metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C44.9D4, C22.5Q16, Q8⋊1Dic11, C22.8SD16, C44.8(C2×C4), (Q8×C11)⋊1C4, (C2×C22).34D4, (C2×C4).40D22, (Q8×C22).1C2, (C2×Q8).1D11, C11⋊3(Q8⋊C4), C2.3(Q8⋊D11), C44⋊C4.10C2, C4.2(C2×Dic11), C4.14(C11⋊D4), (C2×C44).18C22, C2.3(C11⋊Q16), C22.16(C22⋊C4), C2.6(C23.D11), C22.18(C11⋊D4), (C2×C11⋊C8).5C2, SmallGroup(352,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊Dic11
G = < a,b,c,d | a4=c22=1, b2=a2, d2=c11, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >
Smallest permutation representation of Q8⋊Dic11
►Regular action on 352 points
Generators in S352
(1 88 181 32)(2 67 182 33)(3 68 183 34)(4 69 184 35)(5 70 185 36)(6 71 186 37)(7 72 187 38)(8 73 188 39)(9 74 189 40)(10 75 190 41)(11 76 191 42)(12 77 192 43)(13 78 193 44)(14 79 194 23)(15 80 195 24)(16 81 196 25)(17 82 197 26)(18 83 198 27)(19 84 177 28)(20 85 178 29)(21 86 179 30)(22 87 180 31)(45 277 176 205)(46 278 155 206)(47 279 156 207)(48 280 157 208)(49 281 158 209)(50 282 159 210)(51 283 160 211)(52 284 161 212)(53 285 162 213)(54 286 163 214)(55 265 164 215)(56 266 165 216)(57 267 166 217)(58 268 167 218)(59 269 168 219)(60 270 169 220)(61 271 170 199)(62 272 171 200)(63 273 172 201)(64 274 173 202)(65 275 174 203)(66 276 175 204)(89 153 122 263)(90 154 123 264)(91 133 124 243)(92 134 125 244)(93 135 126 245)(94 136 127 246)(95 137 128 247)(96 138 129 248)(97 139 130 249)(98 140 131 250)(99 141 132 251)(100 142 111 252)(101 143 112 253)(102 144 113 254)(103 145 114 255)(104 146 115 256)(105 147 116 257)(106 148 117 258)(107 149 118 259)(108 150 119 260)(109 151 120 261)(110 152 121 262)(221 349 309 298)(222 350 310 299)(223 351 311 300)(224 352 312 301)(225 331 313 302)(226 332 314 303)(227 333 315 304)(228 334 316 305)(229 335 317 306)(230 336 318 307)(231 337 319 308)(232 338 320 287)(233 339 321 288)(234 340 322 289)(235 341 323 290)(236 342 324 291)(237 343 325 292)(238 344 326 293)(239 345 327 294)(240 346 328 295)(241 347 329 296)(242 348 330 297)
(1 218 181 268)(2 219 182 269)(3 220 183 270)(4 199 184 271)(5 200 185 272)(6 201 186 273)(7 202 187 274)(8 203 188 275)(9 204 189 276)(10 205 190 277)(11 206 191 278)(12 207 192 279)(13 208 193 280)(14 209 194 281)(15 210 195 282)(16 211 196 283)(17 212 197 284)(18 213 198 285)(19 214 177 286)(20 215 178 265)(21 216 179 266)(22 217 180 267)(23 49 79 158)(24 50 80 159)(25 51 81 160)(26 52 82 161)(27 53 83 162)(28 54 84 163)(29 55 85 164)(30 56 86 165)(31 57 87 166)(32 58 88 167)(33 59 67 168)(34 60 68 169)(35 61 69 170)(36 62 70 171)(37 63 71 172)(38 64 72 173)(39 65 73 174)(40 66 74 175)(41 45 75 176)(42 46 76 155)(43 47 77 156)(44 48 78 157)(89 240 122 328)(90 241 123 329)(91 242 124 330)(92 221 125 309)(93 222 126 310)(94 223 127 311)(95 224 128 312)(96 225 129 313)(97 226 130 314)(98 227 131 315)(99 228 132 316)(100 229 111 317)(101 230 112 318)(102 231 113 319)(103 232 114 320)(104 233 115 321)(105 234 116 322)(106 235 117 323)(107 236 118 324)(108 237 119 325)(109 238 120 326)(110 239 121 327)(133 297 243 348)(134 298 244 349)(135 299 245 350)(136 300 246 351)(137 301 247 352)(138 302 248 331)(139 303 249 332)(140 304 250 333)(141 305 251 334)(142 306 252 335)(143 307 253 336)(144 308 254 337)(145 287 255 338)(146 288 256 339)(147 289 257 340)(148 290 258 341)(149 291 259 342)(150 292 260 343)(151 293 261 344)(152 294 262 345)(153 295 263 346)(154 296 264 347)
