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## G = Q8⋊6Dic10order 320 = 26·5

### 2nd semidirect product of Q8 and Dic10 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Q8⋊6Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Q8×Dic5 — Q8⋊6Dic10
 Lower central C5 — C2×C10 — Q8⋊6Dic10
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q86Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 550 in 200 conjugacy classes, 115 normal (18 characteristic)
C1, C2 [×3], C4 [×8], C4 [×11], 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 [×8], C20 [×8], C20 [×3], C2×C10, C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Dic10 [×6], C2×Dic5 [×8], C2×C20, C2×C20 [×6], C5×Q8 [×4], Q83Q8, C4×Dic5 [×6], C10.D4 [×6], C4⋊Dic5, C4⋊Dic5 [×12], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], Q8×C10, C4×Dic10 [×3], C202Q8 [×3], C4.Dic10 [×6], Q8×Dic5 [×2], Q8×C20, Q86Dic10
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], Q82D5 [×2], C23×D5, C22×Dic10, C2×Q82D5, D4.10D10, Q86Dic10

Smallest permutation representation of Q86Dic10
Regular action on 320 points
Generators in S320
(1 127 260 31)(2 128 241 32)(3 129 242 33)(4 130 243 34)(5 131 244 35)(6 132 245 36)(7 133 246 37)(8 134 247 38)(9 135 248 39)(10 136 249 40)(11 137 250 21)(12 138 251 22)(13 139 252 23)(14 140 253 24)(15 121 254 25)(16 122 255 26)(17 123 256 27)(18 124 257 28)(19 125 258 29)(20 126 259 30)(41 186 270 240)(42 187 271 221)(43 188 272 222)(44 189 273 223)(45 190 274 224)(46 191 275 225)(47 192 276 226)(48 193 277 227)(49 194 278 228)(50 195 279 229)(51 196 280 230)(52 197 261 231)(53 198 262 232)(54 199 263 233)(55 200 264 234)(56 181 265 235)(57 182 266 236)(58 183 267 237)(59 184 268 238)(60 185 269 239)(61 201 105 294)(62 202 106 295)(63 203 107 296)(64 204 108 297)(65 205 109 298)(66 206 110 299)(67 207 111 300)(68 208 112 281)(69 209 113 282)(70 210 114 283)(71 211 115 284)(72 212 116 285)(73 213 117 286)(74 214 118 287)(75 215 119 288)(76 216 120 289)(77 217 101 290)(78 218 102 291)(79 219 103 292)(80 220 104 293)(81 169 157 317)(82 170 158 318)(83 171 159 319)(84 172 160 320)(85 173 141 301)(86 174 142 302)(87 175 143 303)(88 176 144 304)(89 177 145 305)(90 178 146 306)(91 179 147 307)(92 180 148 308)(93 161 149 309)(94 162 150 310)(95 163 151 311)(96 164 152 312)(97 165 153 313)(98 166 154 314)(99 167 155 315)(100 168 156 316)
(1 305 260 177)(2 306 241 178)(3 307 242 179)(4 308 243 180)(5 309 244 161)(6 310 245 162)(7 311 246 163)(8 312 247 164)(9 313 248 165)(10 314 249 166)(11 315 250 167)(12 316 251 168)(13 317 252 169)(14 318 253 170)(15 319 254 171)(16 320 255 172)(17 301 256 173)(18 302 257 174)(19 303 258 175)(20 304 259 176)(21 99 137 155)(22 100 138 156)(23 81 139 157)(24 82 140 158)(25 83 121 159)(26 84 122 160)(27 85 123 141)(28 86 124 142)(29 87 125 143)(30 88 126 144)(31 89 127 145)(32 90 128 146)(33 91 129 147)(34 92 130 148)(35 93 131 149)(36 94 132 150)(37 95 133 151)(38 96 134 152)(39 97 135 153)(40 98 136 154)(41 111 270 67)(42 112 271 68)(43 113 272 69)(44 114 273 70)(45 115 274 71)(46 116 275 