,120,103)(95,127,111,104)(96,128,112,105)(97,129,113,106)(98,130,114,107)(99,121,115,108)(100,122,116,109)(131,150,157,166)(132,141,158,167)(133,142,159,168)(134,143,160,169)(135,144,151,170)(136,145,152,161)(137,146,153,162)(138,147,154,163)(139,148,155,164)(140,149,156,165)(171,187,200,209)(172,188,191,210)(173,189,192,201)(174,190,193,202)(175,181,194,203)(176,182,195,204)(177,183,196,205)(178,184,197,206)(179,185,198,207)(180,186,199,208)(211,249,240,227)(212,250,231,228)(213,241,232,229)(214,242,233,230)(215,243,234,221)(216,244,235,222)(217,245,236,223)(218,246,237,224)(219,247,238,225)(220,248,239,226)(251,267,280,289)(252,268,271,290)(253,269,272,281)(254,270,273,282)(255,261,274,283)(256,262,275,284)(257,263,276,285)(258,264,277,286)(259,265,278,287)(260,266,279,288), (1,76,31,60)(2,77,32,51)(3,78,33,52)(4,79,34,53)(5,80,35,54)(6,71,36,55)(7,72,37,56)(8,73,38,57)(9,74,39,58)(10,75,40,59)(11,84,49,61)(12,85,50,62)(13,86,41,63)(14,87,42,64)(15,88,43,65)(16,89,44,66)(17,90,45,67)(18,81,46,68)(19,82,47,69)(20,83,48,70)(21,274,306,255)(22,275,307,256)(23,276,308,257)(24,277,309,258)(25,278,310,259)(26,279,301,260)(27,280,302,251)(28,271,303,252)(29,272,304,253)(30,273,305,254)(91,152,117,136)(92,153,118,137)(93,154,119,138)(94,155,120,139)(95,156,111,140)(96,157,112,131)(97,158,113,132)(98,159,114,133)(99,160,115,134)(100,151,116,135)(101,162,124,146)(102,163,125,147)(103,164,126,148)(104,165,127,149)(105,166,128,150)(106,167,129,141)(107,168,130,142)(108,169,121,143)(109,170,122,144)(110,161,123,145)(171,244,200,222)(172,245,191,223)(173,246,192,224)(174,247,193,225)(175,248,194,226)(176,249,195,227)(177,250,196,228)(178,241,197,229)(179,242,198,230)(180,243,199,221)(181,220,203,239)(182,211,204,240)(183,212,205,231)(184,213,206,232)(185,214,207,233)(186,215,208,234)(187,216,209,235)(188,217,210,236)(189,218,201,237)(190,219,202,238)(261,300,283,319)(262,291,284,320)(263,292,285,311)(264,293,286,312)(265,294,287,313)(266,295,288,314)(267,296,289,315)(268,297,290,316)(269,298,281,317)(270,299,282,318), (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)(11,269,16,264)(12,268,17,263)(13,267,18,262)(14,266,19,261)(15,265,20,270)(21,87,26,82)(22,86,27,81)(23,85,28,90)(24,84,29,89)(25,83,30,88)(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)(51,299,56,294)(52,298,57,293)(53,297,58,292)(54,296,59,291)(55,295,60,300)(61,304,66,309)(62,303,67,308)(63,302,68,307)(64,301,69,306)(65,310,70,305)(71,314,76,319)(72,313,77,318)(73,312,78,317)(74,311,79,316)(75,320,80,315)(91,179,96,174)(92,178,97,173)(93,177,98,172)(94,176,99,171)(95,175,100,180)(101,184,106,189)(102,183,107,188)(103,182,108,187)(104,181,109,186)(105,190,110,185)(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)(131,219,136,214)(132,218,137,213)(133,217,138,212)(134,216,139,211)(135,215,140,220)(141,224,146,229)(142,223,147,228)(143,222,148,227)(144,221,149,226)(145,230,150,225)(151,234,156,239)(152,233,157,238)(153,232,158,237)(154,231,159,236)(155,240,160,235)(161,242,166,247)(162,241,167,246)(163,250,168,245)(164,249,169,244)(165,248,170,243) );
G=PermutationGroup([[(1,100),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,243),(22,244),(23,245),(24,246),(25,247),(26,248),(27,249),(28,250),(29,241),(30,242),(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),(81,169),(82,170),(83,161),(84,162),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(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)], [(1,47,31,19),(2,48,32,20),(3,49,33,11),(4,50,34,12),(5,41,35,13),(6,42,36,14),(7,43,37,15),(8,44,38,16),(9,45,39,17),(10,46,40,18),(21,319,306,300),(22,320,307,291),(23,311,308,292),(24,312,309,293),(25,313,310,294),(26,314,301,295),(27,315,302,296),(28,316,303,297),(29,317,304,298),(30,318,305,299),(51,70,77,83),(52,61,78,84),(53,62,79,85),(54,63,80,86),(55,64,71,87),(56,65,72,88),(57,66,73,89),(58,67,74,90),(59,68,75,81),(60,69,76,82),(91,123,117,110),(92,124,118,101),(93,125,119,102),(94,126,120,103),(95,127,111,104),(96,128,112,105),(97,129,113,106),(98,130,114,107),(99,121,115,108),(100,122,116,109),(131,150,157,166),(132,141,158,167),(133,142,159,168),(134,143,160,169),(135,144,151,170),(136,145,152,161),(137,146,153,162),(138,147,154,163),(139,148,155,164),(140,149,156,165),(171,187,200,209),(172,188,191,210),(173,189,192,201),(174,190,193,202),(175,181,194,203),(176,182,195,204),(177,183,196,205),(178,184,197,206),(179,185,198,207),(180,186,199