direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×Q8⋊C4, Q8⋊1C44, C44.61D4, C22.6Q16, C22.10SD16, C4⋊C4.1C22, (C2×C88).3C2, (C2×C8).1C22, C4.2(C2×C44), (Q8×C11)⋊4C4, C44.29(C2×C4), (C2×C22).47D4, C4.12(D4×C11), (C2×Q8).2C22, (Q8×C22).7C2, C2.1(C11×Q16), C2.2(C11×SD16), C22.9(D4×C11), C22.25(C22⋊C4), (C2×C44).115C22, (C11×C4⋊C4).8C2, (C2×C4).18(C2×C22), C2.7(C11×C22⋊C4), SmallGroup(352,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×Q8⋊C4
G = < a,b,c,d | a11=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >
Smallest permutation representation of C11×Q8⋊C4
►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)
(1 127 157 121)(2 128 158 111)(3 129 159 112)(4 130 160 113)(5 131 161 114)(6 132 162 115)(7 122 163 116)(8 123 164 117)(9 124 165 118)(10 125 155 119)(11 126 156 120)(12 34 24 57)(13 35 25 58)(14 36 26 59)(15 37 27 60)(16 38 28 61)(17 39 29 62)(18 40 30 63)(19 41 31 64)(20 42 32 65)(21 43 33 66)(22 44 23 56)(45 70 344 82)(46 71 345 83)(47 72 346 84)(48 73 347 85)(49 74 348 86)(50 75 349 87)(51 76 350 88)(52 77 351 78)(53 67 352 79)(54 68 342 80)(55 69 343 81)(89 145 100 133)(90 146 101 134)(91 147 102 135)(92 148 103 136)(93 149 104 137)(94 150 105 138)(95 151 106 139)(96 152 107 140)(97 153 108 141)(98 154 109 142)(99 144 110 143)(166 230 197 242)(167 231 198 232)(168 221 188 233)(169 222 189 234)(170 223 190 235)(171 224 191 236)(172 225 192 237)(173 226 193 238)(174 227 194 239)(175 228 195 240)(176 229 196 241)(177 211 200 247)(178 212 201 248)(179 213 202 249)(180 214 203 250)(181 215 204 251)(182 216 205 252)(183 217 206 253)(184 218 207 243)(185 219 208 244)(186 220 209 245)(187 210 199 246)(254 318 285 330)(255 319 286 320)(256 309 276 321)(257 310 277 322)(258 311 278 323)(259 312 279 324)(260 313 280 325)(261 314 281 326)(262 315 282 327)(263 316 283 328)(264 317 284 329)(265 299 288 335)(266 300 289 336)(267 301 290 337)(268 302 291 338)(269 303 292 339)(270 304 293 340)(271 305 294 341)(272 306 295 331)(273 307 296 332)(274 308 297 333)(275 298 287 334)
(1 197 157 166)(2 198 158 167)(3 188 159 168)(4 189 160 169)(5 190 161 170)(6 191 162 171)(7 192 163 172)(8 193 164 173)(9 194 165 174)(10 195 155 175)(11 196 156 176)(12 321 24 309)(13 322 25 310)(14 323 26 311)(15 324 27 312)(16 325 28 313)(17 326 29 314)(18 327 30 315)(19 328 31 316)(20 329 32 317)(21 330 33 318)(22 320 23 319)(34 276 57 256)(35 277 58 257)(36 278 59 258)(37 279 60 259)(38 280 61 260)(39 281 62 261)(40 282 63 262)(41 283 64 263)(42 284 65 264)(43 285 66 254)(44 286 56 255)(45 288 344 265)(46 289 345 266)(47 290 346 267)(48 291 347 268)(49 292 348 269)(50 293 349 270)(51 294 350 271)(52 295 351 272)(53 296 352 273)(54 297 342 274)(55 287 343 275)(67 307 79 332)(68 308 80 333)(69 298 81 334)(70 299 82 335)(71 300 83 336)(72 301 84 337)(73 302 85 338)(74 303 86 339)(75 304 87 340)(76 305 88 341)(77 306 78 331)(89 247 100 211)(90 248 101 212)(91 249 102 213)(92 250 103 214)(93 251 104 215)(94 252 105 216)(95 253 106 217)(96 243 107 218)(97 244 108 219)(98 245 109 220)(99 246 110 210)(111 232 128 231)(112 233 129 221)(113 234 130 222)(114 235 131 223)(115 236 132 224)(116 237 122 225)(117 238 123 226)(118 239 124 227)(119 240 125 228)(120 241 126 229)(121 242 127 230)(133 177 145 200)(134 178 146 201)(135 179 147 202)(136 180 148 203)(137 181 149 204)(138 182 150 205)(139 183 151 206)(140 184 152 207)(141 185 153 208)(142 186 154 209)(143 187 144 199)
