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G = Q85Dic10order 320 = 26·5

1st semidirect product of Q8 and Dic10 acting through Inn(Q8)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q85Dic10, C42.123D10, C10.82- 1+4, (C5×Q8)⋊6Q8, C52(Q83Q8), C20⋊Q8.12C2, (C4×Q8).11D5, C20.45(C2×Q8), C4⋊C4.292D10, (Q8×C20).12C2, (C2×Q8).197D10, (Q8×Dic5).11C2, C4.18(C2×Dic10), C10.16(C22×Q8), (C2×C20).493C23, (C4×C20).165C22, (C2×C10).113C24, (C4×Dic10).22C2, C20.6Q8.11C2, C4.Dic10.11C2, Dic5.62(C4○D4), C4⋊Dic5.304C22, (Q8×C10).213C22, (C2×Dic5).51C23, (C4×Dic5).90C22, C2.18(C22×Dic10), C10.D4.9C22, C22.138(C23×D5), C2.11(Q8.10D10), (C2×Dic10).249C22, C2.28(D5×C4○D4), C10.143(C2×C4○D4), (C5×C4⋊C4).341C22, (C2×C4).167(C22×D5), SmallGroup(320,1241)

Series: Derived Chief Lower central Upper central

C1C2×C10 — Q85Dic10
C1C5C10C2×C10C2×Dic5C4×Dic5Q8×Dic5 — Q85Dic10
C5C2×C10 — Q85Dic10
C1C22C4×Q8

Generators and relations for Q85Dic10
 G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >

Subgroups: 550 in 200 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C4 [×6], C4 [×13], C22, C5, C2×C4, C2×C4 [×6], C2×C4 [×8], Q8 [×4], Q8 [×6], C10 [×3], C42 [×3], C42 [×6], C4⋊C4 [×3], C4⋊C4 [×19], C2×Q8, C2×Q8 [×3], Dic5 [×2], Dic5 [×7], C20 [×6], C20 [×4], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×6], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q83Q8, C4×Dic5 [×6], C10.D4, C10.D4 [×9], C4⋊Dic5 [×9], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], Q8×C10, C4×Dic10 [×3], C20.6Q8 [×3], C20⋊Q8 [×3], C4.Dic10 [×3], Q8×Dic5 [×2], Q8×C20, Q85Dic10
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D5, C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22×Q8, C2×C4○D4, 2- 1+4, Dic10 [×4], C22×D5 [×7], Q83Q8, C2×Dic10 [×6], C23×D5, C22×Dic10, Q8.10D10, D5×C4○D4, Q85Dic10

