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G = Q16⋊Dic5order 320 = 26·5

3rd semidirect product of Q16 and Dic5 acting via Dic5/C10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q163Dic5, C40.71(C2×C4), (C5×Q16)⋊11C4, (C2×C8).94D10, (C2×Q16).6D5, C408C4.4C2, C8.5(C2×Dic5), C56(Q16⋊C4), C10.128(C4×D4), (C10×Q16).6C2, (Q8×Dic5).9C2, Q8.2(C2×Dic5), C406C4.11C2, C2.15(D4×Dic5), (C2×Q8).120D10, C22.119(D4×D5), C20.105(C4○D4), C4.35(D42D5), C4.6(C22×Dic5), (C2×C20).458C23, C20.135(C22×C4), (C2×C40).149C22, C2.7(Q16⋊D5), (C2×Dic5).243D4, Q8⋊Dic5.16C2, (Q8×C10).87C22, C10.76(C8.C22), C4⋊Dic5.181C22, (C4×Dic5).60C22, (C5×Q8).24(C2×C4), (C2×C10).369(C2×D4), (C2×C4).546(C22×D5), (C2×C52C8).163C22, SmallGroup(320,811)

Series: Derived Chief Lower central Upper central

C1C20 — Q16⋊Dic5
C1C5C10C2×C10C2×C20C4×Dic5Q8×Dic5 — Q16⋊Dic5
C5C10C20 — Q16⋊Dic5
C1C22C2×C4C2×Q16

Generators and relations for Q16⋊Dic5
 G = < a,b,c,d | a8=c10=1, b2=a4, d2=c5, bab-1=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 310 in 108 conjugacy classes, 57 normal (21 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×8], C22, C5, C8 [×2], C8, C2×C4, C2×C4 [×6], Q8 [×4], Q8 [×2], C10, C10 [×2], C42 [×3], C4⋊C4 [×4], C2×C8, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×4], C20 [×2], C20 [×4], C2×C10, C8⋊C4, Q8⋊C4 [×2], C4.Q8, C4×Q8 [×2], C2×Q16, C52C8, C40 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×4], C5×Q8 [×2], Q16⋊C4, C2×C52C8, C4×Dic5, C4×Dic5 [×2], C4⋊Dic5 [×2], C4⋊Dic5 [×2], C2×C40, C5×Q16 [×4], Q8×C10 [×2], C408C4, C406C4, Q8⋊Dic5 [×2], Q8×Dic5 [×2], C10×Q16, Q16⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C8.C22 [×2], C2×Dic5 [×6], C22×D5, Q16⋊C4, D4×D5, D42D5, C22×Dic5, Q16⋊D5 [×2], D4×Dic5, Q16⋊Dic5