(1 254 98 308)(2 255 99 298)(3 256 89 299)(4 257 90 300)(5 258 91 301)(6 259 92 302)(7 260 93 303)(8 261 94 304)(9 262 95 305)(10 263 96 306)(11 264 97 307)(12 211 344 168)(13 212 345 169)(14 213 346 170)(15 214 347 171)(16 215 348 172)(17 216 349 173)(18 217 350 174)(19 218 351 175)(20 219 352 176)(21 220 342 166)(22 210 343 167)(23 246 55 198)(24 247 45 188)(25 248 46 189)(26 249 47 190)(27 250 48 191)(28 251 49 192)(29 252 50 193)(30 253 51 194)(31 243 52 195)(32 244 53 196)(33 245 54 197)(34 177 82 233)(35 178 83 234)(36 179 84 235)(37 180 85 236)(38 181 86 237)(39 182 87 238)(40 183 88 239)(41 184 78 240)(42 185 79 241)(43 186 80 242)(44 187 81 232)(56 199 69 231)(57 200 70 221)(58 201 71 222)(59 202 72 223)(60 203 73 224)(61 204 74 225)(62 205 75 226)(63 206 76 227)(64 207 77 228)(65 208 67 229)(66 209 68 230)(100 335 159 276)(101 336 160 277)(102 337 161 278)(103 338 162 279)(104 339 163 280)(105 340 164 281)(106 341 165 282)(107 331 155 283)(108 332 156 284)(109 333 157 285)(110 334 158 286)(111 319 143 287)(112 309 133 288)(113 310 134 289)(114 311 135 290)(115 312 136 291)(116 313 137 292)(117 314 138 293)(118 315 139 294)(119 316 140 295)(120 317 141 296)(121 318 142 297)(122 325 149 269)(123 326 150 270)(124 327 151 271)(125 328 152 272)(126 329 153 273)(127 330 154 274)(128 320 144 275)(129 321 145 265)(130 322 146 266)(131 323 147 267)(132 324 148 268)
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), (1,127,157,121)(2,128,158,111)(3,129,159,112)(4,130,160,113)(5,131,161,114)(6,132,162,115)(7,122,163,116)(8,123,164,117)(9,124,165,118)(10,125,155,119)(11,126,156,120)(12,34,24,57)(13,35,25,58)(14,36,26,59)(15,37,27,60)(16,38,28,61)(17,39,29,62)(18,40,30,63)(19,41,31,64)(20,42,32,65)(21,43,33,66)(22,44,23,56)(45,70,344,82)(46,71,345,83)(47,72,346,84)(48,73,347,85)(49,74,348,86)(50,75,349,87)(51,76,350,88)(52,77,351,78)(53,67,352,79)(54,68,342,80)(55,69,343,81)(89,145,100,133)(90,146,101,134)(91,147,102,135)(92,148,103,136)(93,149,104,137)(94,150,105,138)(95,151,106,139)(96,152,107,140)(97,153,108,141)(98,154,109,142)(99,144,110,143)(166,230,197,242)(167,231,198,232)(168,221,188,233)(169,222,189,234)(170,223,190,235)(171,224,191,236)(172,225,192,237)(173,226,193,238)(174,227,194,239)(175,228,195,240)(176,229,196,241)(177,211,200,247)(178,212,201,248)(179,213,202,249)(180,214,203,250)(181,215,204,251)(182,216,205,252)(183,217,206,253)(184,218,207,243)(185,219,208,244)(186,220,209,245)(187,210,199,246)(254,318,285,330)(255,319,286,320)(256,309,276,321)(257,310,277,322)(258,311,278,323)(259,312,279,324)(260,313,280,325)(261,314,281,326)(262,315,282,327)(263,316,283,328)(264,317,284,329)(265,299,288,335)(266,300,289,336)(267,301,290,337)(268,302,291,338)(269,303,292,339)(270,304,293,340)(271,305,294,341)(272,306,295,331)(273,307,296,332)(274,308,297,333)(275,298,287,334), (1,197,157,166)(2,198,158,167)(3,188,159,168)(4,189,160,169)(5,190,161,170)(6,191,162,171)(7,192,163,172