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## G = Dic5.Q16order 320 = 26·5

### 2nd non-split extension by Dic5 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5.Q16
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊C8 — Dic5.Q16
 Lower central C5 — C2×C10 — C2×C20 — Dic5.Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Dic5.Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=a5c4, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd-1=a5b, dcd-1=a5bc-1 >

Subgroups: 266 in 64 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, C2×C8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C2×C10, C4⋊C8, C4⋊Q8, C5⋊C8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4.6Q16, C4×Dic5, C10.D4, C2×C5⋊C8, C2×Dic10, Q8×C10, Dic5⋊C8, Dic5⋊Q8, Dic5.Q16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, SD16, Q16, F5, C4.D4, Q8⋊C4, C2×F5, C4.6Q16, C22⋊F5, Q8⋊F5, C23.F5, Dic5.Q16

Character table of Dic5.Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D 20E 20F size 1 1 1 1 4 8 10 10 10 10 40 4 20 20 20 20 20 20 20 20 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 1 1 1 1 -1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -i -i -i i i i i -i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 -i i i -i -i i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 i -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 i i i -i -i -i -i i 1 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 0 2 2 -2 -2 0 2 0 0 0 0 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 0 -2 -2 2 2 0 2 0 0 0 0 0 0 0 0 2 2 2 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 2 -2 0 0 0 2 0 √2 -√2 -√2 √2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 0 0 0 0 -2 2 0 2 -√2 0 0 0 0 -√2 √2 √2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 0 0 0 0 -2 2 0 2 √2 0 0 0 0 √2 -√2 -√2 2 -2 -2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ14 2 2 -2 -2 0 0 2 -2 0 0 0 2 0 -√2 √2 √2 -√2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ15 2 2 -2 -2 0 0 -2 2 0 0 0 2 0 -√-2 √-2 -√-2 √-2 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 -2 2 -2 0 0 0 0 2 -2 0 2 -√-2 0 0 0 0 √-2 -√-2 √-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ17 2 2 -2 -2 0 0 -2 2 0 0 0 2 0 √-2 -√-2 √-2 -√-2 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 0 0 2 -2 0 2 √-2 0 0 0 0 -√-2 √-2 -√-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ19 4 4 4 4 4 4 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 4 4 4 -4 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 4 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 √5 1 1 √5 -√5 -√5 orthogonal lifted from C22⋊F5 ρ23 4 4 4 4 -4 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -√5 1 1 -√5 √5 √5 orthogonal lifted from C22⋊F5 ρ24 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ52+2ζ5+1 -√5 √5 2ζ54+2ζ53+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 complex lifted from C23.F5 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ53+2ζ5+1 √5 -√5 2ζ54+2ζ52+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 complex lifted from C23.F5 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ54+2ζ52+1 √5 -√5 2ζ53+2ζ5+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 complex lifted from C23.F5 ρ27 4 -4 -4 4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 -1 1 2ζ54+2ζ53+1 -√5 √5 2ζ52+2ζ5+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 complex lifted from C23.F5 ρ28 8 8 -8 -8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from Q8⋊F5, Schur index 2 ρ29 8 -8 8 -8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 symplectic lifted from Q8⋊F5, Schur index 2

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

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

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

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

Matrix representation of Dic5.Q16 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 40 0 0 0 0 1 0 40 0 0 0 17 18 10 0 0 0 8 13 22 31
,
 40 36 0 0 0 0 25 1 0 0 0 0 0 0 1 14 40 2 0 0 2 6 20 35 0 0 18 3 21 26 0 0 35 14 35 13
,
 1 0 0 0 0 0 16 40 0 0 0 0 0 0 26 11 18 2 0 0 30 9 15 34 0 0 40 13 7 13 0 0 27 10 40 40
,
 30 34 0 0 0 0 35 11 0 0 0 0 0 0 19 5 28 0 0 0 11 12 40 0 0 0 13 34 9 0 0 0 2 7 32 1

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,1,17,8,0,0,40,0,18,13,0,0,0,40,10,22,0,0,0,0,0,31],[40,25,0,0,0,0,36,1,0,0,0,0,0,0,1,2,18,35,0,0,14,6,3,14,0,0,40,20,21,35,0,0,2,35,26,13],[1,16,0,0,0,0,0,40,0,0,0,0,0,0,26,30,40,27,0,0,11,9,13,10,0,0,18,15,7,40,0,0,2,34,13,40],[30,35,0,0,0,0,34,11,0,0,0,0,0,0,19,11,13,2,0,0,5,12,34,7,0,0,28,40,9,32,0,0,0,0,0,1] >;

Dic5.Q16 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,269);
// by ID

G=gap.SmallGroup(320,269);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,100,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=1,b^2=a^5,d^2=a^5*c^4,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=d*b*d^-1=a^5*b,d*c*d^-1=a^5*b*c^-1>;
// generators/relations

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