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## G = Dic5⋊4Q16order 320 = 26·5

### 1st semidirect product of Dic5 and Q16 acting through Inn(Dic5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic5⋊4Q16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Q8×Dic5 — Dic5⋊4Q16
 Lower central C5 — C10 — C20 — Dic5⋊4Q16
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic54Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=cac-1=dad-1=a-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 342 in 110 conjugacy classes, 51 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×9], C22, C5, C8 [×3], C2×C4, C2×C4 [×6], Q8 [×2], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, Q16 [×4], C2×Q8, C2×Q8, Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×3], C2×C10, C4×C8, Q8⋊C4, Q8⋊C4, C2.D8, C4×Q8 [×2], C2×Q16, C52C8 [×2], C40, Dic10 [×2], Dic10, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C4×Q16, C2×C52C8, C4×Dic5, C4×Dic5 [×2], C10.D4, C4⋊Dic5, C4⋊Dic5, C5⋊Q16 [×4], C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.D8, C8×Dic5, C20.44D4, C5×Q8⋊C4, Dic53Q8, C2×C5⋊Q16, Q8×Dic5, Dic54Q16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, Q16 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×Q16, C4○D8, C4×D5 [×2], C22×D5, C4×Q16, C2×C4×D5, D4×D5, D42D5, Dic54D4, SD163D5, D5×Q16, Dic54Q16

Smallest permutation representation of Dic54Q16
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 16 36 294 28 314 48 307)(2 15 37 293 29 313 49 306)(3 14 38 292 30 312 50 305)(4 13 39 291 21 311 41 304)(5 12 40 300 22 320 42 303)(6 11 31 299 23 319 43 302)(7 20 32 298 24 318 44 301)(8 19 33 297 25 317 45 310)(9 18 34 296 26 316 46 309)(10 17 35 295 27 315 47 308)(51 287 83 267 63 274 71 254)(52 286 84 266 64 273 72 253)(53 285 85 265 65 272 73 252)(54 284 86 264 66 271 74 251)(55 283 87 263 67 280 75 260)(56 282 88 262 68 279 76 259)(57 281 89 261 69 278 77 258)(58 290 90 270 70 277 78 257)(59 289 81 269 61 276 79 256)(60 288 82 268 62 275 80 255)(91 219 123 242 103 222 111 239)(92 218 124 241 104 221 112 238)(93 217 125 250 105 230 113 237)(94 216 126 249 106 229 114 236)(95 215 127 248 107 228 115 235)(96 214 128 247 108 227 116 234)(97 213 129 246 109 226 117 233)(98 212 130 245 110 225 118 232)(99 211 121 244 101 224 119 231)(100 220 122 243 102 223 120 240)(131 187 151 207 143 174 170 194)(132 186 152 206 144 173 161 193)(133 185 153 205 145 172 162 192)(134 184 154 204 146 171 163 191)(135 183 155 203 147 180 164 200)(136 182 156 202 148 179 165 199)(137 181 157 201 149 178 166 198)(138 190 158 210 150 177 167 197)(139 189 159 209 141 176 168 196)(140 188 160 208 142 175 169 195)
(1 214 28 227)(2 213 29 226)(3 212 30 225)(4 211 21 224)(5 220 22 223)(6 219 23 222)(7 218 24 221)(8 217 25 230)(9 216 26 229)(10 215 27 228)(11 91 319 103)(12 100 320 102)(13 99 311 101)(14 98 312 110)(15 97 313 109)(16 96 314 108)(17 95 315 107)(18 94 316 106)(19 93 317 105)(20 92 318 104)(31 239 43 242)(32 238 44 241)(33 237 45 250)(34 236 46 249)(35 235 47 248)(36 234 48 247)(37 233 49 246)(38 232 50 245)(39 231 41 244)(40 240 42 243)(51 187 63 174)(52 186 64 173)(53 185 65 172)(54 184 66 171)(55 183 67 180)(56 182 68 179)(57 181 69 178)(58 190 70 177)(59 189 61 176)(60 188 62 175)(71 207 83 194)(72 206 84 193)(73 205 85 192)(74 204 86 191)(75 203 87 200)(76 202 88 199)(77 201 89 198)(78 210 90 197)(79 209 81 196)(80 208 82 195)(111 302 123 299)(112 301 124 298)(113 310 125 297)(114 309 126 296)(115 308 127 295)(116 307 128 294)(117 306 129 293)(118 305 130 292)(119 304 121 291)(120 303 122 300)(131 274 143 287)(132 273 144 286)(133 272 145 285)(134 271 146 284)(135 280 147 283)(136 279 148 282)(137 278 149 281)(138 277 150 290)(139 276 141 289)(140 275 142 288)(151 267 170 254)(152 266 161 253)(153 265 162 252)(154 264 163 251)(155 263 164 260)(156 262 165 259)(157 261 166 258)(158 270 167 257)(159 269 168 256)(160 268 169 255)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 Q16 C4○D4 D10 D10 D10 C4○D8 C4×D5 D4⋊2D5 D4×D5 SD16⋊3D5 D5×Q16 kernel Dic5⋊4Q16 C10.D8 C8×Dic5 C20.44D4 C5×Q8⋊C4 Dic5⋊3Q8 C2×C5⋊Q16 Q8×Dic5 C5⋊Q16 C2×Dic5 Q8⋊C4 Dic5 C20 C4⋊C4 C2×C8 C2×Q8 C10 Q8 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 2 2 4 2 2 2 2 4 8 2 2 4 4

Matrix representation of Dic54Q16 in GL4(𝔽41) generated by

 35 40 0 0 1 0 0 0 0 0 40 0 0 0 0 40
,
 35 23 0 0 18 6 0 0 0 0 32 0 0 0 0 32
,
 13 39 0 0 2 28 0 0 0 0 0 17 0 0 12 17
,
 13 39 0 0 2 28 0 0 0 0 12 39 0 0 11 29
G:=sub<GL(4,GF(41))| [35,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[35,18,0,0,23,6,0,0,0,0,32,0,0,0,0,32],[13,2,0,0,39,28,0,0,0,0,0,12,0,0,17,17],[13,2,0,0,39,28,0,0,0,0,12,11,0,0,39,29] >;

Dic54Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4Q_{16}
% in TeX

G:=Group("Dic5:4Q16");
// GroupNames label

G:=SmallGroup(320,417);
// by ID

G=gap.SmallGroup(320,417);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,135,268,570,297,136,12550]);
// 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=d*a*d^-1=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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