)(8,193,164,173)(9,194,165,174)(10,195,155,175)(11,196,156,176)(12,321,24,309)(13,322,25,310)(14,323,26,311)(15,324,27,312)(16,325,28,313)(17,326,29,314)(18,327,30,315)(19,328,31,316)(20,329,32,317)(21,330,33,318)(22,320,23,319)(34,276,57,256)(35,277,58,257)(36,278,59,258)(37,279,60,259)(38,280,61,260)(39,281,62,261)(40,282,63,262)(41,283,64,263)(42,284,65,264)(43,285,66,254)(44,286,56,255)(45,288,344,265)(46,289,345,266)(47,290,346,267)(48,291,347,268)(49,292,348,269)(50,293,349,270)(51,294,350,271)(52,295,351,272)(53,296,352,273)(54,297,342,274)(55,287,343,275)(67,307,79,332)(68,308,80,333)(69,298,81,334)(70,299,82,335)(71,300,83,336)(72,301,84,337)(73,302,85,338)(74,303,86,339)(75,304,87,340)(76,305,88,341)(77,306,78,331)(89,247,100,211)(90,248,101,212)(91,249,102,213)(92,250,103,214)(93,251,104,215)(94,252,105,216)(95,253,106,217)(96,243,107,218)(97,244,108,219)(98,245,109,220)(99,246,110,210)(111,232,128,231)(112,233,129,221)(113,234,130,222)(114,235,131,223)(115,236,132,224)(116,237,122,225)(117,238,123,226)(118,239,124,227)(119,240,125,228)(120,241,126,229)(121,242,127,230)(133,177,145,200)(134,178,146,201)(135,179,147,202)(136,180,148,203)(137,181,149,204)(138,182,150,205)(139,183,151,206)(140,184,152,207)(141,185,153,208)(142,186,154,209)(143,187,144,199), (1,254,98,308)(2,255,99,298)(3,256,89,299)(4,257,90,300)(5,258,91,301)(6,259,92,302)(7,260,93,303)(8,261,94,304)(9,262,95,305)(10,263,96,306)(11,264,97,307)(12,211,344,168)(13,212,345,169)(14,213,346,170)(15,214,347,171)(16,215,348,172)(17,216,349,173)(18,217,350,174)(19,218,351,175)(20,219,352,176)(21,220,342,166)(22,210,343,167)(23,246,55,198)(24,247,45,188)(25,248,46,189)(26,249,47,190)(27,250,48,191)(28,251,49,192)(29,252,50,193)(30,253,51,194)(31,243,52,195)(32,244,53,196)(33,245,54,197)(34,177,82,233)(35,178,83,234)(36,179,84,235)(37,180,85,236)(38,181,86,237)(39,182,87,238)(40,183,88,239)(41,184,78,240)(42,185,79,241)(43,186,80,242)(44,187,81,232)(56,199,69,231)(57,200,70,221)(58,201,71,222)(59,202,72,223)(60,203,73,224)(61,204,74,225)(62,205,75,226)(63,206,76,227)(64,207,77,228)(65,208,67,229)(66,209,68,230)(100,335,159,276)(101,336,160,277)(102,337,161,278)(103,338,162,279)(104,339,163,280)(105,340,164,281)(106,341,165,282)(107,331,155,283)(108,332,156,284)(109,333,157,285)(110,334,158,286)(111,319,143,287)(112,309,133,288)(113,310,134,289)(114,311,135,290)(115,312,136,291)(116,313,137,292)(117,314,138,293)(118,315,139,294)(119,316,140,295)(120,317,141,296)(121,318,142,297)(122,325,149,269)(123,326,150,270)(124,327,151,271)(125,328,152,272)(126,329,153,273)(127,330,154,274)(128,320,144,275)(129,321,145,265)(130,322,146,266)(131,323,147,267)(132,324,148,268)>;
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), (1,127,157,121)(2,128,158,111)(3,129,159,112)(4,130,160,113)(5,131,161,114)(6,132,162,115)(7,122,163,116)(8,123,164,117)(9,124,165,118)(10,125,155,119)(11,126,156,120)(12,34,24,57)(13,35,25,58)(14,36,26,59)(15,37,27,60)(16,38,28,61)(17,39,29,62)(18,40,30,63)(19,41,31,64)(20,42,32,65)(21,43,33,66)(22,44,23,56)(45,70,344,82)(46,71,345,83)(47,72,346,84)(48,73,347,85)(49,74,348,86)(50,75,349,87)(51,76,350,88)(52,77,351,78)(53,67,352,79)(54,68