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## G = Dic11⋊Q8order 352 = 25·11

### 2nd semidirect product of Dic11 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C22 — Dic11⋊Q8
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C4×Dic11 — Dic11⋊Q8
 Lower central C11 — C2×C22 — Dic11⋊Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Dic11⋊Q8
G = < a,b,c,d | a22=c4=1, b2=a11, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a11b, bd=db, dcd-1=c-1 >

Subgroups: 314 in 68 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C2×C4, Q8, C11, C42, C4⋊C4, C2×Q8, C2×Q8, C22, C22, C4⋊Q8, Dic11, Dic11, C44, C44, C2×C22, Dic22, C2×Dic11, C2×C44, C2×C44, Q8×C11, C4×Dic11, Dic11⋊C4, C2×Dic22, Q8×C22, Dic11⋊Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, D11, C4⋊Q8, D22, C11⋊D4, C22×D11, Q8×D11, C2×C11⋊D4, Dic11⋊Q8

Smallest permutation representation of Dic11⋊Q8
Regular action on 352 points
Generators in S352
(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)(309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352)
(1 249 12 260)(2 248 13 259)(3 247 14 258)(4 246 15 257)(5 245 16 256)(6 244 17 255)(7 243 18 254)(8 264 19 253)(9 263 20 252)(10 262 21 251)(11 261 22 250)(23 64 34 53)(24 63 35 52)(25 62 36 51)(26 61 37 50)(27 60 38 49)(28 59 39 48)(29 58 40 47)(30 57 41 46)(31 56 42 45)(32 55 43 66)(33 54 44 65)(67 314 78 325)(68 313 79 324)(69 312 80 323)(70 311 81 322)(71 310 82 321)(72 309 83 320)(73 330 84 319)(74 329 85 318)(75 328 86 317)(76 327 87 316)(77 326 88 315)(89 283 100 272)(90 282 101 271)(91 281 102 270)(92 280 103 269)(93 279 104 268)(94 278 105 267)(95 277 106 266)(96 276 107 265)(97 275 108 286)(98 274 109 285)(99 273 110 284)(111 133 122 144)(112 154 123 143)(113 153 124 142)(114 152 125 141)(115 151 126 140)(116 150 127 139)(117 149 128 138)(118 148 129 137)(119 147 130 136)(120 146 131 135)(121 145 132 134)(155 341 166 352)(156 340 167 351)(157 339 168 350)(158 338 169 349)(159 337 170 348)(160 336 171 347)(161 335 172 346)(162 334 173 345)(163 333 174 344)(164 332 175 343)(165 331 176 342)(177 220 188 209)(178 219 189 208)(179 218 190 207)(180 217 191 206)(181 216 192 205)(182 215 193 204)(183 214 194 203)(184 213 195 202)(185 212 196 201)(186 211 197 200)(187 210 198 199)(221 296 232 307)(222 295 233 306)(223 294 234 305)(224 293 235 304)(225 292 236 303)(226 291 237 302)(227 290 238 301)(228 289 239 300)(229 288 240 299)(230 287 241 298)(231 308 242 297)