(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)
(1 101 12 90)(2 100 13 89)(3 99 14 110)(4 98 15 109)(5 97 16 108)(6 96 17 107)(7 95 18 106)(8 94 19 105)(9 93 20 104)(10 92 21 103)(11 91 22 102)(23 152 34 141)(24 151 35 140)(25 150 36 139)(26 149 37 138)(27 148 38 137)(28 147 39 136)(29 146 40 135)(30 145 41 134)(31 144 42 133)(32 143 43 154)(33 142 44 153)(45 221 56 232)(46 242 57 231)(47 241 58 230)(48 240 59 229)(49 239 60 228)(50 238 61 227)(51 237 62 226)(52 236 63 225)(53 235 64 224)(54 234 65 223)(55 233 66 222)(67 252 78 263)(68 251 79 262)(69 250 80 261)(70 249 81 260)(71 248 82 259)(72 247 83 258)(73 246 84 257)(74 245 85 256)(75 244 86 255)(76 243 87 254)(77 264 88 253)(111 193 122 182)(112 192 123 181)(113 191 124 180)(114 190 125 179)(115 189 126 178)(116 188 127 177)(117 187 128 198)(118 186 129 197)(119 185 130 196)(120 184 131 195)(121 183 132 194)(155 330 166 319)(156 329 167 318)(157 328 168 317)(158 327 169 316)(159 326 170 315)(160 325 171 314)(161 324 172 313)(162 323 173 312)(163 322 174 311)(164 321 175 310)(165 320 176 309)(199 333 210 344)(200 332 211 343)(201 331 212 342)(202 352 213 341)(203 351 214 340)(204 350 215 339)(205 349 216 338)(206 348 217 337)(207 347 218 336)(208 346 219 335)(209 345 220 334)(265 288 276 299)(266 287 277 298)(267 308 278 297)(268 307 279 296)(269 306 280 295)(270 305 281 294)(271 304 282 293)(272 303 283 292)(273 302 284 291)(274 301 285 290)(275 300 286 289)
G:=sub<Sym(352)| (1,88,181,32)(2,67,182,33)(3,68,183,34)(4,69,184,35)(5,70,185,36)(6,71,186,37)(7,72,187,38)(8,73,188,39)(9,74,189,40)(10,75,190,41)(11,76,191,42)(12,77,192,43)(13,78,193,44)(14,79,194,23)(15,80,195,24)(16,81,196,25)(17,82,197,26)(18,83,198,27)(19,84,177,28)(20,85,178,29)(21,86,179,30)(22,87,180,31)(45,277,176,205)(46,278,155,206)(47,279,156,207)(48,280,157,208)(49,281,158,209)(50,282,159,210)(51,283,160,211)(52,284,161,212)(53,285,162,213)(54,286,163,214)(55,265,164,215)(56,266,165,216)(57,267,166,217)(58,268,167,218)(59,269,168,219)(60,270,169,220)(61,271,170,199)(62,272,171,200)(63,273,172,201)(64,274,173,202)(65,275,174,203)(66,276,175,204)(89,153,122,263)(90,154,123,264)(91,133,124,243)(92,134,125,244)(93,135,126,245)(94,136,127,246)(95,137,128,247)(96,138,129,248)(97,139,130,249)(98,140,131,250)(99,141,132,251)(100,142,111,252)(101,143,112,253)(102,144,113,254)(103,145,114,255)(104,146,115,256)(105,147,116,257)(106,148,117,258)(107,149,118,259)(108,150,119,260)(109,151,120,261)(110,152,121,262)(221,349,309,298)(222,350,310,299)(223,351,311,300)(224,352,312,301)(225,331,313,302)(226,332,314,303)(227,333,315,304)(228,334,316,305)(229,335,317,306)(230,336,318,307)(231,337,319,308)(232,338,320,287)(233,339,321,288)(234,340,322,289)(235,341,323,290)(236,342,324,291)(237,343,325,292)(238,344,326,293)(239,345,327,294)(240,346,328,295)(241,347,329,296)(242,348,330,297), (1,218,181,268)(2,219,182,269)(3,220,183,270)(4,199,184,271)(5,200,185,272)(6,201,186,273)(7,202,187,274)(8,203,188,275)(9,204,189,276)(10