Copied to
clipboard

## G = C2×C4×Dic11order 352 = 25·11

### Direct product of C2×C4 and Dic11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C11 — C2×C4×Dic11
 Chief series C1 — C11 — C22 — C2×C22 — C2×Dic11 — C22×Dic11 — C2×C4×Dic11
 Lower central C11 — C2×C4×Dic11
 Upper central C1 — C22×C4

Generators and relations for C2×C4×Dic11
G = < a,b,c,d | a2=b4=c22=1, d2=c11, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 378 in 108 conjugacy classes, 81 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C11, C42, C22×C4, C22×C4, C22, C22, C2×C42, Dic11, C44, C2×C22, C2×C22, C2×Dic11, C2×C44, C22×C22, C4×Dic11, C22×Dic11, C22×C44, C2×C4×Dic11
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, D11, C2×C42, Dic11, D22, C4×D11, C2×Dic11, C22×D11, C4×Dic11, C2×C4×D11, C22×Dic11, C2×C4×Dic11

Smallest permutation representation of C2×C4×Dic11
Regular action on 352 points
Generators in S352
(1 346)(2 347)(3 348)(4 349)(5 350)(6 351)(7 352)(8 331)(9 332)(10 333)(11 334)(12 335)(13 336)(14 337)(15 338)(16 339)(17 340)(18 341)(19 342)(20 343)(21 344)(22 345)(23 68)(24 69)(25 70)(26 71)(27 72)(28 73)(29 74)(30 75)(31 76)(32 77)(33 78)(34 79)(35 80)(36 81)(37 82)(38 83)(39 84)(40 85)(41 86)(42 87)(43 88)(44 67)(45 136)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 151)(61 152)(62 153)(63 154)(64 133)(65 134)(66 135)(89 308)(90 287)(91 288)(92 289)(93 290)(94 291)(95 292)(96 293)(97 294)(98 295)(99 296)(100 297)(101 298)(102 299)(103 300)(104 301)(105 302)(106 303)(107 304)(108 305)(109 306)(110 307)(111 253)(112 254)(113 255)(114 256)(115 257)(116 258)(117 259)(118 260)(119 261)(120 262)(121 263)(122 264)(123 243)(124 244)(125 245)(126 246)(127 247)(128 248)(129 249)(130 250)(131 251)(132 252)(155 199)(156 200)(157 201)(158 202)(159 203)(160 204)(161 205)(162 206)(163 207)(164 208)(165 209)(166 210)(167 211)(168 212)(169 213)(170 214)(171 215)(172 216)(173 217)(174 218)(175 219)(176 220)(177 329)(178 330)(179 309)(180 310)(181 311)(182 312)(183 313)(184 314)(185 315)(186 316)(187 317)(188 318)(189 319)(190 320)(191 321)(192 322)(193 323)(194 324)(195 325)(196 326)(197 327)(198 328)(221 276)(222 277)(223 278)(224 279)(225 280)(226 281)(227 282)(228 283)(229 284)(230 285)(231 286)(232 265)(233 266)(234 267)(235 268)(236 269)(237 270)(238 271)(239 272)(240 273)(241 274)(242 275)
(1 150 107 173)(2 151 108 174)(3 152 109 175)(4 153 110 176)(5 154 89 155)(6 133 90 156)(7 134 91 157)(8 135 92 158)(9 136 93 159)(10 137 94 160)(11 138 95 161)(12 139 96 162)(13 140 97 163)(14 141 98 164)(15 142 99 165)(16 143 100 166)(17 144 101 167)(18 145 102 168)(19 146 103 169)(20 147 104 170)(21 148 105 171)(22 149 106 172)(23 191 223 245)(24 192 224 246)(25 193 225 247)(26 194 226 248)(27 195 227 249)(28 196 228 250)(29 197 229 251)(30 198 230 252)(31 177 231 253)(32 178 232 254)(33 179 233 255)(34 180 234 256)(35 181 235 257)(36 182 236 258)(37 183 237 259)(38 184 238 260)(39 185 239 261)(40 186 240 262)(41 187 241 263)(42 188 242 264)(43 189 221 243)(44 190 222 244)(45 290 203 332)(46 291 204 333)(47 292 205 334)(48 293 206 335)(49 294 207 336)(50 295 208 337)(51 296 209 338)(52 297 210 339)(53 298 211 340)(54 299 212 341)(55 300 213 342)(56 301 214 343)(57 302 215 344)(58 303 216 345)(59 304 217 346)(60 305 218 347)(61 306 219 348)(62 307 220 349)(63 308 199 350)(64 287 200 351)(65 288 201 352)(66 289 202 331)(67 320 277 124)(68 321 278 125)(69 322 279 126)(70 323 280 127)(71 324 281 128)(72 325 282 129)(73 326 283 130)(74 327 284 131)(75 328 285 132)(76 329 286 111)(77 330 265 112)(78 309 266 113)(79 310 267 114)(80 311 268 115)(81 312 269 116)(82 313 270 117)(83 314 271 118)(84 315 272 119)(85 316 273 120)(86 317 274 121)(87 318 275 122)(88 319 276 123)
(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)
(1 239 12 228)(2 238 13 227)(3 237 14 226)(4 236 15 225)(5 235 16 224)(6 234 17 223)(7 233 18 222)(8 232 19 221)(9 231 20 242)(10 230 21 241)(11 229 22 240)(23 90 34 101)(24 89 35 100)(25 110 36 99)(26 109 37 98)(27 108 38 97)(28 107 39 96)(29 106 40 95)(30 105 41 94)(31 104 42 93)(32 103 43 92)(33 102 44 91)(45 111 56 122)(46 132 57 121)(47 131 58 120)(48 130 59 119)(49 129 60 118)(50 128 61 117)(51 127 62 116)(52 126 63 115)(53 125 64 114)(54 124 65 113)(55 123 66 112)(67 288 78 299)(68 287 79 298)(69 308 80 297)(70 307 81 296)(71 306 82 295)(72 305 83 294)(73 304 84 293)(74 303 85 292)(75 302 86 291)(76 301 87 290)(77 300 88 289)(133 256 144 245)(134 255 145 244)(135 254 146 243)(136 253 147 264)(137 252 148 263)(138 251 149 262)(139 250 150 261)(140 249 151 260)(141 248 152 259)(142 247 153 258)(143 246 154 257)(155 181 166 192)(156 180 167 191)(157 179 168 190)(158 178 169 189)(159 177 170 188)(160 198 171 187)(161 197 172 186)(162 196 173 185)(163 195 174 184)(164 194 175 183)(165 193 176 182)(199 311 210 322)(200 310 211 321)(201 309 212 320)(202 330 213 319)(203 329 214 318)(204 328 215 317)(205 327 216 316)(206 326 217 315)(207 325 218 314)(208 324 219 313)(209 323 220 312)(265 342 276 331)(266 341 277 352)(267 340 278 351)(268 339 279 350)(269 338 280 349)(270 337 281 348)(271 336 282 347)(272 335 283 346)(273 334 284 345)(274 333 285 344)(275 332 286 343)

