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G = Dic53Q16order 320 = 26·5

2nd semidirect product of Dic5 and Q16 acting via Q16/Q8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic53Q16, (C5×Q8).7D4, (C2×C8).36D10, C55(C42Q16), (C2×Q16).4D5, C2.17(D5×Q16), C20.184(C2×D4), (C10×Q16).8C2, C10.27(C2×Q16), (Q8×Dic5).7C2, Q8.2(C5⋊D4), (C2×Q8).118D10, C22.275(D4×D5), C20.103(C4○D4), C4.13(D42D5), (C2×C20).456C23, (C2×C40).250C22, Dic5⋊Q8.8C2, (C2×Dic5).242D4, Q8⋊Dic5.14C2, C20.8Q8.12C2, (Q8×C10).85C22, C10.119(C4⋊D4), C2.26(Q16⋊D5), C20.44D4.12C2, C10.75(C8.C22), C4⋊Dic5.179C22, (C4×Dic5).59C22, C2.28(Dic5⋊D4), (C2×Dic10).135C22, C4.46(C2×C5⋊D4), (C2×C5⋊Q16).9C2, (C2×C10).367(C2×D4), (C2×C4).544(C22×D5), (C2×C52C8).162C22, SmallGroup(320,809)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic53Q16
C1C5C10C20C2×C20C4×Dic5Q8×Dic5 — Dic53Q16
C5C10C2×C20 — Dic53Q16
C1C22C2×C4C2×Q16

Generators and relations for Dic53Q16
 G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 358 in 108 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C4 [×2], C4 [×8], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×2], Q8 [×5], C10 [×3], C42 [×2], C4⋊C4 [×4], C2×C8, C2×C8, Q16 [×4], C2×Q8 [×2], C2×Q8, Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], C2×C10, Q8⋊C4 [×2], C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×Q16, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8 [×3], C42Q16, C2×C52C8, C4×Dic5, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C4⋊Dic5, C5⋊Q16 [×2], C2×C40, C5×Q16 [×2], C2×Dic10, Q8×C10 [×2], C20.8Q8, C20.44D4, Q8⋊Dic5, C2×C5⋊Q16, Dic5⋊Q8, Q8×Dic5, C10×Q16, Dic53Q16
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, C5⋊D4 [×2], C22×D5, C42Q16, D4×D5, D42D5, C2×C5⋊D4, D5×Q16, Q16⋊D5, Dic5⋊D4, Dic53Q16