,205,190,277)(11,206,191,278)(12,207,192,279)(13,208,193,280)(14,209,194,281)(15,210,195,282)(16,211,196,283)(17,212,197,284)(18,213,198,285)(19,214,177,286)(20,215,178,265)(21,216,179,266)(22,217,180,267)(23,49,79,158)(24,50,80,159)(25,51,81,160)(26,52,82,161)(27,53,83,162)(28,54,84,163)(29,55,85,164)(30,56,86,165)(31,57,87,166)(32,58,88,167)(33,59,67,168)(34,60,68,169)(35,61,69,170)(36,62,70,171)(37,63,71,172)(38,64,72,173)(39,65,73,174)(40,66,74,175)(41,45,75,176)(42,46,76,155)(43,47,77,156)(44,48,78,157)(89,240,122,328)(90,241,123,329)(91,242,124,330)(92,221,125,309)(93,222,126,310)(94,223,127,311)(95,224,128,312)(96,225,129,313)(97,226,130,314)(98,227,131,315)(99,228,132,316)(100,229,111,317)(101,230,112,318)(102,231,113,319)(103,232,114,320)(104,233,115,321)(105,234,116,322)(106,235,117,323)(107,236,118,324)(108,237,119,325)(109,238,120,326)(110,239,121,327)(133,297,243,348)(134,298,244,349)(135,299,245,350)(136,300,246,351)(137,301,247,352)(138,302,248,331)(139,303,249,332)(140,304,250,333)(141,305,251,334)(142,306,252,335)(143,307,253,336)(144,308,254,337)(145,287,255,338)(146,288,256,339)(147,289,257,340)(148,290,258,341)(149,291,259,342)(150,292,260,343)(151,293,261,344)(152,294,262,345)(153,295,263,346)(154,296,264,347), (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), (1,101,12,90)(2,100,13,89)(3,99,14,110)(4,98,15,109)(5,97,16,108)(6,96,17,107)(7,95,18,106)(8,94,19,105)(9,93,20,104)(10,92,21,103)(11,91,22,102)(23,152,34,141)(24,151,35,140)(25,150,36,139)(26,149,37,138)(27,148,38,137)(28,147,39,136)(29,146,40,135)(30,145,41,134)(31,144,42,133)(32,143,43,154)(33,142,44,153)(45,221,56,232)(46,242,57,231)(47,241,58,230)(48,240,59,229)(49,239,60,228)(50,238,61,227)(51,237,62,226)(52,236,63,225)(53,235,64,224)(54,234,65,223)(55,233,66,222)(67,252,78,263)(68,251,79,262)(69,250,80,261)(70,249,81,260)(71,248,82,259)(72,247,83,258)(73,246,84,257)(74,245,85,256)(75,244,86,255)(76,243,87,254)(77,264,88,253)(111,193,122,182)(112,192,123,181)(113,191,124,180)(114,190,125,179)(115,189,126,178)(116,188,127,177)(117,187,128,198)(118,186,129,197)(119,185,130,196)(120,184,131,195)(121,183,132,194)(155,330,166,319)(156,329,167,318)(157,328,168,317)(158,327,169,316)(159,326,170,315)(160,325,171,314)(161,324,172,313)(162,323,173,312)(163,322,174,311)(164,321,175,310)(165,320,176,309)(199,333,210,344)(200,332,211,343)(201,331,212,342)(202,352,213,341)(203,351,214,340)(204,350,215,339)(205,349,216,338)(206,348,217,337)(207,347,218,336)(208,346,219,335)(209,345,220,334)(265,288,276,299)(266,287,277,298)(267,308,278,297)(268,307,279,296)(269,306,280,295)(270,305,281,294)(271,304,282,293)(272,303,283,292)(273,302,284,291)(274,301,285,290)(275,300,286,289)>;
G:=Group( (1,88,181,32)(2,67,182,33)(3,68,183,34)(4,69,184,35)(5,70,185,36)(6,71,186,37)(7,72,187,38)(8,73,188,39)(9,74,189,40)(10,75,190,41)(11,76,191,42)(12,77,192,43)(13,78,193,44)(14,79,194,23)(15,80,195,24)(16,81,196,25)(17,82,197,26)(18,83,198,27)(19,84,177,28)(20,85,178,29)(21,86,179,30)(22,87,180,31)(45,277,176,205)(46,278,155,206)(47,279,156,207)(48,280,157,208)(49,281,158,209)(50,282,159,210)(51,283,160,211)(52,284,161,212)(53,285,162,213)(54,286,163,214)(55,265,164,215)(56,266