direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C11×C4.Q8, C88⋊6C4, C8⋊2C44, C44.9Q8, C22.11SD16, C4⋊C4.2C22, C4.6(C2×C44), (C2×C8).6C22, C4.1(Q8×C11), (C2×C88).16C2, C44.43(C2×C4), (C2×C22).48D4, C22.12(C4⋊C4), C2.3(C11×SD16), C22.10(D4×C11), (C2×C44).117C22, C2.3(C11×C4⋊C4), (C11×C4⋊C4).9C2, (C2×C4).20(C2×C22), SmallGroup(352,55)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C11×C4.Q8
G = < a,b,c,d | a11=b4=1, c4=b2, d2=b-1c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C11×C4.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)
(1 103 157 91)(2 104 158 92)(3 105 159 93)(4 106 160 94)(5 107 161 95)(6 108 162 96)(7 109 163 97)(8 110 164 98)(9 100 165 99)(10 101 155 89)(11 102 156 90)(12 24 48 350)(13 25 49 351)(14 26 50 352)(15 27 51 342)(16 28 52 343)(17 29 53 344)(18 30 54 345)(19 31 55 346)(20 32 45 347)(21 33 46 348)(22 23 47 349)(34 68 62 80)(35 69 63 81)(36 70 64 82)(37 71 65 83)(38 72 66 84)(39 73 56 85)(40 74 57 86)(41 75 58 87)(42 76 59 88)(43 77 60 78)(44 67 61 79)(111 147 135 130)(112 148 136 131)(113 149 137 132)(114 150 138 122)(115 151 139 123)(116 152 140 124)(117 153 141 125)(118 154 142 126)(119 144 143 127)(120 145 133 128)(121 146 134 129)(166 226 202 238)(167 227 203 239)(168 228 204 240)(169 229 205 241)(170 230 206 242)(171 231 207 232)(172 221 208 233)(173 222 209 234)(174 223 199 235)(175 224 200 236)(176 225 201 237)(177 218 189 243)(178 219 190 244)(179 220 191 245)(180 210 192 246)(181 211 193 247)(182 212 194 248)(183 213 195 249)(184 214 196 250)(185 215 197 251)(186 216 198 252)(187 217 188 253)(254 326 290 314)(255 327 291 315)(256 328 292 316)(257 329 293 317)(258 330 294 318)(259 320 295 319)(260 321 296 309)(261 322 297 310)(262 323 287 311)(263 324 288 312)(264 325 289 313)(265 331 277 306)(266 332 278 307)(267 333 279 308)(268 334 280 298)(269 335 281 299)(270 336 282 300)(271 337 283 301)(272 338 284 302)(273 339 285 303)(274 340 286 304)(275 341 276 305)
(1 226 122 245 157 238 150 220)(2 227 123 246 158 239 151 210)(3 228 124 247 159 240 152 211)(4 229 125 248 160 241 153 212)(5 230 126 249 161 242 154 213)(6 231 127 250 162 232 144 214)(7 221 128 251 163 233 145 215)(8 222 129 252 164 234 146 216)(9 223 130 253 165 235 147 217)(10 224 131 243 155 236 148 218)(11 225 132 244 156 237 149 219)(12 305 59 323 48 341 42 311)(13 306 60 324 49 331 43 312)(14 307 61 325 50 332 44 313)(15 308 62 326 51 333 34 314)(16 298 63 327 52 334 35 315)(17 299 64 328 53 335 36 316)(18 300 65 329 54 336 37 317)(19 301 66 330 55 337 38 318)(20 302 56 320 45 338 39 319)(21 303 57 321 46 339 40 309)(22 304 58 322 47 340 41 310)(23 274 87 297 349 286 75 261)(24 275 88 287 350 276 76 262)(25 265 78 288 351 277 77 263)(26 266 79 289 352 278 67 264)(27 267 80 290 342 279 68 254)(28 268 81 291 343 280 69 255)(29 269 82 292 344 281 70 256)(30 270 83 293 345 282 71 257)(31 271 84 294 346 283 72 258)(32 272 85 295 347 284 73 259)(33 273 86 296 348 285 74 260)(89 175 136 189 101 200 112 177)(90 176 137 190 102 201 113 178)(91 166 138 191 103 202 114 179)(92 167 139 192 104 203 115 180)(93 168 140 193 105 204 116 181)(94 169 141 194 106 205 117 182)(95 170 142 195 107 206 118 183)(96 171 143 196 108 207 119 184)(97 172 133 197 109 208 120 185)(98 173 134 198 110 209 121 186)(99 174 135 188 100 199 111 187)
