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## G = Dic5.12Q16order 320 = 26·5

### 4th non-split extension by Dic5 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — Dic5.12Q16
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C20⋊C8 — Dic5.12Q16
 Lower central C5 — C10 — C20 — Dic5.12Q16
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for Dic5.12Q16
G = < a,b,c,d | a10=c8=1, b2=a5, d2=a5c4, bab-1=a-1, cac-1=a3, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 242 in 70 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C2×C5⋊C8, Q8×C10, C4×C5⋊C8, C20⋊C8, Q8×Dic5, Dic5.12Q16
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), SD16, Q16, F5, C22⋊C8, Q8⋊C4, C4≀C2, C5⋊C8, C2×F5, Q8⋊C8, C2×C5⋊C8, C22.F5, C22⋊F5, Q8⋊F5, Q82F5, C23.2F5, Dic5.12Q16

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 8A ··· 8H 8I 8J 8K 8L 10A 10B 10C 20A ··· 20F order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 10 10 20 ··· 20 size 1 1 1 1 2 2 4 4 5 5 5 5 10 10 20 20 4 10 ··· 10 20 20 20 20 4 4 4 8 ··· 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + - + + - - + - + image C1 C2 C2 C2 C4 C4 C8 D4 SD16 Q16 M4(2) C4≀C2 F5 C2×F5 C5⋊C8 C22.F5 C22⋊F5 Q8⋊F5 Q8⋊2F5 kernel Dic5.12Q16 C4×C5⋊C8 C20⋊C8 Q8×Dic5 C4⋊Dic5 Q8×C10 C5×Q8 C2×Dic5 Dic5 Dic5 C20 C10 C2×Q8 C2×C4 Q8 C4 C22 C2 C2 # reps 1 1 1 1 2 2 8 2 2 2 2 4 1 1 2 2 2 1 1

Matrix representation of Dic5.12Q16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 1 1 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 5 7 31 10 0 0 2 26 5 36 0 0 24 3 34 39 0 0 20 10 15 17
,
 15 26 0 0 0 0 15 15 0 0 0 0 0 0 32 25 6 30 0 0 22 5 16 7 0 0 11 2 36 17 0 0 34 15 39 9
,
 20 38 0 0 0 0 38 21 0 0 0 0 0 0 22 0 3 3 0 0 38 19 38 0 0 0 0 38 19 38 0 0 3 3 0 22

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,5,2,24,20,0,0,7,26,3,10,0,0,31,5,34,15,0,0,10,36,39,17],[15,15,0,0,0,0,26,15,0,0,0,0,0,0,32,22,11,34,0,0,25,5,2,15,0,0,6,16,36,39,0,0,30,7,17,9],[20,38,0,0,0,0,38,21,0,0,0,0,0,0,22,38,0,3,0,0,0,19,38,3,0,0,3,38,19,0,0,0,3,0,38,22] >;

Dic5.12Q16 in GAP, Magma, Sage, TeX

{\rm Dic}_5._{12}Q_{16}
% in TeX

G:=Group("Dic5.12Q16");
// GroupNames label

G:=SmallGroup(320,268);
// by ID

G=gap.SmallGroup(320,268);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,100,1123,570,136,6278,3156]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations

׿
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