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## G = Q8×Dic11order 352 = 25·11

### Direct product of Q8 and Dic11

Series: Derived Chief Lower central Upper central

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

Generators and relations for Q8×Dic11
G = < a,b,c,d | a4=c22=1, b2=a2, d2=c11, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 266 in 70 conjugacy classes, 51 normal (14 characteristic)
C1, C2, C4, C4, C22, C2×C4, C2×C4, Q8, C11, C42, C4⋊C4, C2×Q8, C22, C4×Q8, Dic11, Dic11, C44, C2×C22, C2×Dic11, C2×Dic11, C2×C44, Q8×C11, C4×Dic11, C44⋊C4, Q8×C22, Q8×Dic11
Quotients: C1, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C4○D4, D11, C4×Q8, Dic11, D22, C2×Dic11, C22×D11, Q8×D11, D44⋊C2, C22×Dic11, Q8×Dic11

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + - + + - - + image C1 C2 C2 C2 C4 Q8 C4○D4 D11 D22 Dic11 Q8×D11 D44⋊C2 kernel Q8×Dic11 C4×Dic11 C44⋊C4 Q8×C22 Q8×C11 Dic11 C22 C2×Q8 C2×C4 Q8 C2 C2 # reps 1 3 3 1 8 2 2 5 15 20 5 5

Matrix representation of Q8×Dic11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 86 44 0 0 20 3
,
 88 0 0 0 0 88 0 0 0 0 41 64 0 0 21 48
,
 45 88 0 0 22 47 0 0 0 0 1 0 0 0 0 1
,
 13 12 0 0 60 76 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,86,20,0,0,44,3],[88,0,0,0,0,88,0,0,0,0,41,21,0,0,64,48],[45,22,0,0,88,47,0,0,0,0,1,0,0,0,0,1],[13,60,0,0,12,76,0,0,0,0,1,0,0,0,0,1] >;

Q8×Dic11 in GAP, Magma, Sage, TeX

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

G:=Group("Q8xDic11");
// GroupNames label

G:=SmallGroup(352,140);
// by ID

G=gap.SmallGroup(352,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,103,188,86,11525]);
// Polycyclic

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

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