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

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

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

112 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4X 11A ··· 11E 22A ··· 22AI 44A ··· 44AN order 1 2 ··· 2 4 ··· 4 4 ··· 4 11 ··· 11 22 ··· 22 44 ··· 44 size 1 1 ··· 1 1 ··· 1 11 ··· 11 2 ··· 2 2 ··· 2 2 ··· 2

112 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + - + + image C1 C2 C2 C2 C4 C4 D11 Dic11 D22 D22 C4×D11 kernel C2×C4×Dic11 C4×Dic11 C22×Dic11 C22×C44 C2×Dic11 C2×C44 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 5 20 10 5 40

Matrix representation of C2×C4×Dic11 in GL4(𝔽89) generated by

 88 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 34 0 0 0 0 88 0 0 0 0 1 0 0 0 0 1
,
 88 0 0 0 0 1 0 0 0 0 1 88 0 0 88 2
,
 34 0 0 0 0 88 0 0 0 0 71 45 0 0 62 18
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[34,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[88,0,0,0,0,1,0,0,0,0,1,88,0,0,88,2],[34,0,0,0,0,88,0,0,0,0,71,62,0,0,45,18] >;

C2×C4×Dic11 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_{11}
% in TeX

G:=Group("C2xC4xDic11");
// GroupNames label

G:=SmallGroup(352,117);
// by ID

G=gap.SmallGroup(352,117);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,86,11525]);
// Polycyclic

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

׿
×
𝔽