,165,216)(57,267,166,217)(58,268,167,218)(59,269,168,219)(60,270,169,220)(61,271,170,199)(62,272,171,200)(63,273,172,201)(64,274,173,202)(65,275,174,203)(66,276,175,204)(89,153,122,263)(90,154,123,264)(91,133,124,243)(92,134,125,244)(93,135,126,245)(94,136,127,246)(95,137,128,247)(96,138,129,248)(97,139,130,249)(98,140,131,250)(99,141,132,251)(100,142,111,252)(101,143,112,253)(102,144,113,254)(103,145,114,255)(104,146,115,256)(105,147,116,257)(106,148,117,258)(107,149,118,259)(108,150,119,260)(109,151,120,261)(110,152,121,262)(221,349,309,298)(222,350,310,299)(223,351,311,300)(224,352,312,301)(225,331,313,302)(226,332,314,303)(227,333,315,304)(228,334,316,305)(229,335,317,306)(230,336,318,307)(231,337,319,308)(232,338,320,287)(233,339,321,288)(234,340,322,289)(235,341,323,290)(236,342,324,291)(237,343,325,292)(238,344,326,293)(239,345,327,294)(240,346,328,295)(241,347,329,296)(242,348,330,297), (1,218,181,268)(2,219,182,269)(3,220,183,270)(4,199,184,271)(5,200,185,272)(6,201,186,273)(7,202,187,274)(8,203,188,275)(9,204,189,276)(10,205,190,277)(11,206,191,278)(12,207,192,279)(13,208,193,280)(14,209,194,281)(15,210,195,282)(16,211,196,283)(17,212,197,284)(18,213,198,285)(19,214,177,286)(20,215,178,265)(21,216,179,266)(22,217,180,267)(23,49,79,158)(24,50,80,159)(25,51,81,160)(26,52,82,161)(27,53,83,162)(28,54,84,163)(29,55,85,164)(30,56,86,165)(31,57,87,166)(32,58,88,167)(33,59,67,168)(34,60,68,169)(35,61,69,170)(36,62,70,171)(37,63,71,172)(38,64,72,173)(39,65,73,174)(40,66,74,175)(41,45,75,176)(42,46,76,155)(43,47,77,156)(44,48,78,157)(89,240,122,328)(90,241,123,329)(91,242,124,330)(92,221,125,309)(93,222,126,310)(94,223,127,311)(95,224,128,312)(96,225,129,313)(97,226,130,314)(98,227,131,315)(99,228,132,316)(100,229,111,317)(101,230,112,318)(102,231,113,319)(103,232,114,320)(104,233,115,321)(105,234,116,322)(106,235,117,323)(107,236,118,324)(108,237,119,325)(109,238,120,326)(110,239,121,327)(133,297,243,348)(134,298,244,349)(135,299,245,350)(136,300,246,351)(137,301,247,352)(138,302,248,331)(139,303,249,332)(140,304,250,333)(141,305,251,334)(142,306,252,335)(143,307,253,336)(144,308,254,337)(145,287,255,338)(146,288,256,339)(147,289,257,340)(148,290,258,341)(149,291,259,342)(150,292,260,343)(151,293,261,344)(152,294,262,345)(153,295,263,346)(154,296,264,347), (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), (1,101,12,90)(2,100,13,89)(3,99,14,110)(4,98,15,109)(5,97,16,108)(6,96,17,107)(7,95,18,106)(8,94,19,105)(9,93,20,104)(10,92,21,103)(11,91,22,102)(23,152,34,141)(24,151,35,140)(25,150,36,139)(26,149,37,138)(27,148,38,137)(28,147,39,136)(29,146,40,135)(30,145,41,134)(31,144,42,133)(32,143,43,154)(33,142,44,153)(45,221,56,232)(46,242,57,231)(47,241,58,230)(48,240,59,229)(49,239,60,228)(50,238,61,227)(51,237,62,226)(52,236,63,225)(53,235,64,224)(54,234,65,223)(55,233,66,222)(67,252,78,263)(68,251,79,262)(69,250,80,261)(70,249,81,260)(71,248,82,259)(72,247,83,258)(73,246,84,257)(74,245,85,256)(75,244,86,255)(76,243,87,254)(77,264,88,253)(111,193,122,182)(112,192,123,181)(113,191,124,180)(114,190,125,179)(115,189,126,178)(116,188,127,177)(117,187,128,198)(118,186,129,197)(119,185,130,196)(120,184,131,195)(121,183,132,194)(155,330,