Q8:5Dic10 - GroupNames

G = Q₈⋊₅Dic₁₀ order 320 = 2⁶·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈⋊₅Dic₁₀, C₄².₁₂₃D₁₀, C₁₀.₈2_-¹⁺⁴, (C₅×Q₈)⋊₆Q₈, C₅⋊₂(Q₈⋊₃Q₈), 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

Derived series

C₁ — C₂×C₁₀ — Q₈⋊₅Dic₁₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×Dic₅ — C₄×Dic₅ — Q₈×Dic₅ — Q₈⋊₅Dic₁₀

Lower central

C₅ — C₂×C₁₀ — Q₈⋊₅Dic₁₀

Upper central

C₁ — C₂² — C₄×Q₈

Generators and relations for Q₈⋊₅Dic₁₀
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, C₄, C₄, C₂², C₅, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, Q₈, C₁₀, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂×Q₈, C₂×Q₈, Dic₅, Dic₅, C₂₀, C₂₀, C₂×C₁₀, C₄×Q₈, C₄×Q₈, C₄².C₂, C₄⋊Q₈, Dic₁₀, C₂×Dic₅, C₂×Dic₅, C₂×C₂₀, C₂×C₂₀, C₅×Q₈, Q₈⋊₃Q₈, C₄×Dic₅, C₁₀.D₄, C₁₀.D₄, C₄⋊Dic₅, C₄×C₂₀, C₅×C₄⋊C₄, C₂×Dic₁₀, Q₈×C₁₀, C₄×Dic₁₀, C₂₀.₆Q₈, C₂₀⋊Q₈, C₄.Dic₁₀, Q₈×Dic₅, Q₈×C₂₀, Q₈⋊₅Dic₁₀
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₅, C₂×Q₈, C₄○D₄, C₂⁴, D₁₀, C₂²×Q₈, C₂×C₄○D₄, 2_-¹⁺⁴, Dic₁₀, C₂²×D₅, Q₈⋊₃Q₈, C₂×Dic₁₀, C₂³×D₅, C₂²×Dic₁₀, Q₈.₁₀D₁₀, D₅×C₄○D₄, Q₈⋊₅Dic₁₀

Smallest permutation representation of Q₈⋊₅Dic₁₀
►Regular action on 320 points

Generators in S₃₂₀

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

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

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

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

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

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

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	1	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	-	+		+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₅	C₄○D₄	D₁₀	D₁₀	D₁₀	Dic₁₀	2_-¹⁺⁴	Q₈.₁₀D₁₀	D₅×C₄○D₄
kernel	Q₈⋊₅Dic₁₀	C₄×Dic₁₀	C₂₀.₆Q₈	C₂₀⋊Q₈	C₄.Dic₁₀	Q₈×Dic₅	Q₈×C₂₀	C₅×Q₈	C₄×Q₈	Dic₅	C₄²	C₄⋊C₄	C₂×Q₈	Q₈	C₁₀	C₂	C₂
# reps	1	3	3	3	3	2	1	4	2	4	6	6	2	16	1	4	4

Matrix representation of Q₈⋊₅Dic₁₀ ►in GL₆(𝔽₄₁)

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	39
0	0	0	0	1	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	35
0	0	0	0	20	23

0	40	0	0	0	0
1	7	0	0	0	0
0	0	40	9	0	0
0	0	18	1	0	0
0	0	0	0	9	18
0	0	0	0	32	32

14	27	0	0	0	0
11	27	0	0	0	0
0	0	12	24	0	0
0	0	23	29	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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] >;

Q₈⋊₅Dic₁₀ 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

G = Q8⋊5Dic10 order 320 = 26·5

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

G = Q₈⋊₅Dic₁₀ order 320 = 2⁶·5

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