(1 58 144 316)(2 59 145 317)(3 60 146 318)(4 61 147 319)(5 62 148 320)(6 63 149 321)(7 64 150 322)(8 65 151 323)(9 66 152 324)(10 45 153 325)(11 46 154 326)(12 47 133 327)(13 48 134 328)(14 49 135 329)(15 50 136 330)(16 51 137 309)(17 52 138 310)(18 53 139 311)(19 54 140 312)(20 55 141 313)(21 56 142 314)(22 57 143 315)(23 127 81 243)(24 128 82 244)(25 129 83 245)(26 130 84 246)(27 131 85 247)(28 132 86 248)(29 111 87 249)(30 112 88 250)(31 113 67 251)(32 114 68 252)(33 115 69 253)(34 116 70 254)(35 117 71 255)(36 118 72 256)(37 119 73 257)(38 120 74 258)(39 121 75 259)(40 122 76 260)(41 123 77 261)(42 124 78 262)(43 125 79 263)(44 126 80 264)(89 227 350 179)(90 228 351 180)(91 229 352 181)(92 230 331 182)(93 231 332 183)(94 232 333 184)(95 233 334 185)(96 234 335 186)(97 235 336 187)(98 236 337 188)(99 237 338 189)(100 238 339 190)(101 239 340 191)(102 240 341 192)(103 241 342 193)(104 242 343 194)(105 221 344 195)(106 222 345 196)(107 223 346 197)(108 224 347 198)(109 225 348 177)(110 226 349 178)(155 205 281 299)(156 206 282 300)(157 207 283 301)(158 208 284 302)(159 209 285 303)(160 210 286 304)(161 211 265 305)(162 212 266 306)(163 213 267 307)(164 214 268 308)(165 215 269 287)(166 216 270 288)(167 217 271 289)(168 218 272 290)(169 219 273 291)(170 220 274 292)(171 199 275 293)(172 200 276 294)(173 201 277 295)(174 202 278 296)(175 203 279 297)(176 204 280 298)
(1 332 144 93)(2 333 145 94)(3 334 146 95)(4 335 147 96)(5 336 148 97)(6 337 149 98)(7 338 150 99)(8 339 151 100)(9 340 152 101)(10 341 153 102)(11 342 154 103)(12 343 133 104)(13 344 134 105)(14 345 135 106)(15 346 136 107)(16 347 137 108)(17 348 138 109)(18 349 139 110)(19 350 140 89)(20 351 141 90)(21 352 142 91)(22 331 143 92)(23 291 81 219)(24 292 82 220)(25 293 83 199)(26 294 84 200)(27 295 85 201)(28 296 86 202)(29 297 87 203)(30 298 88 204)(31 299 67 205)(32 300 68 206)(33 301 69 207)(34 302 70 208)(35 303 71 209)(36 304 72 210)(37 305 73 211)(38 306 74 212)(39 307 75 213)(40 308 76 214)(41 287 77 215)(42 288 78 216)(43 289 79 217)(44 290 80 218)(45 240 325 192)(46 241 326 193)(47 242 327 194)(48 221 328 195)(49 222 329 196)(50 223 330 197)(51 224 309 198)(52 225 310 177)(53 226 311 178)(54 227 312 179)(55 228 313 180)(56 229 314 181)(57 230 315 182)(58 231 316 183)(59 232 317 184)(60 233 318 185)(61 234 319 186)(62 235 320 187)(63 236 321 188)(64 237 322 189)(65 238 323 190)(66 239 324 191)(111 279 249 175)(112 280 250 176)(113 281 251 155)(114 282 252 156)(115 283 253 157)(116 284 254 158)(117 285 255 159)(118 286 256 160)(119 265 257 161)(120 266 258 162)(121 267 259 163)(122 268 260 164)(123 269 261 165)(124 270 262 166)(125 271 263 167)(126 272 264 168)(127 273 243 169)(128 274 244 170)(129 275 245 171)(130 276 246 172)(131 277 247 173)(132 278 248 174)

