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## G = Q8⋊Dic10order 320 = 26·5

### 1st semidirect product of Q8 and Dic10 acting via Dic10/Dic5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Q8⋊Dic10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C4×Dic5 — Q8×Dic5 — Q8⋊Dic10
 Lower central C5 — C10 — C2×C20 — Q8⋊Dic10
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Q8⋊Dic10
G = < a,b,c,d | a4=c20=1, b2=a2, d2=c10, bab-1=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd-1=c-1 >

Subgroups: 342 in 96 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4×Q8, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊Q8, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C20.Q8, C20.8Q8, C406C4, Q8⋊Dic5, C5×Q8⋊C4, C20⋊Q8, Q8×Dic5, Q8⋊Dic10
Quotients: C1, C2, C22, D4, Q8, C23, D5, SD16, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C2×SD16, C8.C22, Dic10, C22×D5, Q8⋊Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D5×SD16, Q16⋊D5, Q8⋊Dic10

Smallest permutation representation of Q8⋊Dic10
Regular action on 320 points
Generators in S320
(1 291 279 119)(2 120 280 292)(3 293 261 101)(4 102 262 294)(5 295 263 103)(6 104 264 296)(7 297 265 105)(8 106 266 298)(9 299 267 107)(10 108 268 300)(11 281 269 109)(12 110 270 282)(13 283 271 111)(14 112 272 284)(15 285 273 113)(16 114 274 286)(17 287 275 115)(18 116 276 288)(19 289 277 117)(20 118 278 290)(21 239 69 50)(22 51 70 240)(23 221 71 52)(24 53 72 222)(25 223 73 54)(26 55 74 224)(27 225 75 56)(28 57 76 226)(29 227 77 58)(30 59 78 228)(31 229 79 60)(32 41 80 230)(33 231 61 42)(34 43 62 232)(35 233 63 44)(36 45 64 234)(37 235 65 46)(38 47 66 236)(39 237 67 48)(40 49 68 238)(81 189 311 217)(82 218 312 190)(83 191 313 219)(84 220 314 192)(85 193 315 201)(86 202 316 194)(87 195 317 203)(88 204 318 196)(89 197 319 205)(90 206 320 198)(91 199 301 207)(92 208 302 200)(93 181 303 209)(94 210 304 182)(95 183 305 211)(96 212 306 184)(97 185 307 213)(98 214 308 186)(99 187 309 215)(100 216 310 188)(121 157 249 176)(122 177 250 158)(123 159 251 178)(124 179 252 160)(125 141 253 180)(126 161 254 142)(127 143 255 162)(128 163 256 144)(129 145 257 164)(130 165 258 146)(131 147 259 166)(132 167 260 148)(133 149 241 168)(134 169 242 150)(135 151 243 170)(136 171 244 152)(137 153 245 172)(138 173 246 154)(139 155 247 174)(140 175 248 156)
(1 40 279 68)(2 239 280 50)(3 22 261 70)(4 221 262 52)(5 24 263 72)(6 223 264 54)(7 26 265 74)(8 225 266 56)(9 28 267 76)(10 227 268 58)(11 30 269 78)(12 229 270 60)(13 32 271 80)(14 231 272 42)(15 34 273 62)(16 233 274 44)(17 36 275 64)(18 235 276 46)(19 38 277 66)(20 237 278 48)(21 292 69 120)(23 294 71 102)(25 296 73 104)(27 298 75 106)(29 300 77 108)(31 282 79 110)(33 284 61 112)(35 286 63 114)(37 288 65 116)(39 290 67 118)(41 283 230 111)(43 285 232 113)(45 287 234 115)(47 289 236 117)(49 291 238 119)(51 293 240 101)(53 295 222 103)(55 297 224 105)(57 299 226 107)(59 281 228 109)(81 167 311 148)(82 133 312 241)(83 169 313 150)(84 135 314 243)(85 171 315 152)(86 137 316 245)(87 173 