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## G = Dic5⋊Q16order 320 = 26·5

### 1st semidirect product of Dic5 and Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5⋊Q16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Dic5⋊3Q8 — Dic5⋊Q16
 Lower central C5 — C10 — C2×C20 — Dic5⋊Q16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic5⋊Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=cac-1=a-1, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 390 in 108 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×7], C10 [×3], C42 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, Q16 [×4], C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, Q8⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16 [×2], C52C8, C40, Dic10 [×2], Dic10 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C42Q16, Dic20 [×2], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×3], C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10 [×2], Q8×C10, C10.Q16, C20.8Q8, C5×Q8⋊C4, Dic53Q8, C2×Dic20, C2×C5⋊Q16, Dic5⋊Q8, Dic5⋊Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, Q16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×Q16, C8.C22, C22×D5, C42Q16, C4○D20, D4×D5 [×2], D10⋊D4, SD16⋊D5, D5×Q16, Dic5⋊Q16

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

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

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

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

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 D4 D5 Q16 C4○D4 D10 D10 D10 C4○D20 C8.C22 D4×D5 D4×D5 SD16⋊D5 D5×Q16 kernel Dic5⋊Q16 C10.Q16 C20.8Q8 C5×Q8⋊C4 Dic5⋊3Q8 C2×Dic20 C2×C5⋊Q16 Dic5⋊Q8 Dic10 C2×Dic5 Q8⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×Q8 C4 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 Dic5⋊Q16 in GL4(𝔽41) generated by

 0 40 0 0 1 7 0 0 0 0 1 0 0 0 0 1
,
 18 16 0 0 13 23 0 0 0 0 40 0 0 0 0 40
,
 28 14 0 0 23 13 0 0 0 0 0 26 0 0 11 17
,
 17 40 0 0 1 24 0 0 0 0 23 16 0 0 13 18
G:=sub<GL(4,GF(41))| [0,1,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[18,13,0,0,16,23,0,0,0,0,40,0,0,0,0,40],[28,23,0,0,14,13,0,0,0,0,0,11,0,0,26,17],[17,1,0,0,40,24,0,0,0,0,23,13,0,0,16,18] >;

Dic5⋊Q16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,420);
// by ID

G=gap.SmallGroup(320,420);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,422,135,184,570,297,136,12550]);
// Polycyclic

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

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