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G = Dic5.12Q16order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.4M4(2), Dic5.12Q16, Dic5.22SD16, Q8⋊(C5⋊C8), C52(Q8⋊C8), (C5×Q8)⋊1C8, C20.5(C2×C8), C10.12C4≀C2, (C2×Q8).4F5, (Q8×C10).4C4, C20⋊C8.2C2, C4⋊Dic5.10C4, C2.3(Q82F5), C2.3(Q8⋊F5), (Q8×Dic5).14C2, C10.13(C22⋊C8), C4.2(C22.F5), C10.6(Q8⋊C4), (C2×Dic5).111D4, C22.42(C22⋊F5), C2.6(C23.2F5), (C4×Dic5).189C22, C4.2(C2×C5⋊C8), (C4×C5⋊C8).2C2, (C2×C4).72(C2×F5), (C2×C20).40(C2×C4), (C2×C10).43(C22⋊C4), SmallGroup(320,268)

Series: Derived Chief Lower central Upper central

C1C20 — Dic5.12Q16
C1C5C10C2×C10C2×Dic5C4×Dic5C20⋊C8 — Dic5.12Q16
C5C10C20 — Dic5.12Q16
C1C22C2×C4C2×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 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×3], C2×C4, C2×C4 [×4], Q8 [×2], Q8, C10 [×3], C42 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×Q8, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C4×C8, C4⋊C8, C4×Q8, C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, Q8⋊C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C2×C5⋊C8 [×2], Q8×C10, C4×C5⋊C8, C20⋊C8, Q8×Dic5, Dic5.12Q16
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), SD16, Q16, F5, C22⋊C8, Q8⋊C4, C4≀C2, C5⋊C8 [×2], 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)(11 273 16 278)(12 272 17 277)(13 271 18 276)(14 280 19 275)(15 279 20 274)(21 60 26 55)(22 59 27 54)(23 58 28 53)(24 57 29 52)(25 56 30 51)(31 86 36 81)(32 85 37 90)(33 84 38 89)(34 83 39 88)(35 82 40 87)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(91 143 96 148)(92 142 97 147)(93 141 98 146)(94 150 99 145)(95 149 100 144)(101 140 106 135)(102 139 107 134)(103 138 108 133)(104 137 109 132)(105 136 110 131)(111 165 116 170)(112 164 117 169)(113 163 118 168)(114 162 119 167)(115 161 120 166)(121 157 126 152)(122 156 127 151)(123 155 128 160)(124 154 129 159)(125 153 130 158)(171 230 176 225)(172 229 177 224)(173 228 178 223)(174 227 179 222)(175 226 180 221)(181 213 186 218)(182 212 187 217)(183 211 188 216)(184 220 189 215)(185 219 190 214)(191 241 196 246)(192 250 197 245)(193 249 198 244)(194 248 199 243)(195 247 200 242)(201 232 206 237)(202 231 207 236)(203 240 208 235)(204 239 209 234)(205 238 210 233)(251 310 256 305)(252 309 257 304)(253 308 258 303)(254 307 259 302)(255 306 260 301)(261 293 266 298)(262 292 267 297)(263 291 268 296)(264 300 269 295)(265 299 270 294)(281 312 286 317)(282 311 287 316)(283 320 288 315)(284 319 289 314)(285 318 290 313)
(1 13 85 270 23 314 78 256)(2 20 84 263 24 311 77 259)(3 17 83 266 25 318 76 252)(4 14 82 269 26 315 75 255)(5 11 81 262 27 312 74 258)(6 18 90 265 28 319 73 251)(7 15 89 268 29 316 72 254)(8 12 88 261 30 313 71 257)(9 19 87 264 21 320 80 260)(10 16 86 267 22 317 79 253)(31 292 54 286 49 303 68 273)(32 299 53 289 50 310 67 276)(33 296 52 282 41 307 66 279)(34 293 51 285 42 304 65 272)(35 300 60 288 43 301 64 275)(36 297 59 281 44 308 63 278)(37 294 58 284 45 305 62 271)(38 291 57 287 46 302 