Smallest permutation representation of Q16⋊Dic5
Regular action on 320 points
Generators in S320
(1 150 14 159 39 165 48 134)(2 141 15 160 40 166 49 135)(3 142 16 151 31 167 50 136)(4 143 17 152 32 168 41 137)(5 144 18 153 33 169 42 138)(6 145 19 154 34 170 43 139)(7 146 20 155 35 161 44 140)(8 147 11 156 36 162 45 131)(9 148 12 157 37 163 46 132)(10 149 13 158 38 164 47 133)(21 185 296 196 305 202 316 176)(22 186 297 197 306 203 317 177)(23 187 298 198 307 204 318 178)(24 188 299 199 308 205 319 179)(25 189 300 200 309 206 320 180)(26 190 291 191 310 207 311 171)(27 181 292 192 301 208 312 172)(28 182 293 193 302 209 313 173)(29 183 294 194 303 210 314 174)(30 184 295 195 304 201 315 175)(51 107 85 91 76 125 67 116)(52 108 86 92 77 126 68 117)(53 109 87 93 78 127 69 118)(54 110 88 94 79 128 70 119)(55 101 89 95 80 129 61 120)(56 102 90 96 71 130 62 111)(57 103 81 97 72 121 63 112)(58 104 82 98 73 122 64 113)(59 105 83 99 74 123 65 114)(60 106 84 100 75 124 66 115)(211 287 247 271 231 270 230 251)(212 288 248 272 232 261 221 252)(213 289 249 273 233 262 222 253)(214 290 250 274 234 263 223 254)(215 281 241 275 235 264 224 255)(216 282 242 276 236 265 225 256)(217 283 243 277 237 266 226 257)(218 284 244 278 238 267 227 258)(219 285 245 279 239 268 228 259)(220 286 246 280 240 269 229 260)
(1 79 39 54)(2 80 40 55)(3 71 31 56)(4 72 32 57)(5 73 33 58)(6 74 34 59)(7 75 35 60)(8 76 36 51)(9 77 37 52)(10 78 38 53)(11 85 45 67)(12 86 46 68)(13 87 47 69)(14 88 48 70)(15 89 49 61)(16 90 50 62)(17 81 41 63)(18 82 42 64)(19 83 43 65)(20 84 44 66)(21 256 305 276)(22 257 306 277)(23 258 307 278)(24 259 308 279)(25 260 309 280)(26 251 310 271)(27 252 301 272)(28 253 302 273)(29 254 303 274)(30 255 304 275)(91 162 116 147)(92 163 117 148)(93 164 118 149)(94 165 119 150)(95 166 120 141)(96 167 111 142)(97 168 112 143)(98 169 113 144)(99 170 114 145)(100 161 115 146)(101 135 129 160)(102 136 130 151)(103 137 121 152)(104 138 122 153)(105 139 123 154)(106 140 124 155)(107 131 125 156)(108 132 126 157)(109 133 127 158)(110 134 128 159)(171 211 191 231)(172 212 192 232)(173 213 193 233)(174 214 194 234)(175 215 195 235)(176 216 196 236)(177 217 197 237)(178 218 198 238)(179 219 199 239)(180 220 200 240)(181 221 208 248)(182 222 209 249)(183 223 210 250)(184 224 201 241)(185 225 202 242)(186 226 203 243)(187 227 204 244)(188 228 205 245)(189 229 206 246)(190 230 207 247)(261 312 288 292)(262 313 289 293)(263 314 290 294)(264 315 281 295)(265 316 282 296)(266 317 283 297)(267 318 284 298)(268 319 285 299)(269 320 286 300)(270 311 287 291)
(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)
(1 176 6 171)(2 175 7 180)(3 174 8 179)(4 173 9 178)(5 172 10 177)(11 188 16 183)(12 187 17 182)(13 186 18 181)(14 185 19 190)(15 184 20 189)(21 170 26 165)(22 169 27 164)(23 168 28 163)(24 167 29 162)(25 166 30 161)(31 194 36 199)(32 193 37 198)(33 192 38 197)(34 191 39 196)(35 200 40 195)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 219 56 214)(52 218 57 213)(53 217 58 212)(54 216 59 211)(55 215 60 220)(61 224 66 229)(62 223 67 228)(63 222 68 227)(64 221 69 226)(65 230 70 225)(71 234 76 239)(72 233 77 238)(73 232 78 237)(74 231 79 236)(75 240 80 235)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 259 96 254)(92 258 97 253)(93 257 98 252)(94 256 99 251)(95 255 100 260)(101 264 106 269)(102 263 107 268)(103 262 108 267)(104 261 109 266)(105 270 110 265)(111 274 116 279)(112 273 117 278)(113 272 118 277)(114 271 119 276)(115 280 120 275)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 299 136 294)(132 298 137 293)(133 297 138 292)(134 296 139 291)(135 295 140 300)(141 304 146 309)(142 303 147 308)(143 302 148 307)(144 301 149 306)(145 310 150 305)(151 314 156 319)(152 313 157 318)(153 312 158 317)(154 311 159 316)(155 320 160 315)

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

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

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

50 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444444455888810···102020202020···2040···40
size11112244441010101020202020224420202···244448···84···4

50 irreducible representations

dim11111112222224444
type+++++++++-+--+
imageC1C2C2C2C2C2C4D4D5C4○D4D10Dic5D10C8.C22D42D5D4×D5Q16⋊D5
kernelQ16⋊Dic5C408C4C406C4Q8⋊Dic5Q8×Dic5C10×Q16C5×Q16C2×Dic5C2×Q16C20C2×C8Q16C2×Q8C10C4C22C2
# reps11122182222842228

Matrix representation of Q16⋊Dic5 in GL6(𝔽41)

100000
010000
00001921
00002022
0011101921
0031302022
,
4000000
0400000
00393520
0032162132
000193832
002232925
,
7400000
100000
0004000
0013500
0000040
0000135
,
28230000
14130000
001101137
0016402130
00152126
0010263729

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,31,0,0,0,0,10,30,0,0,19,20,19,20,0,0,21,22,21,22],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,32,0,22,0,0,9,16,19,32,0,0,35,21,38,9,0,0,20,32,32,25],[7,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,0,0,0,40,35],[28,14,0,0,0,0,23,13,0,0,0,0,0,0,1,16,15,10,0,0,10,40,2,26,0,0,11,21,12,37,0,0,37,30,6,29] >;

Q16⋊Dic5 in GAP, Magma, Sage, TeX

Q_{16}\rtimes {\rm Dic}_5
% in TeX

G:=Group("Q16:Dic5");
// GroupNames label

G:=SmallGroup(320,811);
// by ID

G=gap.SmallGroup(320,811);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,758,219,184,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=c^10=1,b^2=a^4,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

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