(1 254 138 308)(2 255 139 298)(3 256 140 299)(4 257 141 300)(5 258 142 301)(6 259 143 302)(7 260 133 303)(8 261 134 304)(9 262 135 305)(10 263 136 306)(11 264 137 307)(12 235 76 187)(13 236 77 177)(14 237 67 178)(15 238 68 179)(16 239 69 180)(17 240 70 181)(18 241 71 182)(19 242 72 183)(20 232 73 184)(21 233 74 185)(22 234 75 186)(23 209 58 252)(24 199 59 253)(25 200 60 243)(26 201 61 244)(27 202 62 245)(28 203 63 246)(29 204 64 247)(30 205 65 248)(31 206 66 249)(32 207 56 250)(33 208 57 251)(34 220 342 166)(35 210 343 167)(36 211 344 168)(37 212 345 169)(38 213 346 170)(39 214 347 171)(40 215 348 172)(41 216 349 173)(42 217 350 174)(43 218 351 175)(44 219 352 176)(45 231 85 196)(46 221 86 197)(47 222 87 198)(48 223 88 188)(49 224 78 189)(50 225 79 190)(51 226 80 191)(52 227 81 192)(53 228 82 193)(54 229 83 194)(55 230 84 195)(89 324 148 265)(90 325 149 266)(91 326 150 267)(92 327 151 268)(93 328 152 269)(94 329 153 270)(95 330 154 271)(96 320 144 272)(97 321 145 273)(98 322 146 274)(99 323 147 275)(100 311 130 276)(101 312 131 277)(102 313 132 278)(103 314 122 279)(104 315 123 280)(105 316 124 281)(106 317 125 282)(107 318 126 283)(108 319 127 284)(109 309 128 285)(110 310 129 286)(111 341 165 287)(112 331 155 288)(113 332 156 289)(114 333 157 290)(115 334 158 291)(116 335 159 292)(117 336 160 293)(118 337 161 294)(119 338 162 295)(120 339 163 296)(121 340 164 297)
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), (1,103,157,91)(2,104,158,92)(3,105,159,93)(4,106,160,94)(5,107,161,95)(6,108,162,96)(7,109,163,97)(8,110,164,98)(9,100,165,99)(10,101,155,89)(11,102,156,90)(12,24,48,350)(13,25,49,351)(14,26,50,352)(15,27,51,342)(16,28,52,343)(17,29,53,344)(18,30,54,345)(19,31,55,346)(20,32,45,347)(21,33,46,348)(22,23,47,349)(34,68,62,80)(35,69,63,81)(36,70,64,82)(37,71,65,83)(38,72,66,84)(39,73,56,85)(40,74,57,86)(41,75,58,87)(42,76,59,88)(43,77,60,78)(44,67,61,79)(111,147,135,130)(112,148,136,131)(113,149,137,132)(114,150,138,122)(115,151,139,123)(116,152,140,124)(117,153,141,125)(118,154,142,126)(119,144,143,127)(120,145,133,128)(121,146,134,129)(166,226,202,238)(167,227,203,239)(168,228,204,240)(169,229,205,241)(170,230,206,242)(171,231,207,232)(172,221,208,233)(173,222,209,234)(174,223,199,235)(175,224,200,236)(176,225,201,237)(177,218,189,243)(178,219,190,244)(179,220,191,245)(180,210,192,246)(181,211,193,247)(182,212,194,248)(183,213,195,249)(184,214,196,250)(185,215,197,251)(186,216,198,252)(187,217,188,253)(254,326,290,314)(255,327,291,315)(256,328,292,316)(257,329,293,317)(258,330,294,318)(259,320,295,319)(260,321,296,309)(261,322,297,310)(262,323,287,311)(263,324,288,312)(264,325,289,313)(265,331,277,306)(266,332,278,307)(267,333,279,308)(268,334,280,298)(269,335,281,299)(270,336,282,300)(271,337,283,301)(272,338,284,302)(273,339,285,303)(274,340,286,304)(275,341,276,305), (1,226,122,245,157,238,150,220)(2,227,123,246,158,239,151,210)(3,228,124,247,159,240,152,211)(4,229,125,248,160