G:=sub<Sym(352)| (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)(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,249,12,260)(2,248,13,259)(3,247,14,258)(4,246,15,257)(5,245,16,256)(6,244,17,255)(7,243,18,254)(8,264,19,253)(9,263,20,252)(10,262,21,251)(11,261,22,250)(23,64,34,53)(24,63,35,52)(25,62,36,51)(26,61,37,50)(27,60,38,49)(28,59,39,48)(29,58,40,47)(30,57,41,46)(31,56,42,45)(32,55,43,66)(33,54,44,65)(67,314,78,325)(68,313,79,324)(69,312,80,323)(70,311,81,322)(71,310,82,321)(72,309,83,320)(73,330,84,319)(74,329,85,318)(75,328,86,317)(76,327,87,316)(77,326,88,315)(89,283,100,272)(90,282,101,271)(91,281,102,270)(92,280,103,269)(93,279,104,268)(94,278,105,267)(95,277,106,266)(96,276,107,265)(97,275,108,286)(98,274,109,285)(99,273,110,284)(111,133,122,144)(112,154,123,143)(113,153,124,142)(114,152,125,141)(115,151,126,140)(116,150,127,139)(117,149,128,138)(118,148,129,137)(119,147,130,136)(120,146,131,135)(121,145,132,134)(155,341,166,352)(156,340,167,351)(157,339,168,350)(158,338,169,349)(159,337,170,348)(160,336,171,347)(161,335,172,346)(162,334,173,345)(163,333,174,344)(164,332,175,343)(165,331,176,342)(177,220,188,209)(178,219,189,208)(179,218,190,207)(180,217,191,206)(181,216,192,205)(182,215,193,204)(183,214,194,203)(184,213,195,202)(185,212,196,201)(186,211,197,200)(187,210,198,199)(221,296,232,307)(222,295,233,306)(223,294,234,305)(224,293,235,304)(225,292,236,303)(226,291,237,302)(227,290,238,301)(228,289,239,300)(229,288,240,299)(230,287,241,298)(231,308,242,297), (1,58,144,316)(2,59,145,317)(3,60,146,318)(4,61,147,319)(5,62,148,320)(6,63,149,321)(7,64,150,322)(8,65,151,323)(9,66,152,324)(10,45,153,325)(11,46,154,326)(12,47,133,327)(13,48,134,328)(14,49,135,329)(15,50,136,330)(16,51,137,309)(17,52,138,310)(18,53,139,311)(19,54,140,312)(20,55,141,313)(21,56,142,314)(22,57,143,315)(23,127,81,243)(24,128,82,244)(25,129,83,245)(26,130,84,246)(27,131,85,247)(28,132,86,248)(29,111,87,249)(30,112,88,250)(31,113,67,251)(32,114,68,252)(33,115,69,253)(34,116,70,254)(35,117,71,255)(36,118,72,256)(37,119,73,257)(38,120,74,258)(39,121,75,259)(40,122,76,260)(41,123,77,261)(42,124,78,262)(43,125,79,263)(44,126,80,264)(89,227,350,179)(90,228,351,180)(91,229,352,181)(92,230,331,182)(93,231,332,183)(94,232,333,184)(95,233,334,185)(96,234,335,186)(97,235,336,187)(98,236,337,188)(99,237,338,189)(100,238,339,190)(101,239,340,191)(102,240,341,192)(103,241,342,193)(104,242,343,194)(105,221,344,195)(106,222,345,196)(107,223,346,197)(108,224,347,198)(109,225,348,177)(110,226,349,178)(155,205,281,299)(156,206,282,300)(157,207,283,301)(158,208,284,302)(159,209,285,303)(160,210,286,304)(161,211,265,305)(162,212,266,306)(163,213,267,307)(164,214,268,308)(165,215,269,287)(166,216,270,288)(167,217,271,289)(168,218,272,290)(169,219,273,291)(170,220,274,292)(171,199,275,293)(172,200,276,294)(173,201,277,295)(174,202,278,296)(175,203,279,297)(176,204,280,298), (1,332,144,93)(2,333,145,94)(3,334,146,95)(4,335,147,96)(5,336,148,97)(6,337,149,98)(7,338,150,99)(8,339,151,100)(9,340,152,101)(10,341,153,102)(11,342,154,103)(12,343,133,104)(13,344,134,105)(14,345,135,106)(15,346,136,107)(16,347,137,108)(17,348,138,109)(18,349,139,110)(19,350,140,89)(20,351,141,90)(21,352,142,91)(22,331,143,92)(23,291,81,219)(24,292,82,220)(25,293,83,199)(26,294,84,200)(27,295,85,201)(28,296,86,202)(29,297,87,203)(30,298,88,204)(31,299,67,205)(32,300,68,206)(33,301,69,207)(34,302,70,208)(35,303,71,209)(36,304,72,210)(37,305,73,211)(38,306,74,212)(39,307,75,213)(40,308,76,214)(41,287,77,215)(42,288,78,216)(43,289,79,217)(44,290,80,218)(45,240,325,192)(46,241,326,193)(47,242,327,194)(48,221,328,195)(49,222,329,196)(50,223,330,197)(51,224,309,198)(52,225,310,177)(53,226,311,178)(54,227,312,179)(55,228,313,180)(56,229,314,181)(57,230,315,182)(58,231,316,183)(59,232,317,184)(60,233,318,185)(61,234,319,186)(62,235,320,187)(63,236,321,188)(64,237,322,189)(65,238,323,190)(66,239,324,191)(111,279,249,175)(112,280,250,176)(113,281,251,155)(114,282,252,156)(115,283,253,157)(116,284,254,158)(117,285,255,159)(118,286,256,160)(119,265,257,161)(120,266,258,162)(121,267,259,163)(122,268,260,164)(123,269,261,165)(124,270,262,166)(125,271,263,167)(126,272,264,168)(127,273,243,169)(128,274,244,170)(129,275,245,171)(130,276,246,172)(131,277,247,173)(132,278,248,174)>;

