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G = C23×Dic11  order 352 = 25·11

Direct product of C23 and Dic11

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C23×Dic11, C24.2D11, C22.14C24, C23.36D22, C11⋊2(C23×C4), (C22×C22)⋊5C4, C22⋊2(C22×C4), (C23×C22).3C2, C2.2(C23×D11), (C2×C22).69C23, (C22×C22).47C22, C22.33(C22×D11), (C2×C22)⋊9(C2×C4), SmallGroup(352,186)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C11 — C23×Dic11
Chief series
C1 — C11 — C22 — Dic11 — C2×Dic11 — C22×Dic11 — C23×Dic11
Lower central
C11 — C23×Dic11
Upper central
C1 — C24

Generators and relations for C23×Dic11
 G = < a,b,c,d,e | a2=b2=c2=d22=1, e2=d11, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 746 in 236 conjugacy classes, 185 normal (7 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C11, C22×C4, C24, C22, C22, C23×C4, Dic11, C2×C22, C2×Dic11, C22×C22, C22×Dic11, C23×C22, C23×Dic11
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, D11, C23×C4, Dic11, D22, C2×Dic11, C22×D11, C22×Dic11, C23×D11, C23×Dic11

Smallest permutation representation of C23×Dic11
►Regular action on 352 points
Generators in S352
(1 157)(2 158)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 155)(22 156)(23 140)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 133)(39 134)(40 135)(41 136)(42 137)(43 138)(44 139)(45 128)(46 129)(47 130)(48 131)(49 132)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 110)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)(85 106)(86 107)(87 108)(88 109)(177 342)(178 343)(179 344)(180 345)(181 346)(182 347)(183 348)(184 349)(185 350)(186 351)(187 352)(188 331)(189 332)(190 333)(191 334)(192 335)(193 336)(194 337)(195 338)(196 339)(197 340)(198 341)(199 309)(200 310)(201 311)(202 312)(203 313)(204 314)(205 315)(206 316)(207 317)(208 318)(209 319)(210 320)(211 321)(212 322)(213 323)(214 324)(215 325)(216 326)(217 327)(218 328)(219 329)(220 330)(221 298)(222 299)(223 300)(224 301)(225 302)(226 303)(227 304)(228 305)(229 306)(230 307)(231 308)(232 287)(233 288)(234 289)(235 290)(236 291)(237 292)(238 293)(239 294)(240 295)(241 296)(242 297)(243 265)(244 266)(245 267)(246 268)(247 269)(248 270)(249 271)(250 272)(251 273)(252 274)(253 275)(254 276)(255 277)(256 278)(257 279)(258 280)(259 281)(260 282)(261 283)(262 284)(263 285)(264 286)

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

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

(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)

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

112 conjugacy classes

class 1 2A···2O4A···4P11A···11E22A···22BW
order12···24···411···1122···22
size11···111···112···22···2

112 irreducible representations

dim1111222
type++++-+
imageC1C2C2C4D11Dic11D22
kernelC23×Dic11C22×Dic11C23×C22C22×C22C24C23C23
# reps11411654035

Matrix representation of C23×Dic11 ►in GL5(𝔽89)

880000
088000
00100
000880
000088
,
10000
088000
008800
000880
000088
,
880000
01000
00100
000880
000088
,
10000
01000
00100
000088
000142
,
10000
01000
00100
0004343
0001746

G:=sub<GL(5,GF(89))| [88,0,0,0,0,0,88,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,88],[1,0,0,0,0,0,88,0,0,0,0,0,88,0,0,0,0,0,88,0,0,0,0,0,88],[88,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,88],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,88,42],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,43,17,0,0,0,43,46] >;
 

C23×Dic11 in GAP, Magma, Sage, TeX 

C_2^3\times {\rm Dic}_{11}
 
% in TeX
 
G:=Group("C2^3xDic11");
 
// GroupNames label
 
G:=SmallGroup(352,186);
 
// by ID
 
G=gap.SmallGroup(352,186);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,96,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^22=1,e^2=d^11,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
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