,241,153,212)(5,230,126,249,161,242,154,213)(6,231,127,250,162,232,144,214)(7,221,128,251,163,233,145,215)(8,222,129,252,164,234,146,216)(9,223,130,253,165,235,147,217)(10,224,131,243,155,236,148,218)(11,225,132,244,156,237,149,219)(12,305,59,323,48,341,42,311)(13,306,60,324,49,331,43,312)(14,307,61,325,50,332,44,313)(15,308,62,326,51,333,34,314)(16,298,63,327,52,334,35,315)(17,299,64,328,53,335,36,316)(18,300,65,329,54,336,37,317)(19,301,66,330,55,337,38,318)(20,302,56,320,45,338,39,319)(21,303,57,321,46,339,40,309)(22,304,58,322,47,340,41,310)(23,274,87,297,349,286,75,261)(24,275,88,287,350,276,76,262)(25,265,78,288,351,277,77,263)(26,266,79,289,352,278,67,264)(27,267,80,290,342,279,68,254)(28,268,81,291,343,280,69,255)(29,269,82,292,344,281,70,256)(30,270,83,293,345,282,71,257)(31,271,84,294,346,283,72,258)(32,272,85,295,347,284,73,259)(33,273,86,296,348,285,74,260)(89,175,136,189,101,200,112,177)(90,176,137,190,102,201,113,178)(91,166,138,191,103,202,114,179)(92,167,139,192,104,203,115,180)(93,168,140,193,105,204,116,181)(94,169,141,194,106,205,117,182)(95,170,142,195,107,206,118,183)(96,171,143,196,108,207,119,184)(97,172,133,197,109,208,120,185)(98,173,134,198,110,209,121,186)(99,174,135,188,100,199,111,187), (1,254,138,308)(2,255,139,298)(3,256,140,299)(4,257,141,300)(5,258,142,301)(6,259,143,302)(7,260,133,303)(8,261,134,304)(9,262,135,305)(10,263,136,306)(11,264,137,307)(12,235,76,187)(13,236,77,177)(14,237,67,178)(15,238,68,179)(16,239,69,180)(17,240,70,181)(18,241,71,182)(19,242,72,183)(20,232,73,184)(21,233,74,185)(22,234,75,186)(23,209,58,252)(24,199,59,253)(25,200,60,243)(26,201,61,244)(27,202,62,245)(28,203,63,246)(29,204,64,247)(30,205,65,248)(31,206,66,249)(32,207,56,250)(33,208,57,251)(34,220,342,166)(35,210,343,167)(36,211,344,168)(37,212,345,169)(38,213,346,170)(39,214,347,171)(40,215,348,172)(41,216,349,173)(42,217,350,174)(43,218,351,175)(44,219,352,176)(45,231,85,196)(46,221,86,197)(47,222,87,198)(48,223,88,188)(49,224,78,189)(50,225,79,190)(51,226,80,191)(52,227,81,192)(53,228,82,193)(54,229,83,194)(55,230,84,195)(89,324,148,265)(90,325,149,266)(91,326,150,267)(92,327,151,268)(93,328,152,269)(94,329,153,270)(95,330,154,271)(96,320,144,272)(97,321,145,273)(98,322,146,274)(99,323,147,275)(100,311,130,276)(101,312,131,277)(102,313,132,278)(103,314,122,279)(104,315,123,280)(105,316,124,281)(106,317,125,282)(107,318,126,283)(108,319,127,284)(109,309,128,285)(110,310,129,286)(111,341,165,287)(112,331,155,288)(113,332,156,289)(114,333,157,290)(115,334,158,291)(116,335,159,292)(117,336,160,293)(118,337,161,294)(119,338,162,295)(120,339,163,296)(121,340,164,297)>;
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), (1,103,157,91)(2,104,158,92)(3,105,159,93)(4,106,160,94)(5,107,161,95)(6,108,162,96)(7,109,163,97)(8,110,164,98)(9,100,165,99)(10,101,155,89)(11,102,156,90)(12,24,48,350)(13,25,49,351)(14,26,50,352)(15,27,51,342)(16,28,52,343)(17,29,53,344)(18,30,54,345)(19,31,55,346)(20,32,45,347)(21,33,46,348)(22,23,47,349)(34,68,62,80)(35,69,63,81)(36,70,64,82)(37,71,65,83)(38,72,66,84)(39,73,56,85)(40,74,57,86)(41,75,58,87)(42,76,59,88)(43,77,60