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)(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352), (1,249,12,260)(2,248,13,259)(3,247,14,258)(4,246,15,257)(5,245,16,256)(6,244,17,255)(7,243,18,254)(8,264,19,253)(9,263,20,252)(10,262,21,251)(11,261,22,250)(23,64,34,53)(24,63,35,52)(25,62,36,51)(26,61,37,50)(27,60,38,49)(28,59,39,48)(29,58,40,47)(30,57,41,46)(31,56,42,45)(32,55,43,66)(33,54,44,65)(67,314,78,325)(68,313,79,324)(69,312,80,323)(70,311,81,322)(71,310,82,321)(72,309,83,320)(73,330,84,319)(74,329,85,318)(75,328,86,317)(76,327,87,316)(77,326,88,315)(89,283,100,272)(90,282,101,271)(91,281,102,270)(92,280,103,269)(93,279,104,268)(94,278,105,267)(95,277,106,266)(96,276,107,265)(97,275,108,286)(98,274,109,285)(99,273,110,284)(111,133,122,144)(112,154,123,143)(113,153,124,142)(114,152,125,141)(115,151,126,140)(116,150,127,139)(117,149,128,138)(118,148,129,137)(119,147,130,136)(120,146,131,135)(121,145,132,134)(155,341,166,352)(156,340,167,351)(157,339,168,350)(158,338,169,349)(159,337,170,348)(160,336,171,347)(161,335,172,346)(162,334,173,345)(163,333,174,344)(164,332,175,343)(165,331,176,342)(177,220,188,209)(178,219,189,208)(179,218,190,207)(180,217,191,206)(181,216,192,205)(182,215,193,204)(183,214,194,203)(184,213,195,202)(185,212,196,201)(186,211,197,200)(187,210,198,199)(221,296,232,307)(222,295,233,306)(223,294,234,305)(224,293,235,304)(225,292,236,303)(226,291,237,302)(227,290,238,301)(228,289,239,300)(229,288,240,299)(230,287,241,298)(231,308,242,297), (1,58,144,316)(2,59,145,317)(3,60,146,318)(4,61,147,319)(5,62,148,320)(6,63,149,321)(7,64,150,322)(8,65,151,323)(9,66,152,324)(10,45,153,325)(11,46,154,326)(12,47,133,327)(13,48,134,328)(14,49,135,329)(15,50,136,330)(16,51,137,309)(17,52,138,310)(18,53,139,311)(19,54,140,312)(20,55,141,313)(21,56,142,314)(22,57,143,315)(23,127,81,243)(24,128,82,244)(25,129,83,245)(26,130,84,246)(27,131,85,247)(28,132,86,248)(29,111,87,249)(30,112,88,250)(31,113,67,251)(32,114,68,252)(33,115,69,253)(34,116,70,254)(35,117,71,255)(36,118,72,256)(37,119,73,257)(38,120,74,258)(39,121,75,259)(40,122,76,260)(41,123,77,261)(42,124,78,262)(43,125,79,263)(44,126,80,264)(89,227,350,179)(90,228,351,180)(91,229,352,181)(92,230,331,182)(93,231,332,183)(94,232,333,184)(95,233,334,185)(96,234,335,186)(97,235,336,187)(98,236,337,188)(99,237,338,189)(100,238,339,190)(101,239,340,191)(102,240,341,192)(103,241,342,193)(104,242,343,194)(105,221,344,195)(106,222,345,196)(107,223,346,197)(108,224,347,198)(109,225,348,177)(110,226,349,178)(155,205,281,299)(156,206,282,300)(157,207,283,301)(158,208,284,302)(159,209,285,303)(160,210,286,304)(161,211,265,305)(162,212,266,306)(163,213,267,307)(164,214,268,308)(165,215,269,287)(166,216,270,288)(167,217,271,289)(168,218,272,290)(169,219,273,291)(170,220,274,292)(171,199,275,293)(172,200,276,294)(173,201,277,295)(174,202,278,296)(175,203,279,297)(176,204,280,298), (1,332,144,93)(2,333,145,94)(3,334,146,95)(4,335,147,96)(5,336,148,97)(6,337,149,98)(7,338,150,99)(8,339,151,100)(9,340,152,101)(10,341,153,102)(11,342,154,103)(12,343,133,104)(13,344,134,105)(14,345,135,106)(15,346,136,107)(16,347,137,108)(17,348,138,109)(18,349,139,110)(19,350,140,89)(20,351,141,90)(21,352,142,91)(22,331,143,92)(23,291,81,219)(24,292,82,220)(25,293,83,199)(26,294,84,200)(27,295,85,201)(28,296,86,202)(29,297,87,203)(30,298,88,204)(31,299,67,205)(32,300,68,206)(33,301,69,207)(34,302,70,208)(35,303,71,209)(36,304,72,210)(37,305,73,211)(38,306,74,212)(39,307,75,213)(40,308,76,214)(41,287,77,215)(42,288,78,216)(43,289,79,217)(44,290,80,218)(45,240,325,192)(46,241,326,193)(47,242,327,194)(48,221,328,195)(49,222,329,196)(50,223,330,197)(51,224,309,198)(52,225,310,177)(53,226,311,178)(54,227,312,179)(55,228,313,180)(56,229,314,181)(57,230,315,182)(58,231,316,183)(59,232,317,184)(60,233,318,185)(61,234,319,186)(62,235,320,187)(63,236,321,188)(64,237,322,189)(65,238,323,190)(66,239,324,191)(111,279,249,175)(112,280,250,176)(113,281,251,155)(114,282,252,156)(115,283,253,157)(116,284,254,158)(117,285,255,159)(118,286,256,160)(119,265,257,161)(120,266,258,162)(121,267,259,163)(122,268,260,164)(123,269,261,165)(124,270,262,166)(125,271,263,167)(126,272,264,168)(127,273,243,169)(128,274,244,170)(129,275,245,171)(130,276,246,172)(131,277,247,173)(132,278,248,174) );

