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G = Dic55Q16order 320 = 26·5

2nd semidirect product of Dic5 and Q16 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic55Q16, Dic2012C4, C54(C4×Q16), C8.15(C4×D5), C2.3(D5×Q16), C40.57(C2×C4), C10.83(C4×D4), C2.D8.9D5, C4⋊C4.167D10, (C2×C8).226D10, C10.20(C2×Q16), (C8×Dic5).1C2, C22.88(D4×D5), C20.36(C4○D4), C10.26(C4○D8), C2.3(D83D5), (C2×C40).78C22, C4.8(Q82D5), (C2×Dic20).9C2, C10.Q16.7C2, (C2×C20).289C23, C20.106(C22×C4), (C2×Dic5).276D4, Dic53Q8.7C2, Dic10.24(C2×C4), C2.13(D208C4), (C2×Dic10).90C22, (C4×Dic5).263C22, C4.44(C2×C4×D5), (C5×C2.D8).4C2, (C2×C10).294(C2×D4), (C5×C4⋊C4).82C22, (C2×C4).392(C22×D5), (C2×C52C8).238C22, SmallGroup(320,500)

Series: Derived Chief Lower central Upper central

C1C20 — Dic55Q16
C1C5C10C20C2×C20C4×Dic5Dic53Q8 — Dic55Q16
C5C10C20 — Dic55Q16
C1C22C2×C4C2.D8

Generators and relations for Dic55Q16
 G = < a,b,c,d | a10=c8=1, b2=a5, d2=c4, bab-1=dad-1=a-1, ac=ca, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 358 in 110 conjugacy classes, 51 normal (27 characteristic)
C1, C2 [×3], C4 [×2], C4 [×9], C22, C5, C8 [×2], C8, C2×C4, C2×C4 [×6], Q8 [×6], C10 [×3], C42 [×3], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, Q16 [×4], C2×Q8 [×2], Dic5 [×2], Dic5 [×5], C20 [×2], C20 [×2], C2×C10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C52C8, C40 [×2], Dic10 [×4], Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C4×Q16, Dic20 [×4], C2×C52C8, C4×Dic5, C4×Dic5 [×2], C10.D4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10 [×2], C10.Q16 [×2], C8×Dic5, C5×C2.D8, Dic53Q8 [×2], C2×Dic20, Dic55Q16
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, Q82D5, D208C4, D83D5, D5×Q16, Dic55Q16