Smallest permutation representation of Q85Dic10
Regular action on 320 points
Generators in S320
(1 320 174 225)(2 301 175 226)(3 302 176 227)(4 303 177 228)(5 304 178 229)(6 305 179 230)(7 306 180 231)(8 307 161 232)(9 308 162 233)(10 309 163 234)(11 310 164 235)(12 311 165 236)(13 312 166 237)(14 313 167 238)(15 314 168 239)(16 315 169 240)(17 316 170 221)(18 317 171 222)(19 318 172 223)(20 319 173 224)(21 86 272 125)(22 87 273 126)(23 88 274 127)(24 89 275 128)(25 90 276 129)(26 91 277 130)(27 92 278 131)(28 93 279 132)(29 94 280 133)(30 95 261 134)(31 96 262 135)(32 97 263 136)(33 98 264 137)(34 99 265 138)(35 100 266 139)(36 81 267 140)(37 82 268 121)(38 83 269 122)(39 84 270 123)(40 85 271 124)(41 101 77 144)(42 102 78 145)(43 103 79 146)(44 104 80 147)(45 105 61 148)(46 106 62 149)(47 107 63 150)(48 108 64 151)(49 109 65 152)(50 110 66 153)(51 111 67 154)(52 112 68 155)(53 113 69 156)(54 114 70 157)(55 115 71 158)(56 116 72 159)(57 117 73 160)(58 118 74 141)(59 119 75 142)(60 120 76 143)(181 295 248 202)(182 296 249 203)(183 297 250 204)(184 298 251 205)(185 299 252 206)(186 300 253 207)(187 281 254 208)(188 282 255 209)(189 283 256 210)(190 284 257 211)(191 285 258 212)(192 286 259 213)(193 287 260 214)(194 288 241 215)(195 289 242 216)(196 290 243 217)(197 291 244 218)(198 292 245 219)(199 293 246 220)(200 294 247 201)
(1 258 174 191)(2 192 175 259)(3 260 176 193)(4 194 177 241)(5 242 178 195)(6 196 179 243)(7 244 180 197)(8 198 161 245)(9 246 162 199)(10 200 163 247)(11 248 164 181)(12 182 165 249)(13 250 166 183)(14 184 167 251)(15 252 168 185)(16 186 169 253)(17 254 170 187)(18 188 171 255)(19 256 172 189)(20 190 173 257)(21 157 272 114)(22 115 273 158)(23 159 274 116)(24 117 275 160)(25 141 276 118)(26 119 277 142)(27 143 278 120)(28 101 279 144)(29 145 280 102)(30 103 261 146)(31 147 262 104)(32 105 263 148)(33 149 264 106)(34 107 265 150)(35 151 266 108)(36 109 267 152)(37 153 268 110)(38 111 269 154)(39 155 270 112)(40 113 271 156)(41 132 77 93)(42 94 78 133)(43 134 79 95)(44 96 80 135)(45 136 61 97)(46 98 62 137)(47 138 63 99)(48 100 64 139)(49 140 65 81)(50 82 66 121)(51 122 67 83)(52 84 68 123)(53 124 69 85)(54 86 70 125)(55 126 71 87)(56 88 72 127)(57 128 73 89)(58 90 74 129)(59 130 75 91)(60 92 76 131)(201 234 294 309)(202 310 295 235)(203 236 296 311)(204 312 297 237)(205 238 298 313)(206 314 299 239)(207 240 300 315)(208 316 281 221)(209 222 282 317)(210 318 283 223)(211 224 284 319)(212 320 285 225)(213 226 286 301)(214 302 287 227)(215 228 288 303)(216 304 289 229)(217 230 290 305)(218 306 291 231)(219 232 292 307)(220 308 293 233)
(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 58 11 48)(2 57 12 47)(3 56 13 46)(4 55 14 45)(5 54 15 44)(6 53 16 43)(7 52 17 42)(8 51 18 41)(9 50 19 60)(10 49 20 59)(21 299 31 289)(22 298 32 288)(23 297 33 287)(24 296 34 286)(25 295 35 285)(26 294 36 284)(27 293 37 283)(28 292 38 282)(29 291 39 281)(30 290 40 300)(61 177 71 167)(62 176 72 166)(63 175 73 165)(64 174 74 164)(65 173 75 163)(66 172 76 162)(67 171 77 161)(68 170 78 180)(69 169 79 179)(70 168 80 178)(81 257 91 247)(82 256 92 246)(83 255 93 245)(84 254 94 244)(85 253 95 243)(86 252 96 242)(87 251 97 241)(88 250 98 260)(89 249 99 259)(90 248 100 258)(101 307 111 317)(102 306 112 316)(103 305 113 315)(104 304 114 314)(105 303 115 313)(106 302 116 312)(107 301 117 311)(108 320 118 310)(109 319 119 309)(110 318 120 308)(121 189 131 199)(122 188 132 198)(123 187 133 197)(124 186 134 196)(125 185 135 195)(126 184 136 194)(127 183 137 193)(128 182 138 192)(129 181 139 191)(130 200 140 190)(141 235 151 225)(142 234 152 224)(143 233 153 223)(144 232 154 222)(145 231 155 221)(146 230 156 240)(147 229 157 239)(148 228 158 238)(149 227 159 237)(150 226 160 236)(201 267 211 277)(202 266 212 276)(203 265 213 275)(204 264 214 274)(205 263 215 273)(206 262 216 272)(207 261 217 271)(208 280 218 270)(209 279 219 269)(210 278 220 268)

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

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

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

65 conjugacy classes

class 1 2A2B2C4A···4H4I4J4K4L4M4N4O4P···4U5A5B10A···10F20A···20H20I···20AF
order12224···444444444···45510···1020···2020···20
size11112···24441010101020···20222···22···24···4

65 irreducible representations

dim11111112222222444
type+++++++-++++--
imageC1C2C2C2C2C2C2Q8D5C4○D4D10D10D10Dic102- 1+4Q8.10D10D5×C4○D4
kernelQ85Dic10C4×Dic10C20.6Q8C20⋊Q8C4.Dic10Q8×Dic5Q8×C20C5×Q8C4×Q8Dic5C42C4⋊C4C2×Q8Q8C10C2C2
# reps133332142466216144

Matrix representation of Q85Dic10 in GL6(𝔽41)

100000
010000
001000
000100
00004039
000011
,
100000
010000
001000
000100
00001835
00002023
,
0400000
170000
0040900
0018100
0000918
00003232
,
14270000
11270000
00122400
00232900
000010
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,20,0,0,0,0,35,23],[0,1,0,0,0,0,40,7,0,0,0,0,0,0,40,18,0,0,0,0,9,1,0,0,0,0,0,0,9,32,0,0,0,0,18,32],[14,11,0,0,0,0,27,27,0,0,0,0,0,0,12,23,0,0,0,0,24,29,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Q85Dic10 in GAP, Magma, Sage, TeX

Q_8\rtimes_5{\rm Dic}_{10}
% in TeX

G:=Group("Q8:5Dic10");
// GroupNames label

G:=SmallGroup(320,1241);
// by ID

G=gap.SmallGroup(320,1241);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,675,192,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^20=1,b^2=a^2,d^2=c^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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