,208),(211,249,240,227),(212,250,231,228),(213,241,232,229),(214,242,233,230),(215,243,234,221),(216,244,235,222),(217,245,236,223),(218,246,237,224),(219,247,238,225),(220,248,239,226),(251,267,280,289),(252,268,271,290),(253,269,272,281),(254,270,273,282),(255,261,274,283),(256,262,275,284),(257,263,276,285),(258,264,277,286),(259,265,278,287),(260,266,279,288)], [(1,76,31,60),(2,77,32,51),(3,78,33,52),(4,79,34,53),(5,80,35,54),(6,71,36,55),(7,72,37,56),(8,73,38,57),(9,74,39,58),(10,75,40,59),(11,84,49,61),(12,85,50,62),(13,86,41,63),(14,87,42,64),(15,88,43,65),(16,89,44,66),(17,90,45,67),(18,81,46,68),(19,82,47,69),(20,83,48,70),(21,274,306,255),(22,275,307,256),(23,276,308,257),(24,277,309,258),(25,278,310,259),(26,279,301,260),(27,280,302,251),(28,271,303,252),(29,272,304,253),(30,273,305,254),(91,152,117,136),(92,153,118,137),(93,154,119,138),(94,155,120,139),(95,156,111,140),(96,157,112,131),(97,158,113,132),(98,159,114,133),(99,160,115,134),(100,151,116,135),(101,162,124,146),(102,163,125,147),(103,164,126,148),(104,165,127,149),(105,166,128,150),(106,167,129,141),(107,168,130,142),(108,169,121,143),(109,170,122,144),(110,161,123,145),(171,244,200,222),(172,245,191,223),(173,246,192,224),(174,247,193,225),(175,248,194,226),(176,249,195,227),(177,250,196,228),(178,241,197,229),(179,242,198,230),(180,243,199,221),(181,220,203,239),(182,211,204,240),(183,212,205,231),(184,213,206,232),(185,214,207,233),(186,215,208,234),(187,216,209,235),(188,217,210,236),(189,218,201,237),(190,219,202,238),(261,300,283,319),(262,291,284,320),(263,292,285,311),(264,293,286,312),(265,294,287,313),(266,295,288,314),(267,296,289,315),(268,297,290,316),(269,298,281,317),(270,299,282,318)], [(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),(11,269,16,264),(12,268,17,263),(13,267,18,262),(14,266,19,261),(15,265,20,270),(21,87,26,82),(22,86,27,81),(23,85,28,90),(24,84,29,89),(25,83,30,88),(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),(51,299,56,294),(52,298,57,293),(53,297,58,292),(54,296,59,291),(55,295,60,300),(61,304,66,309),(62,303,67,308),(63,302,68,307),(64,301,69,306),(65,310,70,305),(71,314,76,319),(72,313,77,318),(73,312,78,317),(74,311,79,316),(75,320,80,315),(91,179,96,174),(92,178,97,173),(93,177,98,172),(94,176,99,171),(95,175,100,180),(101,184,106,189),(102,183,107,188),(103,182,108,187),(104,181,109,186),(105,190,110,185),(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),(131,219,136,214),(132,218,137,213),(133,217,138,212),(134,216,139,211),(135,215,140,220),(141,224,146,229),(142,223,147,228),(143,222,148,227),(144,221,149,226),(145,230,150,225),(151,234,156,239),(152,233,157,238),(153,232,158,237),(154,231,159,236),(155,240,160,235),(161,242,166,247),(162,241,167,246),(163,250,168,245),(164,249,169,244),(165,248,170,243)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | SD16 | Q16 | D10 | Dic5 | D10 | C5⋊D4 | C5⋊D4 | Q8⋊D5 | C5⋊Q16 |
| kernel | C2×Q8⋊Dic5 | Q8⋊Dic5 | C22×C5⋊2C8 | C2×C4⋊Dic5 | Q8×C2×C10 | Q8×C10 | C2×C20 | C22×C10 | C22×Q8 | C2×C10 | C2×C10 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C23 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 12 | 4 | 4 | 4 |
Matrix representation of C2×Q8⋊Dic5 ►in GL5(𝔽41)
| 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 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 9 |
| 0 | 0 | 0 | 18 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 40 | 0 | 0 |
| 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 31 | 32 |
| 0 | 0 | 0 | 34 | 10 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 34 | 40 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 7 | 1 | 0 | 0 |
| 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 0 | 39 | 38 |
| 0 | 0 | 0 | 1 | 2 |
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],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,18,0,0,0,9,1],[1,0,0,0,0,0,17,1,0,0,0,40,24,0,0,0,0,0,31,34,0,0,0,32,10],[40,0,0,0,0,0,34,1,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,7,34,0,0,0,1,34,0,0,0,0,0,39,1,0,0,0,38,2] >;
C2×Q8⋊Dic5 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes {\rm Dic}_5 % in TeX
G:=Group("C2xQ8:Dic5"); // GroupNames label
G:=SmallGroup(320,851);
// by ID
G=gap.SmallGroup(320,851);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,184,1684,438,102,12550]);
// 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=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations