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## G = C11×C42.C2order 352 = 25·11

### Direct product of C11 and C42.C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C11×C42.C2
 Chief series C1 — C2 — C22 — C2×C22 — C2×C44 — C11×C4⋊C4 — C11×C42.C2
 Lower central C1 — C22 — C11×C42.C2
 Upper central C1 — C2×C22 — C11×C42.C2

Generators and relations for C11×C42.C2
G = < a,b,c,d | a11=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, C2×C4, C11, C42, C4⋊C4, C22, C22, C42.C2, C44, C44, C2×C22, C2×C44, C2×C44, C4×C44, C11×C4⋊C4, C11×C42.C2
Quotients: C1, C2, C22, Q8, C23, C11, C2×Q8, C4○D4, C22, C42.C2, C2×C22, Q8×C11, C22×C22, Q8×C22, C11×C4○D4, C11×C42.C2

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

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 11A ··· 11J 22A ··· 22AD 44A ··· 44BH 44BI ··· 44CV order 1 2 2 2 4 ··· 4 4 4 4 4 11 ··· 11 22 ··· 22 44 ··· 44 44 ··· 44 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C11 C22 C22 Q8 C4○D4 Q8×C11 C11×C4○D4 kernel C11×C42.C2 C4×C44 C11×C4⋊C4 C42.C2 C42 C4⋊C4 C44 C22 C4 C2 # reps 1 1 6 10 10 60 2 4 20 40

Matrix representation of C11×C42.C2 in GL4(𝔽89) generated by

 1 0 0 0 0 1 0 0 0 0 39 0 0 0 0 39
,
 34 0 0 0 0 34 0 0 0 0 0 88 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 88 0
,
 25 1 0 0 88 64 0 0 0 0 23 2 0 0 2 66
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,39,0,0,0,0,39],[34,0,0,0,0,34,0,0,0,0,0,1,0,0,88,0],[0,1,0,0,1,0,0,0,0,0,0,88,0,0,1,0],[25,88,0,0,1,64,0,0,0,0,23,2,0,0,2,66] >;

C11×C42.C2 in GAP, Magma, Sage, TeX

C_{11}\times C_4^2.C_2
% in TeX

G:=Group("C11xC4^2.C2");
// GroupNames label

G:=SmallGroup(352,160);
// by ID

G=gap.SmallGroup(352,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-11,-2,-2,528,1081,1063,3242,410]);
// Polycyclic

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

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