Smallest permutation representation of Dic53Q16
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)
(1 171 6 176)(2 180 7 175)(3 179 8 174)(4 178 9 173)(5 177 10 172)(11 170 16 165)(12 169 17 164)(13 168 18 163)(14 167 19 162)(15 166 20 161)(21 189 26 184)(22 188 27 183)(23 187 28 182)(24 186 29 181)(25 185 30 190)(31 198 36 193)(32 197 37 192)(33 196 38 191)(34 195 39 200)(35 194 40 199)(41 209 46 204)(42 208 47 203)(43 207 48 202)(44 206 49 201)(45 205 50 210)(51 218 56 213)(52 217 57 212)(53 216 58 211)(54 215 59 220)(55 214 60 219)(61 229 66 224)(62 228 67 223)(63 227 68 222)(64 226 69 221)(65 225 70 230)(71 238 76 233)(72 237 77 232)(73 236 78 231)(74 235 79 240)(75 234 80 239)(81 249 86 244)(82 248 87 243)(83 247 88 242)(84 246 89 241)(85 245 90 250)(91 258 96 253)(92 257 97 252)(93 256 98 251)(94 255 99 260)(95 254 100 259)(101 269 106 264)(102 268 107 263)(103 267 108 262)(104 266 109 261)(105 265 110 270)(111 278 116 273)(112 277 117 272)(113 276 118 271)(114 275 119 280)(115 274 120 279)(121 289 126 284)(122 288 127 283)(123 287 128 282)(124 286 129 281)(125 285 130 290)(131 298 136 293)(132 297 137 292)(133 296 138 291)(134 295 139 300)(135 294 140 299)(141 309 146 304)(142 308 147 303)(143 307 148 302)(144 306 149 301)(145 305 150 310)(151 318 156 313)(152 317 157 312)(153 316 158 311)(154 315 159 320)(155 314 160 319)
(1 158 38 148 28 165 48 138)(2 159 39 149 29 166 49 139)(3 160 40 150 30 167 50 140)(4 151 31 141 21 168 41 131)(5 152 32 142 22 169 42 132)(6 153 33 143 23 170 43 133)(7 154 34 144 24 161 44 134)(8 155 35 145 25 162 45 135)(9 156 36 146 26 163 46 136)(10 157 37 147 27 164 47 137)(11 207 291 176 311 196 302 187)(12 208 292 177 312 197 303 188)(13 209 293 178 313 198 304 189)(14 210 294 179 314 199 305 190)(15 201 295 180 315 200 306 181)(16 202 296 171 316 191 307 182)(17 203 297 172 317 192 308 183)(18 204 298 173 318 193 309 184)(19 205 299 174 319 194 310 185)(20 206 300 175 320 195 301 186)(51 111 81 91 61 121 71 101)(52 112 82 92 62 122 72 102)(53 113 83 93 63 123 73 103)(54 114 84 94 64 124 74 104)(55 115 85 95 65 125 75 105)(56 116 86 96 66 126 76 106)(57 117 87 97 67 127 77 107)(58 118 88 98 68 128 78 108)(59 119 89 99 69 129 79 109)(60 120 90 100 70 130 80 110)(211 276 242 256 222 287 231 267)(212 277 243 257 223 288 232 268)(213 278 244 258 224 289 233 269)(214 279 245 259 225 290 234 270)(215 280 246 260 226 281 235 261)(216 271 247 251 227 282 236 262)(217 272 248 252 228 283 237 263)(218 273 249 253 229 284 238 264)(219 274 250 254 230 285 239 265)(220 275 241 255 221 286 240 266)
(1 68 28 58)(2 69 29 59)(3 70 30 60)(4 61 21 51)(5 62 22 52)(6 63 23 53)(7 64 24 54)(8 65 25 55)(9 66 26 56)(10 67 27 57)(11 267 311 256)(12 268 312 257)(13 269 313 258)(14 270 314 259)(15 261 315 260)(16 262 316 251)(17 263 317 252)(18 264 318 253)(19 265 319 254)(20 266 320 255)(31 81 41 71)(32 82 42 72)(33 83 43 73)(34 84 44 74)(35 85 45 75)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(91 168 101 151)(92 169 102 152)(93 170 103 153)(94 161 104 154)(95 162 105 155)(96 163 106 156)(97 164 107 157)(98 165 108 158)(99 166 109 159)(100 167 110 160)(111 131 121 141)(112 132 122 142)(113 133 123 143)(114 134 124 144)(115 135 125 145)(116 136 126 146)(117 137 127 147)(118 138 128 148)(119 139 129 149)(120 140 130 150)(171 227 182 216)(172 228 183 217)(173 229 184 218)(174 230 185 219)(175 221 186 220)(176 222 187 211)(177 223 188 212)(178 224 189 213)(179 225 190 214)(180 226 181 215)(191 247 202 236)(192 248 203 237)(193 249 204 238)(194 250 205 239)(195 241 206 240)(196 242 207 231)(197 243 208 232)(198 244 209 233)(199 245 210 234)(200 246 201 235)(271 296 282 307)(272 297 283 308)(273 298 284 309)(274 299 285 310)(275 300 286 301)(276 291 287 302)(277 292 288 303)(278 293 289 304)(279 294 290 305)(280 295 281 306)

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

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

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

47 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444455888810···102020202020···2040···40
size111122448101020202040224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++-++--+-
imageC1C2C2C2C2C2C2C2D4D4D5Q16C4○D4D10D10C5⋊D4C8.C22D42D5D4×D5D5×Q16Q16⋊D5
kernelDic53Q16C20.8Q8C20.44D4Q8⋊Dic5C2×C5⋊Q16Dic5⋊Q8Q8×Dic5C10×Q16C2×Dic5C5×Q8C2×Q16Dic5C20C2×C8C2×Q8Q8C10C4C22C2C2
# reps111111112224224812244

Matrix representation of Dic53Q16 in GL4(𝔽41) generated by

1000
0100
00040
0017
,
1000
0100
0090
001932
,
17600
34000
00241
004017
,
33100
17800
00241
004017
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,7],[1,0,0,0,0,1,0,0,0,0,9,19,0,0,0,32],[17,34,0,0,6,0,0,0,0,0,24,40,0,0,1,17],[33,17,0,0,1,8,0,0,0,0,24,40,0,0,1,17] >;

Dic53Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_3Q_{16}
% in TeX

G:=Group("Dic5:3Q16");
// GroupNames label

G:=SmallGroup(320,809);
// by ID

G=gap.SmallGroup(320,809);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,422,135,184,570,297,136,12550]);
// Polycyclic

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