,78)(44,67,61,79)(111,147,135,130)(112,148,136,131)(113,149,137,132)(114,150,138,122)(115,151,139,123)(116,152,140,124)(117,153,141,125)(118,154,142,126)(119,144,143,127)(120,145,133,128)(121,146,134,129)(166,226,202,238)(167,227,203,239)(168,228,204,240)(169,229,205,241)(170,230,206,242)(171,231,207,232)(172,221,208,233)(173,222,209,234)(174,223,199,235)(175,224,200,236)(176,225,201,237)(177,218,189,243)(178,219,190,244)(179,220,191,245)(180,210,192,246)(181,211,193,247)(182,212,194,248)(183,213,195,249)(184,214,196,250)(185,215,197,251)(186,216,198,252)(187,217,188,253)(254,326,290,314)(255,327,291,315)(256,328,292,316)(257,329,293,317)(258,330,294,318)(259,320,295,319)(260,321,296,309)(261,322,297,310)(262,323,287,311)(263,324,288,312)(264,325,289,313)(265,331,277,306)(266,332,278,307)(267,333,279,308)(268,334,280,298)(269,335,281,299)(270,336,282,300)(271,337,283,301)(272,338,284,302)(273,339,285,303)(274,340,286,304)(275,341,276,305), (1,226,122,245,157,238,150,220)(2,227,123,246,158,239,151,210)(3,228,124,247,159,240,152,211)(4,229,125,248,160,241,153,212)(5,230,126,249,161,242,154,213)(6,231,127,250,162,232,144,214)(7,221,128,251,163,233,145,215)(8,222,129,252,164,234,146,216)(9,223,130,253,165,235,147,217)(10,224,131,243,155,236,148,218)(11,225,132,244,156,237,149,219)(12,305,59,323,48,341,42,311)(13,306,60,324,49,331,43,312)(14,307,61,325,50,332,44,313)(15,308,62,326,51,333,34,314)(16,298,63,327,52,334,35,315)(17,299,64,328,53,335,36,316)(18,300,65,329,54,336,37,317)(19,301,66,330,55,337,38,318)(20,302,56,320,45,338,39,319)(21,303,57,321,46,339,40,309)(22,304,58,322,47,340,41,310)(23,274,87,297,349,286,75,261)(24,275,88,287,350,276,76,262)(25,265,78,288,351,277,77,263)(26,266,79,289,352,278,67,264)(27,267,80,290,342,279,68,254)(28,268,81,291,343,280,69,255)(29,269,82,292,344,281,70,256)(30,270,83,293,345,282,71,257)(31,271,84,294,346,283,72,258)(32,272,85,295,347,284,73,259)(33,273,86,296,348,285,74,260)(89,175,136,189,101,200,112,177)(90,176,137,190,102,201,113,178)(91,166,138,191,103,202,114,179)(92,167,139,192,104,203,115,180)(93,168,140,193,105,204,116,181)(94,169,141,194,106,205,117,182)(95,170,142,195,107,206,118,183)(96,171,143,196,108,207,119,184)(97,172,133,197,109,208,120,185)(98,173,134,198,110,209,121,186)(99,174,135,188,100,199,111,187), (1,254,138,308)(2,255,139,298)(3,256,140,299)(4,257,141,300)(5,258,142,301)(6,259,143,302)(7,260,133,303)(8,261,134,304)(9,262,135,305)(10,263,136,306)(11,264,137,307)(12,235,76,187)(13,236,77,177)(14,237,67,178)(15,238,68,179)(16,239,69,180)(17,240,70,181)(18,241,71,182)(19,242,72,183)(20,232,73,184)(21,233,74,185)(22,234,75,186)(23,209,58,252)(24,199,59,253)(25,200,60,243)(26,201,61,244)(27,202,62,245)(28,203,63,246)(29,204,64,247)(30,205,65,248)(31,206,66,249)(32,207,56,250)(33,208,57,251)(34,220,342,166)(35,210,343,167)(36,211,344,168)(37,212,345,169)(38,213,346,170)(39,214,347,171)(40,215,348,172)(41,216,349,173)(42,217,350,174)(43,218,351,175)(44,219,352,176)(45,231,85,196)(46,221,86,197)(47,222,87,198)(48,223,88,188)(49,224,78,189)(50,225,79,190)(51,226,80,191)(52,227,81,192)(53,228,82,193)(54,229,83,