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## G = Dic11.Q8order 352 = 25·11

### The non-split extension by Dic11 of 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 — C4⋊C4

Generators and relations for Dic11.Q8
G = < a,b,c,d | a22=c4=1, b2=a11, d2=c2, bab-1=dad-1=a-1, ac=ca, cbc-1=a11b, bd=db, dcd-1=a11c-1 >

Subgroups: 258 in 56 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C4, C22, C2×C4, C2×C4, C11, C42, C4⋊C4, C4⋊C4, C22, C42.C2, Dic11, Dic11, C44, C2×C22, C2×Dic11, C2×C44, C4×Dic11, Dic11⋊C4, C44⋊C4, C11×C4⋊C4, Dic11.Q8
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C4○D4, D11, C42.C2, D22, C22×D11, D445C2, D42D11, Q8×D11, 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)
(1 183 12 194)(2 182 13 193)(3 181 14 192)(4 180 15 191)(5 179 16 190)(6 178 17 189)(7 177 18 188)(8 198 19 187)(9 197 20 186)(10 196 21 185)(11 195 22 184)(23 336 34 347)(24 335 35 346)(25 334 36 345)(26 333 37 344)(27 332 38 343)(28 331 39 342)(29 352 40 341)(30 351 41 340)(31 350 42 339)(32 349 43 338)(33 348 44 337)(45 249 56 260)(46 248 57 259)(47 247 58 258)(48 246 59 257)(49 245 60 256)(50 244 61 255)(51 243 62 254)(52 264 63 253)(53 263 64 252)(54 262 65 251)(55 261 66 250)(67 94 78 105)(68 93 79 104)(69 92 80 103)(70 91 81 102)(71 90 82 101)(72 89 83 100)(73 110 84 99)(74 109 85 98)(75 108 86 97)(76 107 87 96)(77 106 88 95)(111 142 122 153)(112 141 123 152)(113 140 124 151)(114 139 125 150)(115 138 126 149)(116 137 127 148)(117 136 128 147)(118 135 129 146)(119 134 130 145)(120 133 131 144)(121 154 132 143)(155 312 166 323)(156 311 167 322)(157 310 168 321)(158 309 169 320)(159 330 170 319)(160 329 171 318)(161 328 172 317)(162 327 173 316)(163 326 174 315)(164 325 175 314)(165 324 176 313)(199 281 210 270)(200 280 211 269)(201 279 212 268)(202 278 213 267)(203 277 214 266)(204 276 215 265)(205 275 216 286)(206 274 217 285)(207 273 218 284)(208 272 219 283)(209 271 220 282)(221 293 232 304)(222 292 233 303)(223 291 234 302)(224 290 235 301)(225 289 236 300)(226 288 237 299)(227 287 238 298)(228 308 239 297)(229 307 240 296)(230 306 241 295)(231 305 242 294)
(1 348 259 225)(2 349 260 226)(3 350 261 227)(4 351 262 228)(5 352 263 229)(6 331 264 230)(7 332 243 231)(8 333 244 232)(9 334 245 233)(10 335 246 234)(11 336 247 235)(12 337 248 236)(13 338 249 237)(14 339 250 238)(15 340 251 239)(16 341 252 240)(17 342 253 241)(18 343 254 242)(19 344 255 221)(20 345 256 222)(21 346 257 223)(22 347 258 224)(23 58 290 195)(24 59 291 196)(25 60 292 197)(26 61 293 198)(27 62 294 177)(28 63 295 178)(29 64 296 179)(30 65 297 180)(31 66 298 181)(32 45 299 182)(33 46 300 183)(34 47 301 184)(35 48 302 185)(36 49 303 186)(37 50 304 187)(38 51 305 188)(39 52 306 189)(40 53 307 190)(41 54 308 191)(42 55 287 192)(43 56 288 193)(44 57 289 194)(67 169 119 220)(68 170 120 199)(69 171 121 200)(70 172 122 201)(71 173 123 202)(72 174 124 203)(73 175 125 204)(74 176 126 205)(75 155 127 206)(76 156 128 207)(77 157 129 208)(78 158 130 209)(79 159 131 210)(80 160 132 211)(81 161 111 212)(82 162 112 213)(83 163 113 214)(84 164 114 215)(85 165 115 216)(86 166 116 217)(87 167 117 218)(88 168 118 219)(89 326 151 266)(90 327 152 267)(91 328 153 268)(92 329 154 269)(93 330 133 270)(94 309 134 271)(95 310 135 272)(96 311 136 273)(97 312 137 274)(98 313 138 275)(99 314 139 276)(100 315 140 277)(101 316 141 278)(102 317 142 279)(103 318 143 280)(104 319 144 281)(105 320 145 282)(106 321 146 283)(107 322 147 284)(108 323 148 285)(109 324 149 286)(110 325 150 265)
(1 124 259 72)(2 123 260 71)(3 122 261 70)(4 121 262 69)(5 120 263 68)(6 119 264 67)(7 118 243 88)(8 117 244 87)(9 116 245 86)(10 115 246 85)(11 114 247 84)(12 113 248 83)(13 112 249 82)(14 111 250 81)(15 132 251 80)(16 131 252 79)(17 130 253 78)(18 129 254 77)(19 128 255 76)(20 127 256 75)(21 126 257 74)(22 125 258 73)(23 325 290 265)(24 324 291 286)(25 323 292 285)(26 322 293 284)(27 321 294 283)(28 320 295 282)(29 319 296 281)(30 318 297 280)(31 317 298 279)(32 316 299 278)(33 315 300 277)(34 314 301 276)(35 313 302 275)(36 312 303 274)(37 311 304 273)(38 310 305 272)(39 309 306 271)(40 330 307 270)(41 329 308 269)(42 328 287 268)(43 327 288 267)(44 326 289 266)(45 90 182 152)(46 89 183 151)(47 110 184 150)(48 109 185 149)(49 108 186 148)(50 107 187 147)(51 106 188 146)(52 105 189 145)(53 104 190 144)(54 103 191 143)(55 102 192 142)(56 101 193 141)(57 100 194 140)(58 99 195 139)(59 98 196 138)(60 97 197 137)(61 96 198 136)(62 95 177 135)(63 94 178 134)(64 93 179 133)(65 92 180 154)(66 91 181 153)(155 233 206 334)(156 232 207 333)(157 231 208 332)(158 230 209 331)(159 229 210 352)(160 228 211 351)(161 227 212 350)(162 226 213 349)(163 225 214 348)(164 224 215 347)(165 223 216 346)(166 222 217 345)(167 221 218 344)(168 242 219 343)(169 241 220 342)(170 240 199 341)(171 239 200 340)(172 238 201 339)(173 237 202 338)(174 236 203 337)(175 235 204 336)(176 234 205 335)

