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## G = C11×C2.C42order 352 = 25·11

### Direct product of C11 and C2.C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C11×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C22 — C22×C44 — C11×C2.C42
 Lower central C1 — C2 — C11×C2.C42
 Upper central C1 — C22×C22 — C11×C2.C42

Generators and relations for C11×C2.C42
G = < a,b,c,d | a11=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C11, C22×C4, C22, C22, C2.C42, C44, C2×C22, C2×C22, C2×C44, C2×C44, C22×C22, C22×C44, C11×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C11, C42, C22⋊C4, C4⋊C4, C22, C2.C42, C44, C2×C22, C2×C44, D4×C11, Q8×C11, C4×C44, C11×C22⋊C4, C11×C4⋊C4, C11×C2.C42

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

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

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

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

220 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 11A ··· 11J 22A ··· 22BR 44A ··· 44DP order 1 2 ··· 2 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

220 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C11 C22 C44 D4 Q8 D4×C11 Q8×C11 kernel C11×C2.C42 C22×C44 C2×C44 C2.C42 C22×C4 C2×C4 C2×C22 C2×C22 C22 C22 # reps 1 3 12 10 30 120 3 1 30 10

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

 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 2
,
 1 0 0 0 0 1 0 0 0 0 88 0 0 0 0 88
,
 55 0 0 0 0 88 0 0 0 0 43 44 0 0 51 46
,
 1 0 0 0 0 34 0 0 0 0 55 87 0 0 0 34
G:=sub<GL(4,GF(89))| [1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,1,0,0,0,0,88,0,0,0,0,88],[55,0,0,0,0,88,0,0,0,0,43,51,0,0,44,46],[1,0,0,0,0,34,0,0,0,0,55,0,0,0,87,34] >;

C11×C2.C42 in GAP, Magma, Sage, TeX

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

G:=Group("C11xC2.C4^2");
// GroupNames label

G:=SmallGroup(352,44);
// by ID

G=gap.SmallGroup(352,44);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,528,553,1063]);
// Polycyclic

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

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×
𝔽