317 154)(88 139 318 247)(89 175 319 156)(90 121 320 249)(91 177 301 158)(92 123 302 251)(93 179 303 160)(94 125 304 253)(95 161 305 142)(96 127 306 255)(97 163 307 144)(98 129 308 257)(99 165 309 146)(100 131 310 259)(122 207 250 199)(124 209 252 181)(126 211 254 183)(128 213 256 185)(130 215 258 187)(132 217 260 189)(134 219 242 191)(136 201 244 193)(138 203 246 195)(140 205 248 197)(141 210 180 182)(143 212 162 184)(145 214 164 186)(147 216 166 188)(149 218 168 190)(151 220 170 192)(153 202 172 194)(155 204 174 196)(157 206 176 198)(159 208 178 200)
(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)
(1 143 11 153)(2 142 12 152)(3 141 13 151)(4 160 14 150)(5 159 15 149)(6 158 16 148)(7 157 17 147)(8 156 18 146)(9 155 19 145)(10 154 20 144)(21 211 31 201)(22 210 32 220)(23 209 33 219)(24 208 34 218)(25 207 35 217)(26 206 36 216)(27 205 37 215)(28 204 38 214)(29 203 39 213)(30 202 40 212)(41 314 51 304)(42 313 52 303)(43 312 53 302)(44 311 54 301)(45 310 55 320)(46 309 56 319)(47 308 57 318)(48 307 58 317)(49 306 59 316)(50 305 60 315)(61 191 71 181)(62 190 72 200)(63 189 73 199)(64 188 74 198)(65 187 75 197)(66 186 76 196)(67 185 77 195)(68 184 78 194)(69 183 79 193)(70 182 80 192)(81 223 91 233)(82 222 92 232)(83 221 93 231)(84 240 94 230)(85 239 95 229)(86 238 96 228)(87 237 97 227)(88 236 98 226)(89 235 99 225)(90 234 100 224)(101 125 111 135)(102 124 112 134)(103 123 113 133)(104 122 114 132)(105 121 115 131)(106 140 116 130)(107 139 117 129)(108 138 118 128)(109 137 119 127)(110 136 120 126)(161 270 171 280)(162 269 172 279)(163 268 173 278)(164 267 174 277)(165 266 175 276)(166 265 176 275)(167 264 177 274)(168 263 178 273)(169 262 179 272)(170 261 180 271)(241 295 251 285)(242 294 252 284)(243 293 253 283)(244 292 254 282)(245 291 255 281)(246 290 256 300)(247 289 257 299)(248 288 258 298)(249 287 259 297)(250 286 260 296)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 4 4 8 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + - - - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 SD16 C4○D4 D10 D10 D10 Dic10 C8.C22 D4⋊2D5 D4×D5 D5×SD16 Q16⋊D5 kernel Q8⋊Dic10 C20.Q8 C20.8Q8 C40⋊6C4 Q8⋊Dic5 C5×Q8⋊C4 C20⋊Q8 Q8×Dic5 C2×Dic5 C5×Q8 Q8⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×Q8 Q8 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 4 2 2 2 2 8 1 2 2 4 4

Matrix representation of Q8⋊Dic10 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 5 0 0 16 40
,
 40 0 0 0 0 40 0 0 0 0 20 12 0 0 11 21
,
 27 2 0 0 25 11 0 0 0 0 5 17 0 0 1 36
,
 24 18 0 0 34 17 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,5,40],[40,0,0,0,0,40,0,0,0,0,20,11,0,0,12,21],[27,25,0,0,2,11,0,0,0,0,5,1,0,0,17,36],[24,34,0,0,18,17,0,0,0,0,1,0,0,0,0,1] >;

Q8⋊Dic10 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,418);
// by ID

G=gap.SmallGroup(320,418);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,254,219,58,851,438,102,12550]);
// Polycyclic

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

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