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

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

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

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

Matrix representation of Dic11.Q8 in GL6(𝔽89)

 88 0 0 0 0 0 0 88 0 0 0 0 0 0 0 1 0 0 0 0 88 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 14 70 0 0 0 0 1 75 0 0 0 0 0 0 88 7 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 62 47 0 0 0 0 47 27 0 0 0 0 0 0 88 0 0 0 0 0 0 88 0 0 0 0 0 0 34 0 0 0 0 0 5 55
,
 58 23 0 0 0 0 55 31 0 0 0 0 0 0 1 82 0 0 0 0 0 88 0 0 0 0 0 0 34 36 0 0 0 0 0 55

G:=sub<GL(6,GF(89))| [88,0,0,0,0,0,0,88,0,0,0,0,0,0,0,88,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,1,0,0,0,0,70,75,0,0,0,0,0,0,88,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[62,47,0,0,0,0,47,27,0,0,0,0,0,0,88,0,0,0,0,0,0,88,0,0,0,0,0,0,34,5,0,0,0,0,0,55],[58,55,0,0,0,0,23,31,0,0,0,0,0,0,1,0,0,0,0,0,82,88,0,0,0,0,0,0,34,0,0,0,0,0,36,55] >;

Dic11.Q8 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(352,84);
// by ID

G=gap.SmallGroup(352,84);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,55,218,188,86,11525]);
// Polycyclic

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

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