,342,80)(55,69,343,81)(89,145,100,133)(90,146,101,134)(91,147,102,135)(92,148,103,136)(93,149,104,137)(94,150,105,138)(95,151,106,139)(96,152,107,140)(97,153,108,141)(98,154,109,142)(99,144,110,143)(166,230,197,242)(167,231,198,232)(168,221,188,233)(169,222,189,234)(170,223,190,235)(171,224,191,236)(172,225,192,237)(173,226,193,238)(174,227,194,239)(175,228,195,240)(176,229,196,241)(177,211,200,247)(178,212,201,248)(179,213,202,249)(180,214,203,250)(181,215,204,251)(182,216,205,252)(183,217,206,253)(184,218,207,243)(185,219,208,244)(186,220,209,245)(187,210,199,246)(254,318,285,330)(255,319,286,320)(256,309,276,321)(257,310,277,322)(258,311,278,323)(259,312,279,324)(260,313,280,325)(261,314,281,326)(262,315,282,327)(263,316,283,328)(264,317,284,329)(265,299,288,335)(266,300,289,336)(267,301,290,337)(268,302,291,338)(269,303,292,339)(270,304,293,340)(271,305,294,341)(272,306,295,331)(273,307,296,332)(274,308,297,333)(275,298,287,334), (1,197,157,166)(2,198,158,167)(3,188,159,168)(4,189,160,169)(5,190,161,170)(6,191,162,171)(7,192,163,172)(8,193,164,173)(9,194,165,174)(10,195,155,175)(11,196,156,176)(12,321,24,309)(13,322,25,310)(14,323,26,311)(15,324,27,312)(16,325,28,313)(17,326,29,314)(18,327,30,315)(19,328,31,316)(20,329,32,317)(21,330,33,318)(22,320,23,319)(34,276,57,256)(35,277,58,257)(36,278,59,258)(37,279,60,259)(38,280,61,260)(39,281,62,261)(40,282,63,262)(41,283,64,263)(42,284,65,264)(43,285,66,254)(44,286,56,255)(45,288,344,265)(46,289,345,266)(47,290,346,267)(48,291,347,268)(49,292,348,269)(50,293,349,270)(51,294,350,271)(52,295,351,272)(53,296,352,273)(54,297,342,274)(55,287,343,275)(67,307,79,332)(68,308,80,333)(69,298,81,334)(70,299,82,335)(71,300,83,336)(72,301,84,337)(73,302,85,338)(74,303,86,339)(75,304,87,340)(76,305,88,341)(77,306,78,331)(89,247,100,211)(90,248,101,212)(91,249,102,213)(92,250,103,214)(93,251,104,215)(94,252,105,216)(95,253,106,217)(96,243,107,218)(97,244,108,219)(98,245,109,220)(99,246,110,210)(111,232,128,231)(112,233,129,221)(113,234,130,222)(114,235,131,223)(115,236,132,224)(116,237,122,225)(117,238,123,226)(118,239,124,227)(119,240,125,228)(120,241,126,229)(121,242,127,230)(133,177,145,200)(134,178,146,201)(135,179,147,202)(136,180,148,203)(137,181,149,204)(138,182,150,205)(139,183,151,206)(140,184,152,207)(141,185,153,208)(142,186,154,209)(143,187,144,199), (1,254,98,308)(2,255,99,298)(3,256,89,299)(4,257,90,300)(5,258,91,301)(6,259,92,302)(7,260,93,303)(8,261,94,304)(9,262,95,305)(10,263,96,306)(11,264,97,307)(12,211,344,168)(13,212,345,169)(14,213,346,170)(15,214,347,171)(16,215,348,172)(17,216,349,173)(18,217,350,174)(19,218,351,175)(20,219,352,176)(21,220,342,166)(22,210,343,167)(23,246,55,198)(24,247,45,188)(25,248,46,189)(26,249,47,190)(27,250,48,191)(28,251,49,192)(29,252,50,193)(30,253,51,194)(31,243,52,195)(32,244,53,196)(33,245,54,197)(34,177,82,233)(35,178,83,234)(36,179,84,235)(37,180,85,236)(38,181,86,237)(39,182,87,238)(40,183,88,239)(41,184,78,240)(42,185,79,241)(43,186,80,242)(44,187,81,232)(56,199,69,231)(57,200,70,221)(58,201,71,222)(59,202,72,223)(60,203,73,224)(61,204,74,225)(62,205,75,226)(63,206,76,227)(64,207,77,228)(65,208,67,229)(66,209,68,230)(100,335,159