Smallest permutation representation of Dic55Q16
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)(11 155 16 160)(12 154 17 159)(13 153 18 158)(14 152 19 157)(15 151 20 156)(21 172 26 177)(22 171 27 176)(23 180 28 175)(24 179 29 174)(25 178 30 173)(31 201 36 206)(32 210 37 205)(33 209 38 204)(34 208 39 203)(35 207 40 202)(41 192 46 197)(42 191 47 196)(43 200 48 195)(44 199 49 194)(45 198 50 193)(51 221 56 226)(52 230 57 225)(53 229 58 224)(54 228 59 223)(55 227 60 222)(61 212 66 217)(62 211 67 216)(63 220 68 215)(64 219 69 214)(65 218 70 213)(71 241 76 246)(72 250 77 245)(73 249 78 244)(74 248 79 243)(75 247 80 242)(81 232 86 237)(82 231 87 236)(83 240 88 235)(84 239 89 234)(85 238 90 233)(91 261 96 266)(92 270 97 265)(93 269 98 264)(94 268 99 263)(95 267 100 262)(101 252 106 257)(102 251 107 256)(103 260 108 255)(104 259 109 254)(105 258 110 253)(111 281 116 286)(112 290 117 285)(113 289 118 284)(114 288 119 283)(115 287 120 282)(121 272 126 277)(122 271 127 276)(123 280 128 275)(124 279 129 274)(125 278 130 273)(131 301 136 306)(132 310 137 305)(133 309 138 304)(134 308 139 303)(135 307 140 302)(141 292 146 297)(142 291 147 296)(143 300 148 295)(144 299 149 294)(145 298 150 293)(161 319 166 314)(162 318 167 313)(163 317 168 312)(164 316 169 311)(165 315 170 320)
(1 160 40 148 28 165 48 140)(2 151 31 149 29 166 49 131)(3 152 32 150 30 167 50 132)(4 153 33 141 21 168 41 133)(5 154 34 142 22 169 42 134)(6 155 35 143 23 170 43 135)(7 156 36 144 24 161 44 136)(8 157 37 145 25 162 45 137)(9 158 38 146 26 163 46 138)(10 159 39 147 27 164 47 139)(11 202 295 175 315 195 302 182)(12 203 296 176 316 196 303 183)(13 204 297 177 317 197 304 184)(14 205 298 178 318 198 305 185)(15 206 299 179 319 199 306 186)(16 207 300 180 320 200 307 187)(17 208 291 171 311 191 308 188)(18 209 292 172 312 192 309 189)(19 210 293 173 313 193 310 190)(20 201 294 174 314 194 301 181)(51 116 89 96 69 124 71 104)(52 117 90 97 70 125 72 105)(53 118 81 98 61 126 73 106)(54 119 82 99 62 127 74 107)(55 120 83 100 63 128 75 108)(56 111 84 91 64 129 76 109)(57 112 85 92 65 130 77 110)(58 113 86 93 66 121 78 101)(59 114 87 94 67 122 79 102)(60 115 88 95 68 123 80 103)(211 276 248 256 228 283 231 263)(212 277 249 257 229 284 232 264)(213 278 250 258 230 285 233 265)(214 279 241 259 221 286 234 266)(215 280 242 260 222 287 235 267)(216 271 243 251 223 288 236 268)(217 272 244 252 224 289 237 269)(218 273 245 253 225 290 238 270)(219 274 246 254 226 281 239 261)(220 275 247 255 227 282 240 262)
(1 215 28 222)(2 214 29 221)(3 213 30 230)(4 212 21 229)(5 211 22 228)(6 220 23 227)(7 219 24 226)(8 218 25 225)(9 217 26 224)(10 216 27 223)(11 100 315 108)(12 99 316 107)(13 98 317 106)(14 97 318 105)(15 96 319 104)(16 95 320 103)(17 94 311 102)(18 93 312 101)(19 92 313 110)(20 91 314 109)(31 234 49 241)(32 233 50 250)(33 232 41 249)(34 231 42 248)(35 240 43 247)(36 239 44 246)(37 238 45 245)(38 237 46 244)(39 236 47 243)(40 235 48 242)(51 186 69 179)(52 185 70 178)(53 184 61 177)(54 183 62 176)(55 182 63 175)(56 181 64 174)(57 190 65 173)(58 189 66 172)(59 188 67 171)(60 187 68 180)(71 206 89 199)(72 205 90 198)(73 204 81 197)(74 203 82 196)(75 202 83 195)(76 201 84 194)(77 210 85 193)(78 209 86 192)(79 208 87 191)(80 207 88 200)(111 301 129 294)(112 310 130 293)(113 309 121 292)(114 308 122 291)(115 307 123 300)(116 306 124 299)(117 305 125 298)(118 304 126 297)(119 303 127 296)(120 302 128 295)(131 279 149 286)(132 278 150 285)(133 277 141 284)(134 276 142 283)(135 275 143 282)(136 274 144 281)(137 273 145 290)(138 272 146 289)(139 271 147 288)(140 280 148 287)(151 266 166 259)(152 265 167 258)(153 264 168 257)(154 263 169 256)(155 262 170 255)(156 261 161 254)(157 270 162 253)(158 269 163 252)(159 268 164 251)(160 267 165 260)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12224444444444444444558888888810···102020202020···2040···40
size11112244445555101020202020222222101010102···244448···84···4

56 irreducible representations

dim1111111222222224444
type++++++++-++++--
imageC1C2C2C2C2C2C4D4D5Q16C4○D4D10D10C4○D8C4×D5Q82D5D4×D5D83D5D5×Q16
kernelDic55Q16C10.Q16C8×Dic5C5×C2.D8Dic53Q8C2×Dic20Dic20C2×Dic5C2.D8Dic5C20C4⋊C4C2×C8C10C8C4C22C2C2
# reps1211218224242482244

Matrix representation of Dic55Q16 in GL5(𝔽41)

400000
040000
004000
00001
000406
,
320000
032000
003200
0003633
00035
,
10000
002400
0292400
000400
000040
,
400000
0123900
0112900
00058
0003836

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,1,6],[32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,36,3,0,0,0,33,5],[1,0,0,0,0,0,0,29,0,0,0,24,24,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,12,11,0,0,0,39,29,0,0,0,0,0,5,38,0,0,0,8,36] >;

Dic55Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_5Q_{16}
% in TeX

G:=Group("Dic5:5Q16");
// GroupNames label

G:=SmallGroup(320,500);
// by ID

G=gap.SmallGroup(320,500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,135,268,570,297,136,12550]);
// Polycyclic

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

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