166,319)(156,329,167,318)(157,328,168,317)(158,327,169,316)(159,326,170,315)(160,325,171,314)(161,324,172,313)(162,323,173,312)(163,322,174,311)(164,321,175,310)(165,320,176,309)(199,333,210,344)(200,332,211,343)(201,331,212,342)(202,352,213,341)(203,351,214,340)(204,350,215,339)(205,349,216,338)(206,348,217,337)(207,347,218,336)(208,346,219,335)(209,345,220,334)(265,288,276,299)(266,287,277,298)(267,308,278,297)(268,307,279,296)(269,306,280,295)(270,305,281,294)(271,304,282,293)(272,303,283,292)(273,302,284,291)(274,301,285,290)(275,300,286,289) );
G=PermutationGroup([[(1,88,181,32),(2,67,182,33),(3,68,183,34),(4,69,184,35),(5,70,185,36),(6,71,186,37),(7,72,187,38),(8,73,188,39),(9,74,189,40),(10,75,190,41),(11,76,191,42),(12,77,192,43),(13,78,193,44),(14,79,194,23),(15,80,195,24),(16,81,196,25),(17,82,197,26),(18,83,198,27),(19,84,177,28),(20,85,178,29),(21,86,179,30),(22,87,180,31),(45,277,176,205),(46,278,155,206),(47,279,156,207),(48,280,157,208),(49,281,158,209),(50,282,159,210),(51,283,160,211),(52,284,161,212),(53,285,162,213),(54,286,163,214),(55,265,164,215),(56,266,165,216),(57,267,166,217),(58,268,167,218),(59,269,168,219),(60,270,169,220),(61,271,170,199),(62,272,171,200),(63,273,172,201),(64,274,173,202),(65,275,174,203),(66,276,175,204),(89,153,122,263),(90,154,123,264),(91,133,124,243),(92,134,125,244),(93,135,126,245),(94,136,127,246),(95,137,128,247),(96,138,129,248),(97,139,130,249),(98,140,131,250),(99,141,132,251),(100,142,111,252),(101,143,112,253),(102,144,113,254),(103,145,114,255),(104,146,115,256),(105,147,116,257),(106,148,117,258),(107,149,118,259),(108,150,119,260),(109,151,120,261),(110,152,121,262),(221,349,309,298),(222,350,310,299),(223,351,311,300),(224,352,312,301),(225,331,313,302),(226,332,314,303),(227,333,315,304),(228,334,316,305),(229,335,317,306),(230,336,318,307),(231,337,319,308),(232,338,320,287),(233,339,321,288),(234,340,322,289),(235,341,323,290),(236,342,324,291),(237,343,325,292),(238,344,326,293),(239,345,327,294),(240,346,328,295),(241,347,329,296),(242,348,330,297)], [(1,218,181,268),(2,219,182,269),(3,220,183,270),(4,199,184,271),(5,200,185,272),(6,201,186,273),(7,202,187,274),(8,203,188,275),(9,204,189,276),(10,205,190,277),(11,206,191,278),(12,207,192,279),(13,208,193,280),(14,209,194,281),(15,210,195,282),(16,211,196,283),(17,212,197,284),(18,213,198,285),(19,214,177,286),(20,215,178,265),(21,216,179,266),(22,217,180,267),(23,49,79,158),(24,50,80,159),(25,51,81,160),(26,52,82,161),(27,53,83,162),(28,54,84,163),(29,55,85,164),(30,56,86,165),(31,57,87,166),(32,58,88,167),(33,59,67,168),(34,60,68,169),(35,61,69,170),(36,62,70,171),(37,63,71,172),(38,64,72,173),(39,65,73,174),(40,66,74,175),(41,45,75,176),(42,46,76,155),(43,47,77,156),(44,48,78,157),(89,240,122,328),(90,241,123,329),(91,242,124,330),(92,221,125,309),(93,222,126,310),(94,223,127,311),(95,224,128,312),(96,225,129,313),(97,226,130,314),(98,227,131,315),(99,228,132,316),(100,229,111,317),(101,230,112,318),(102,231,113,319),(103,232,114,320),(104,233,115,321),(105,234,116,322),(106,235,117,323),(107,236,118,324),(108,237,119,325),(109,238,120,326),(110,239,121,327),(133,297,243,348),(134,298,244,349),(135,299,245,350),(136