194)(55,230,84,195)(89,324,148,265)(90,325,149,266)(91,326,150,267)(92,327,151,268)(93,328,152,269)(94,329,153,270)(95,330,154,271)(96,320,144,272)(97,321,145,273)(98,322,146,274)(99,323,147,275)(100,311,130,276)(101,312,131,277)(102,313,132,278)(103,314,122,279)(104,315,123,280)(105,316,124,281)(106,317,125,282)(107,318,126,283)(108,319,127,284)(109,309,128,285)(110,310,129,286)(111,341,165,287)(112,331,155,288)(113,332,156,289)(114,333,157,290)(115,334,158,291)(116,335,159,292)(117,336,160,293)(118,337,161,294)(119,338,162,295)(120,339,163,296)(121,340,164,297) );
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)], [(1,103,157,91),(2,104,158,92),(3,105,159,93),(4,106,160,94),(5,107,161,95),(6,108,162,96),(7,109,163,97),(8,110,164,98),(9,100,165,99),(10,101,155,89),(11,102,156,90),(12,24,48,350),(13,25,49,351),(14,26,50,352),(15,27,51,342),(16,28,52,343),(17,29,53,344),(18,30,54,345),(19,31,55,346),(20,32,45,347),(21,33,46,348),(22,23,47,349),(34,68,62,80),(35,69,63,81),(36,70,64,82),(37,71,65,83),(38,72,66,84),(39,73,56,85),(40,74,57,86),(41,75,58,87),(42,76,59,88),(43,77,60,78),(44,67,61,79),(111,147,135,130),(112,148,136,131),(113,149,137,132),(114,150,138,122),(115,151,139,123),(116,152,140,124),(117,153,141,125),(118,154,142,126),(119,144,143,127),(120,145,133,128),(121,146,134,129),(166,226,202,238),(167,227,203,239),(168,228,204,240),(169,229,205,241),(170,230,206,242),(171,231,207,232),(172,221,208,233),(173,222,209,234),(174,223,199,235),(175,224,200,236),(176,225,201,237),(177,218,189,243),(178,219,190,244),(179,220,191,245),(180,210,192,246),(181,211,193,247),(182,212,194,248),(183,213,195,249),(184,214,196,250),(185,215,197,251),(186,216,198,252),(187,217,188,253),(254,326,290,314),(255,327,291,315),(256,328,292,316),(257,329,293,317),(258,330,294,318),(259,320,295,319),(260,321,296,309),(261,322,297,310),(262,323,287,311),(263,324,288,312),(264,325,289,313),(265,331,277,306),(266,332,278,307),(267,333,279,308),(268,334,280,298),(269,335,281,299),(270,336,282,300),(271,337,283,301),(272,338,284,302),(273,339,285,303),(274,340,286,304),(275,341,276,305)], [(1,226,122,245,157,238,150,220),(2,227,123,246,158,239,151,210),(3,228,124,247,159,240,152,211),(4,229,125,248,160,241,153,212),(5,230,126,249,161,242,154,213),(6,231,127,250,162,232,144,214),(7,221,128,251,163,233,145,215),(8,222,129,252,164,234,146,216),(9,223,130,253,165,235,147,217),(10,224,131,243,155,236,148,218),(11,225,132,244,156,237,149,219),(12,305,59,323,48,341,42,311),(13,306,60,324,49,331,43,312),(14,307,61,325,50,332,44,313),(15,308,62,326,51,333,34,314),(16,298,63,327,52,334,35,315),(17,299,64,328,53,335,36,316),(18,300,65,329,54,336,37,317),(19,301,66,330,55,337,38,318),(20,302,56,320,45,338,39,319),(21,303,57,321,46,339,40,309),(22,304,58,322,47,340,41,310),(23,274,87,297,349,286,75,261),(24,275,88,287,350,276,76,262),(25,265,78,288,351,277,77,263),(26,266,79,289,352,278,67,264),(27,267,80,290,342,279,68,254),(28,268,81,291,343,280,69,255),(29,269,82,292,344,281,70,256),(30,270,83,293,345,282,71,257),(31,271,84,294,346,283,72,258),(32,272,85,295,347,284,73,259),(33,273,86,296,