72)(47 117 276 73)(48 118 277 74)(49 119 278 75)(50 120 279 76)(51 101 280 77)(52 102 261 78)(53 103 262 79)(54 104 263 80)(55 105 264 61)(56 106 265 62)(57 107 266 63)(58 108 267 64)(59 109 268 65)(60 110 269 66)(181 202 235 295)(182 203 236 296)(183 204 237 297)(184 205 238 298)(185 206 239 299)(186 207 240 300)(187 208 221 281)(188 209 222 282)(189 210 223 283)(190 211 224 284)(191 212 225 285)(192 213 226 286)(193 214 227 287)(194 215 228 288)(195 216 229 289)(196 217 230 290)(197 218 231 291)(198 219 232 292)(199 220 233 293)(200 201 234 294)
(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)
(1 182 11 192)(2 181 12 191)(3 200 13 190)(4 199 14 189)(5 198 15 188)(6 197 16 187)(7 196 17 186)(8 195 18 185)(9 194 19 184)(10 193 20 183)(21 47 31 57)(22 46 32 56)(23 45 33 55)(24 44 34 54)(25 43 35 53)(26 42 36 52)(27 41 37 51)(28 60 38 50)(29 59 39 49)(30 58 40 48)(61 81 71 91)(62 100 72 90)(63 99 73 89)(64 98 74 88)(65 97 75 87)(66 96 76 86)(67 95 77 85)(68 94 78 84)(69 93 79 83)(70 92 80 82)(101 141 111 151)(102 160 112 150)(103 159 113 149)(104 158 114 148)(105 157 115 147)(106 156 116 146)(107 155 117 145)(108 154 118 144)(109 153 119 143)(110 152 120 142)(121 272 131 262)(122 271 132 261)(123 270 133 280)(124 269 134 279)(125 268 135 278)(126 267 136 277)(127 266 137 276)(128 265 138 275)(129 264 139 274)(130 263 140 273)(161 219 171 209)(162 218 172 208)(163 217 173 207)(164 216 174 206)(165 215 175 205)(166 214 176 204)(167 213 177 203)(168 212 178 202)(169 211 179 201)(170 210 180 220)(221 245 231 255)(222 244 232 254)(223 243 233 253)(224 242 234 252)(225 241 235 251)(226 260 236 250)(227 259 237 249)(228 258 238 248)(229 257 239 247)(230 256 240 246)(281 310 291 320)(282 309 292 319)(283 308 293 318)(284 307 294 317)(285 306 295 316)(286 305 296 315)(287 304 297 314)(288 303 298 313)(289 302 299 312)(290 301 300 311)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P ··· 4U 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + - - + - image C1 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 Dic10 2- 1+4 Q8⋊2D5 D4.10D10 kernel Q8⋊6Dic10 C4×Dic10 C20⋊2Q8 C4.Dic10 Q8×Dic5 Q8×C20 C5×Q8 C4×Q8 C20 C42 C4⋊C4 C2×Q8 Q8 C10 C4 C2 # reps 1 3 3 6 2 1 4 2 4 6 6 2 16 1 4 4

Matrix representation of Q86Dic10 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 1 2 0 0 40 40
,
 1 0 0 0 0 1 0 0 0 0 9 18 0 0 0 32
,
 27 2 0 0 25 11 0 0 0 0 40 0 0 0 0 40
,
 22 13 0 0 10 19 0 0 0 0 9 18 0 0 32 32
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,40,0,0,2,40],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,18,32],[27,25,0,0,2,11,0,0,0,0,40,0,0,0,0,40],[22,10,0,0,13,19,0,0,0,0,9,32,0,0,18,32] >;

Q86Dic10 in GAP, Magma, Sage, TeX

Q_8\rtimes_6{\rm Dic}_{10}
% in TeX

G:=Group("Q8:6Dic10");
// GroupNames label

G:=SmallGroup(320,1242);
// by ID

G=gap.SmallGroup(320,1242);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1571,192,12550]);
// 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,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

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