,276)(101,336,160,277)(102,337,161,278)(103,338,162,279)(104,339,163,280)(105,340,164,281)(106,341,165,282)(107,331,155,283)(108,332,156,284)(109,333,157,285)(110,334,158,286)(111,319,143,287)(112,309,133,288)(113,310,134,289)(114,311,135,290)(115,312,136,291)(116,313,137,292)(117,314,138,293)(118,315,139,294)(119,316,140,295)(120,317,141,296)(121,318,142,297)(122,325,149,269)(123,326,150,270)(124,327,151,271)(125,328,152,272)(126,329,153,273)(127,330,154,274)(128,320,144,275)(129,321,145,265)(130,322,146,266)(131,323,147,267)(132,324,148,268) );
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)], [(1,127,157,121),(2,128,158,111),(3,129,159,112),(4,130,160,113),(5,131,161,114),(6,132,162,115),(7,122,163,116),(8,123,164,117),(9,124,165,118),(10,125,155,119),(11,126,156,120),(12,34,24,57),(13,35,25,58),(14,36,26,59),(15,37,27,60),(16,38,28,61),(17,39,29,62),(18,40,30,63),(19,41,31,64),(20,42,32,65),(21,43,33,66),(22,44,23,56),(45,70,344,82),(46,71,345,83),(47,72,346,84),(48,73,347,85),(49,74,348,86),(50,75,349,87),(51,76,350,88),(52,77,351,78),(53,67,352,79),(54,68,342,80),(55,69,343,81),(89,145,100,133),(90,146,101,134),(91,147,102,135),(92,148,103,136),(93,149,104,137),(94,150,105,138),(95,151,106,139),(96,152,107,140),(97,153,108,141),(98,154,109,142),(99,144,110,143),(166,230,197,242),(167,231,198,232),(168,221,188,233),(169,222,189,234),(170,223,190,235),(171,224,191,236),(172,225,192,237),(173,226,193,238),(174,227,194,239),(175,228,195,240),(176,229,196,241),(177,211,200,247),(178,212,201,248),(179,213,202,249),(180,214,203,250),(181,215,204,251),(182,216,205,252),(183,217,206,253),(184,218,207,243),(185,219,208,244),(186,220,209,245),(187,210,199,246),(254,318,285,330),(255,319,286,320),(256,309,276,321),(257,310,277,322),(258,311,278,323),(259,312,279,324),(260,313,280,325),(261,314,281,326),(262,315,282,327),(263,316,283,328),(264,317,284,329),(265,299,288,335),(266,300,289,336),(267,301,290,337),(268,302,291,338),(269,303,292,339),(270,304,293,340),(271,305,294,341),(272,306,295,331),(273,307,296,332),(274,308,297,333),(275,298,287,334)], [(1,197,157,166),(2,198,158,167),(3,188,159,168),(4,189,160,169),(5,190,161,170),(6,191,162,171),(7,192,163,172),(8,193,164,173),(9,194,165,174),(10,195,155,175),(11,196,156,176),(12,321,24,309),(13,322,25,310),(14,323,26,311),(15,324,27,312),(16,325,28,313),(17,326,29,314),(18,327,30,315),(19,328,31,316),(20,329,32,317),(21,330,33,318),(22,320,23,319),(34,276,57,256),(35,277,58,257),(36,278,59,258),(37,279,60,259),(38,280,61,260),(39,281,62,261),(40,282,63,262),(41,283,64,263),(42,284,65,264),(43,285,66,254),(44,286,56,255),(45,288,344,265),(46,289,345,266),(47,290,346,267),(48,291,347,268),(49,292,348,269),(50,293,349,270),(51,294,350,271),(52,295,351,272),(53,296,352,273),(54,297,342,274),(55,287,343,275),(67,307,79,332),(68,308,80,333),(69,298,81,334),(70,299,82,335),(71,300,83,336),(72,301,84,337),(73,302,85,338),(74,303,86,339),(75,304,87,340),(76,305,88,341),(77,306,78,331),(89,247,100,211),(90,248,101,212),(91,249,102,213),(92,250,103,214),(93,251,104,215),(94,252,105,216),(95,253,106,217),(96,243,107,218),(97,244,108,219),(98,245,109,220),(99,246,110,210),(111,232