61 274)(39 298 56 290 47 309 70 277)(40 295 55 283 48 306 69 280)(91 249 154 178 102 237 170 182)(92 246 153 171 103 234 169 185)(93 243 152 174 104 231 168 188)(94 250 151 177 105 238 167 181)(95 247 160 180 106 235 166 184)(96 244 159 173 107 232 165 187)(97 241 158 176 108 239 164 190)(98 248 157 179 109 236 163 183)(99 245 156 172 110 233 162 186)(100 242 155 175 101 240 161 189)(111 212 143 198 129 223 139 201)(112 219 142 191 130 230 138 204)(113 216 141 194 121 227 137 207)(114 213 150 197 122 224 136 210)(115 220 149 200 123 221 135 203)(116 217 148 193 124 228 134 206)(117 214 147 196 125 225 133 209)(118 211 146 199 126 222 132 202)(119 218 145 192 127 229 131 205)(120 215 144 195 128 226 140 208)
(1 108 28 92)(2 109 29 93)(3 110 30 94)(4 101 21 95)(5 102 22 96)(6 103 23 97)(7 104 24 98)(8 105 25 99)(9 106 26 100)(10 107 27 91)(11 223 317 217)(12 224 318 218)(13 225 319 219)(14 226 320 220)(15 227 311 211)(16 228 312 212)(17 229 313 213)(18 230 314 214)(19 221 315 215)(20 222 316 216)(31 124 44 111)(32 125 45 112)(33 126 46 113)(34 127 47 114)(35 128 48 115)(36 129 49 116)(37 130 50 117)(38 121 41 118)(39 122 42 119)(40 123 43 120)(51 150 70 131)(52 141 61 132)(53 142 62 133)(54 143 63 134)(55 144 64 135)(56 145 65 136)(57 146 66 137)(58 147 67 138)(59 148 68 139)(60 149 69 140)(71 162 83 151)(72 163 84 152)(73 164 85 153)(74 165 86 154)(75 166 87 155)(76 167 88 156)(77 168 89 157)(78 169 90 158)(79 170 81 159)(80 161 82 160)(171 289 190 271)(172 290 181 272)(173 281 182 273)(174 282 183 274)(175 283 184 275)(176 284 185 276)(177 285 186 277)(178 286 187 278)(179 287 188 279)(180 288 189 280)(191 251 209 270)(192 252 210 261)(193 253 201 262)(194 254 202 263)(195 255 203 264)(196 256 204 265)(197 257 205 266)(198 258 206 267)(199 259 207 268)(200 260 208 269)(231 291 248 307)(232 292 249 308)(233 293 250 309)(234 294 241 310)(235 295 242 301)(236 296 243 302)(237 297 244 303)(238 298 245 304)(239 299 246 305)(240 300 247 306)

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

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

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

38 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H4I4J4K4L 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122244444444444458···8888810101020···20
size11112244555510102020410···10202020204448···8

38 irreducible representations

dim1111111222224444488
type+++++-++--+-+
imageC1C2C2C2C4C4C8D4SD16Q16M4(2)C4≀C2F5C2×F5C5⋊C8C22.F5C22⋊F5Q8⋊F5Q82F5
kernelDic5.12Q16C4×C5⋊C8C20⋊C8Q8×Dic5C4⋊Dic5Q8×C10C5×Q8C2×Dic5Dic5Dic5C20C10C2×Q8C2×C4Q8C4C22C2C2
# reps1111228222241122211

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

100000
010000
0004000
0000400
0000040
001111
,
4000000
0400000
00573110
00226536
002433439
0020101517
,
15260000
15150000
003225630
00225167
001123617
003415399
,
20380000
38210000
0022033
003819380
000381938
0033022

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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