,300,246,351),(137,301,247,352),(138,302,248,331),(139,303,249,332),(140,304,250,333),(141,305,251,334),(142,306,252,335),(143,307,253,336),(144,308,254,337),(145,287,255,338),(146,288,256,339),(147,289,257,340),(148,290,258,341),(149,291,259,342),(150,292,260,343),(151,293,261,344),(152,294,262,345),(153,295,263,346),(154,296,264,347)], [(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)], [(1,101,12,90),(2,100,13,89),(3,99,14,110),(4,98,15,109),(5,97,16,108),(6,96,17,107),(7,95,18,106),(8,94,19,105),(9,93,20,104),(10,92,21,103),(11,91,22,102),(23,152,34,141),(24,151,35,140),(25,150,36,139),(26,149,37,138),(27,148,38,137),(28,147,39,136),(29,146,40,135),(30,145,41,134),(31,144,42,133),(32,143,43,154),(33,142,44,153),(45,221,56,232),(46,242,57,231),(47,241,58,230),(48,240,59,229),(49,239,60,228),(50,238,61,227),(51,237,62,226),(52,236,63,225),(53,235,64,224),(54,234,65,223),(55,233,66,222),(67,252,78,263),(68,251,79,262),(69,250,80,261),(70,249,81,260),(71,248,82,259),(72,247,83,258),(73,246,84,257),(74,245,85,256),(75,244,86,255),(76,243,87,254),(77,264,88,253),(111,193,122,182),(112,192,123,181),(113,191,124,180),(114,190,125,179),(115,189,126,178),(116,188,127,177),(117,187,128,198),(118,186,129,197),(119,185,130,196),(120,184,131,195),(121,183,132,194),(155,330,166,319),(156,329,167,318),(157,328,168,317),(158,327,169,316),(159,326,170,315),(160,325,171,314),(161,324,172,313),(162,323,173,312),(163,322,174,311),(164,321,175,310),(165,320,176,309),(199,333,210,344),(200,332,211,343),(201,331,212,342),(202,352,213,341),(203,351,214,340),(204,350,215,339),(205,349,216,338),(206,348,217,337),(207,347,218,336),(208,346,219,335),(209,345,220,334),(265,288,276,299),(266,287,277,298),(267,308,278,297),(268,307,279,296),(269,306,280,295),(270,305,281,294),(271,304,282,293),(272,303,283,292),(273,302,284,291),(274,301,285,290),(275,300,286,289)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 44 | 44 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | SD16 | Q16 | D11 | D22 | Dic11 | C11⋊D4 | C11⋊D4 | Q8⋊D11 | C11⋊Q16 |
| kernel | Q8⋊Dic11 | C2×C11⋊C8 | C44⋊C4 | Q8×C22 | Q8×C11 | C44 | C2×C22 | C22 | C22 | C2×Q8 | C2×C4 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 5 | 5 |
Matrix representation of Q8⋊Dic11 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 4 |
| 0 | 0 | 44 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 74 | 28 |
| 0 | 0 | 11 | 15 |
| 44 | 88 | 0 | 0 |
| 25 | 48 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 60 | 58 | 0 | 0 |
| 53 | 29 | 0 | 0 |
| 0 | 0 | 3 | 63 |
| 0 | 0 | 62 | 86 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,1,44,0,0,4,88],[88,0,0,0,0,88,0,0,0,0,74,11,0,0,28,15],[44,25,0,0,88,48,0,0,0,0,88,0,0,0,0,88],[60,53,0,0,58,29,0,0,0,0,3,62,0,0,63,86] >;
Q8⋊Dic11 in GAP, Magma, Sage, TeX
Q_8\rtimes {\rm Dic}_{11} % in TeX
G:=Group("Q8:Dic11"); // GroupNames label
G:=SmallGroup(352,41);
// by ID
G=gap.SmallGroup(352,41);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,103,579,297,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^22=1,b^2=a^2,d^2=c^11,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations
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