,128,231),(112,233,129,221),(113,234,130,222),(114,235,131,223),(115,236,132,224),(116,237,122,225),(117,238,123,226),(118,239,124,227),(119,240,125,228),(120,241,126,229),(121,242,127,230),(133,177,145,200),(134,178,146,201),(135,179,147,202),(136,180,148,203),(137,181,149,204),(138,182,150,205),(139,183,151,206),(140,184,152,207),(141,185,153,208),(142,186,154,209),(143,187,144,199)], [(1,254,98,308),(2,255,99,298),(3,256,89,299),(4,257,90,300),(5,258,91,301),(6,259,92,302),(7,260,93,303),(8,261,94,304),(9,262,95,305),(10,263,96,306),(11,264,97,307),(12,211,344,168),(13,212,345,169),(14,213,346,170),(15,214,347,171),(16,215,348,172),(17,216,349,173),(18,217,350,174),(19,218,351,175),(20,219,352,176),(21,220,342,166),(22,210,343,167),(23,246,55,198),(24,247,45,188),(25,248,46,189),(26,249,47,190),(27,250,48,191),(28,251,49,192),(29,252,50,193),(30,253,51,194),(31,243,52,195),(32,244,53,196),(33,245,54,197),(34,177,82,233),(35,178,83,234),(36,179,84,235),(37,180,85,236),(38,181,86,237),(39,182,87,238),(40,183,88,239),(41,184,78,240),(42,185,79,241),(43,186,80,242),(44,187,81,232),(56,199,69,231),(57,200,70,221),(58,201,71,222),(59,202,72,223),(60,203,73,224),(61,204,74,225),(62,205,75,226),(63,206,76,227),(64,207,77,228),(65,208,67,229),(66,209,68,230),(100,335,159,276),(101,336,160,277),(102,337,161,278),(103,338,162,279),(104,339,163,280),(105,340,164,281),(106,341,165,282),(107,331,155,283),(108,332,156,284),(109,333,157,285),(110,334,158,286),(111,319,143,287),(112,309,133,288),(113,310,134,289),(114,311,135,290),(115,312,136,291),(116,313,137,292),(117,314,138,293),(118,315,139,294),(119,316,140,295),(120,317,141,296),(121,318,142,297),(122,325,149,269),(123,326,150,270),(124,327,151,271),(125,328,152,272),(126,329,153,273),(127,330,154,274),(128,320,144,275),(129,321,145,265),(130,322,146,266),(131,323,147,267),(132,324,148,268)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44T | 44U | ··· | 44BH | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C11 | C22 | C22 | C22 | C44 | D4 | D4 | SD16 | Q16 | D4×C11 | D4×C11 | C11×SD16 | C11×Q16 |
| kernel | C11×Q8⋊C4 | C11×C4⋊C4 | C2×C88 | Q8×C22 | Q8×C11 | Q8⋊C4 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C44 | C2×C22 | C22 | C22 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 40 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 |
Matrix representation of C11×Q8⋊C4 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 87 |
| 0 | 0 | 1 | 1 |
| 0 | 88 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 45 | 7 |
| 0 | 0 | 3 | 44 |
| 82 | 53 | 0 | 0 |
| 36 | 7 | 0 | 0 |
| 0 | 0 | 75 | 32 |
| 0 | 0 | 30 | 14 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,64,0,0,0,0,64],[88,0,0,0,0,88,0,0,0,0,88,1,0,0,87,1],[0,88,0,0,88,0,0,0,0,0,45,3,0,0,7,44],[82,36,0,0,53,7,0,0,0,0,75,30,0,0,32,14] >;
C11×Q8⋊C4 in GAP, Magma, Sage, TeX
C_{11}\times Q_8\rtimes C_4 % in TeX
G:=Group("C11xQ8:C4"); // GroupNames label
G:=SmallGroup(352,52);
// by ID
G=gap.SmallGroup(352,52);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,1063,5283,2649,117]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations
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