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),(309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352)], [(1,249,12,260),(2,248,13,259),(3,247,14,258),(4,246,15,257),(5,245,16,256),(6,244,17,255),(7,243,18,254),(8,264,19,253),(9,263,20,252),(10,262,21,251),(11,261,22,250),(23,64,34,53),(24,63,35,52),(25,62,36,51),(26,61,37,50),(27,60,38,49),(28,59,39,48),(29,58,40,47),(30,57,41,46),(31,56,42,45),(32,55,43,66),(33,54,44,65),(67,314,78,325),(68,313,79,324),(69,312,80,323),(70,311,81,322),(71,310,82,321),(72,309,83,320),(73,330,84,319),(74,329,85,318),(75,328,86,317),(76,327,87,316),(77,326,88,315),(89,283,100,272),(90,282,101,271),(91,281,102,270),(92,280,103,269),(93,279,104,268),(94,278,105,267),(95,277,106,266),(96,276,107,265),(97,275,108,286),(98,274,109,285),(99,273,110,284),(111,133,122,144),(112,154,123,143),(113,153,124,142),(114,152,125,141),(115,151,126,140),(116,150,127,139),(117,149,128,138),(118,148,129,137),(119,147,130,136),(120,146,131,135),(121,145,132,134),(155,341,166,352),(156,340,167,351),(157,339,168,350),(158,338,169,349),(159,337,170,348),(160,336,171,347),(161,335,172,346),(162,334,173,345),(163,333,174,344),(164,332,175,343),(165,331,176,342),(177,220,188,209),(178,219,189,208),(179,218,190,207),(180,217,191,206),(181,216,192,205),(182,215,193,204),(183,214,194,203),(184,213,195,202),(185,212,196,201),(186,211,197,200),(187,210,198,199),(221,296,232,307),(222,295,233,306),(223,294,234,305),(224,293,235,304),(225,292,236,303),(226,291,237,302),(227,290,238,301),(228,289,239,300),(229,288,240,299),(230,287,241,298),(231,308,242,297)], [(1,58,144,316),(2,59,145,317),(3,60,146,318),(4,61,147,319),(5,62,148,320),(6,63,149,321),(7,64,150,322),(8,65,151,323),(9,66,152,324),(10,45,153,325),(11,46,154,326),(12,47,133,327),(13,48,134,328),(14,49,135,329),(15,50,136,330),(16,51,137,309),(17,52,138,310),(18,53,139,311),(19,54,140,312),(20,55,141,313),(21,56,142,314),(22,57,143,315),(23,127,81,243),(24,128,82,244),(25,129,83,245),(26,130,84,246),(27,131,85,247),(28,132,86,248),(29,111,87,249),(30,112,88,250),(31,113,67,251),(32,114,68,252),(33,115,69,253),(34,116,70,254),(35,117,71,255),(36,118,72,256),(37,119,73,257),(38,120,74,258),(39,121,75,259),(40,122,76,260),(41,123,77,261),(42,124,78,262),(43,125,79,263),(44,126,80,264),(89,227,350,179),(90,228,351,180),(91,229,352,181),(92,230,331,182),(93,231,332,183),(94,232,333,184),(95,233,334,185),(96,234,335,186),(97,235,336,187),(98,236,337,188),(99,237,338,189),(100,238,339,190),(101,239,340,191),(102,240,341,192),(103,241,342,193),(104,242,343,194),(105,221,344,195),(106,222,345,196),(107,223,346,197),(108,224,347,198),(109,225,348,177),(110,226,349,178),(155,205,281,299),(156,206,282,300),(157,207,283,301),(158,208,284,302),(159,209,285,303),(160,210,286,304),(161,211,265,305),(162,212,266,306),(163,213,267,307),(164,214,268,308),(165,215,269,287),(166,216,270,288),(167,217,271,289),(168,218,272,290),(169,219,273,291),(170,220,274,292),(171,199,275,293),(172,200,276,294),(173,201,277,295),(174,202,278,296),(175,203,279,297),(176,204,280,298)], [(1,332,144,93),(2,333,145,94),(3,334,146,95),(4,335,147,96),(5,336,148,97),(6,337,149,98),(7,338,150,99),(8,339,151,100),(9,340,152,101),(10,341,153,102),(11,342,154,103),(12,343,133,104),(13,344,134,105),(14,345,135,106),(15,346,136,107),(16,347,137,108),(17,348,138,109),(18,349,139,110),(19,350,140,89),(20,351,141,90),(21,352,142,91),(22,331,143,92),(23,291,81,219),(24,292,82,220),(25,293,83,199),(26,294,84,200),(27,295,85,201),(28,296,86,202),(29,297,87,203),(30,298,88,204),(31,299,67,205),(32,300,68,206),(33,301,69,207),(34,302,70,208),(35,303,71,209),(36,304,72,210),(37,305,73,211),(38,306,74,212),(39,307,75,213),(40,308,76,214),(41,287,77,215),(42,288,78,216),(43,289,79,217),(44,290,80,218),(45,240,325,192),(46,241,326,193),(47,242,327,194),(48,221,328,195),(49,222,329,196),(50,223,330,197),(51,224,309,198),(52,225,310,177),(53,226,311,178),(54,227,312,179),(55,228,313,180),(56,229,314,181),(57,230,315,182),(58,231,316,183),(59,232,317,184),(60,233,318,185),(61,234,319,186),(62,235,320,187),(63,236,321,188),(64,237,322,189),(65,238,323,190),(66,239,324,191),(111,279,249,175),(112,280,250,176),(113,281,251,155),(114,282,252,156),(115,283,253,157),(116,284,254,158),(117,285,255,159),(118,286,256,160),(119,265,257,161),(120,266,258,162),(121,267,259,163),(122,268,260,164),(123,269,261,165),(124,270,262,166),(125,271,263,167),(126,272,264,168),(127,273,243,169),(128,274,244,170),(129,275,245,171),(130,276,246,172),(131,277,247,173),(132,278,248,174)]])