348,285,74,260),(89,175,136,189,101,200,112,177),(90,176,137,190,102,201,113,178),(91,166,138,191,103,202,114,179),(92,167,139,192,104,203,115,180),(93,168,140,193,105,204,116,181),(94,169,141,194,106,205,117,182),(95,170,142,195,107,206,118,183),(96,171,143,196,108,207,119,184),(97,172,133,197,109,208,120,185),(98,173,134,198,110,209,121,186),(99,174,135,188,100,199,111,187)], [(1,254,138,308),(2,255,139,298),(3,256,140,299),(4,257,141,300),(5,258,142,301),(6,259,143,302),(7,260,133,303),(8,261,134,304),(9,262,135,305),(10,263,136,306),(11,264,137,307),(12,235,76,187),(13,236,77,177),(14,237,67,178),(15,238,68,179),(16,239,69,180),(17,240,70,181),(18,241,71,182),(19,242,72,183),(20,232,73,184),(21,233,74,185),(22,234,75,186),(23,209,58,252),(24,199,59,253),(25,200,60,243),(26,201,61,244),(27,202,62,245),(28,203,63,246),(29,204,64,247),(30,205,65,248),(31,206,66,249),(32,207,56,250),(33,208,57,251),(34,220,342,166),(35,210,343,167),(36,211,344,168),(37,212,345,169),(38,213,346,170),(39,214,347,171),(40,215,348,172),(41,216,349,173),(42,217,350,174),(43,218,351,175),(44,219,352,176),(45,231,85,196),(46,221,86,197),(47,222,87,198),(48,223,88,188),(49,224,78,189),(50,225,79,190),(51,226,80,191),(52,227,81,192),(53,228,82,193),(54,229,83,194),(55,230,84,195),(89,324,148,265),(90,325,149,266),(91,326,150,267),(92,327,151,268),(93,328,152,269),(94,329,153,270),(95,330,154,271),(96,320,144,272),(97,321,145,273),(98,322,146,274),(99,323,147,275),(100,311,130,276),(101,312,131,277),(102,313,132,278),(103,314,122,279),(104,315,123,280),(105,316,124,281),(106,317,125,282),(107,318,126,283),(108,319,127,284),(109,309,128,285),(110,310,129,286),(111,341,165,287),(112,331,155,288),(113,332,156,289),(114,333,157,290),(115,334,158,291),(116,335,159,292),(117,336,160,293),(118,337,161,294),(119,338,162,295),(120,339,163,296),(121,340,164,297)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C11 | C22 | C22 | C44 | Q8 | D4 | SD16 | Q8×C11 | D4×C11 | C11×SD16 |
| kernel | C11×C4.Q8 | C11×C4⋊C4 | C2×C88 | C88 | C4.Q8 | C4⋊C4 | C2×C8 | C8 | C44 | C2×C22 | C22 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 10 | 20 | 10 | 40 | 1 | 1 | 4 | 10 | 10 | 40 |
Matrix representation of C11×C4.Q8 ►in GL4(𝔽89) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 78 | 0 |
| 0 | 0 | 0 | 78 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 88 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 88 | 0 | 0 | 0 |
| 0 | 0 | 69 | 20 |
| 0 | 0 | 69 | 69 |
| 3 | 41 | 0 | 0 |
| 41 | 86 | 0 | 0 |
| 0 | 0 | 13 | 39 |
| 0 | 0 | 39 | 76 |
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,78,0,0,0,0,78],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,88,0],[0,88,0,0,1,0,0,0,0,0,69,69,0,0,20,69],[3,41,0,0,41,86,0,0,0,0,13,39,0,0,39,76] >;
C11×C4.Q8 in GAP, Magma, Sage, TeX
C_{11}\times C_4.Q_8 % in TeX
G:=Group("C11xC4.Q8"); // GroupNames label
G:=SmallGroup(352,55);
// by ID
G=gap.SmallGroup(352,55);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,271,5283,117]);
// Polycyclic
G:=Group<a,b,c,d|a^11=b^4=1,c^4=b^2,d^2=b^-1*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
Export