64 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 11A ··· 11E 22A ··· 22O 44A ··· 44AD order 1 2 2 2 4 4 4 4 4 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 1 1 2 2 4 4 22 22 22 22 44 44 2 ··· 2 2 ··· 2 4 ··· 4

64 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 type + + + + + - + + + - image C1 C2 C2 C2 C2 Q8 D4 D11 D22 C11⋊D4 Q8×D11 kernel Dic11⋊Q8 C4×Dic11 Dic11⋊C4 C2×Dic22 Q8×C22 Dic11 C44 C2×Q8 C2×C4 C4 C2 # reps 1 1 4 1 1 4 2 5 15 20 10

Matrix representation of Dic11⋊Q8 in GL6(𝔽89)

 88 0 0 0 0 0 0 88 0 0 0 0 0 0 2 1 0 0 0 0 16 53 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 85 27 0 0 0 0 62 4 0 0 0 0 0 0 43 27 0 0 0 0 70 46 0 0 0 0 0 0 88 0 0 0 0 0 0 88
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 76 78 0 0 0 0 64 13
,
 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 4 41 0 0 0 0 43 85

G:=sub<GL(6,GF(89))| [88,0,0,0,0,0,0,88,0,0,0,0,0,0,2,16,0,0,0,0,1,53,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[85,62,0,0,0,0,27,4,0,0,0,0,0,0,43,70,0,0,0,0,27,46,0,0,0,0,0,0,88,0,0,0,0,0,0,88],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,76,64,0,0,0,0,78,13],[88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,4,43,0,0,0,0,41,85] >;

Dic11⋊Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{11}\rtimes Q_8
% in TeX

G:=Group("Dic11:Q8");
// GroupNames label

G:=SmallGroup(352,139);
// by ID

G=gap.SmallGroup(352,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,55,362,116,50,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^22=c^4=1,b^2=a^11,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^11*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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