C2^2xC11:C8 - GroupNames

G = C₂²×C₁₁⋊C₈ order 352 = 2⁵·11

Direct product of C₂² and C₁₁⋊C₈

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

Aliases: C₂²×C₁₁⋊C₈, C₄₄.₄₀C₂³, C₂³.₄Dic₁₁, C₂₂⋊₂(C₂×C₈), (C₂×C₂₂)⋊₃C₈, C₁₁⋊₂(C₂²×C₈), C₄₄.₄₂(C₂×C₄), (C₂×C₄₄).₁₂C₄, (C₂×C₄).₉₉D₂₂, (C₂²×C₂₂).₆C₄, (C₂×C₄).₉Dic₁₁, (C₂²×C₄).₉D₁₁, C₂₂.₂₀(C₂²×C₄), (C₂²×C₄₄).₁₂C₂, C₄.₁₄(C₂×Dic₁₁), C₄.₄₀(C₂²×D₁₁), (C₂×C₄₄).₁₁₂C₂², C₂.₁(C₂²×Dic₁₁), C₂².₁₁(C₂×Dic₁₁), (C₂×C₂₂).₃₁(C₂×C₄), SmallGroup(352,115)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₁ — C₂²×C₁₁⋊C₈

Chief series

C₁ — C₁₁ — C₂₂ — C₄₄ — C₁₁⋊C₈ — C₂×C₁₁⋊C₈ — C₂²×C₁₁⋊C₈

Lower central

C₁₁ — C₂²×C₁₁⋊C₈

Upper central

C₁ — C₂²×C₄

Generators and relations for C₂²×C₁₁⋊C₈
G = < a,b,c,d | a²=b²=c¹¹=d⁸=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c^-1 >

Subgroups: 186 in 76 conjugacy classes, 65 normal (11 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₈, C₂×C₄, C₂³, C₁₁, C₂×C₈, C₂²×C₄, C₂₂, C₂₂, C₂²×C₈, C₄₄, C₄₄, C₂×C₂₂, C₁₁⋊C₈, C₂×C₄₄, C₂²×C₂₂, C₂×C₁₁⋊C₈, C₂²×C₄₄, C₂²×C₁₁⋊C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, C₂³, C₂×C₈, C₂²×C₄, D₁₁, C₂²×C₈, Dic₁₁, D₂₂, C₁₁⋊C₈, C₂×Dic₁₁, C₂²×D₁₁, C₂×C₁₁⋊C₈, C₂²×Dic₁₁, C₂²×C₁₁⋊C₈

Smallest permutation representation of C₂²×C₁₁⋊C₈
►Regular action on 352 points

Generators in S₃₅₂

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

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

(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)

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

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

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

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

112 conjugacy classes

class	1	2A	···	2G	4A	···	4H	8A	···	8P	11A	···	11E	22A	···	22AI	44A	···	44AN
order	1	2	···	2	4	···	4	8	···	8	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	C₄	C₄	C₈	D₁₁	Dic₁₁	D₂₂	Dic₁₁	C₁₁⋊C₈
kernel	C₂²×C₁₁⋊C₈	C₂×C₁₁⋊C₈	C₂²×C₄₄	C₂×C₄₄	C₂²×C₂₂	C₂×C₂₂	C₂²×C₄	C₂×C₄	C₂×C₄	C₂³	C₂²
# reps	1	6	1	6	2	16	5	15	15	5	40

Matrix representation of C₂²×C₁₁⋊C₈ ►in GL₅(𝔽₈₉)

1	0	0	0	0
0	88	0	0	0
0	0	88	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	88	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	56	88
0	0	0	57	88

52	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	53	34
0	0	0	64	36

G:=sub<GL(5,GF(89))| [1,0,0,0,0,0,88,0,0,0,0,0,88,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,88,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,56,57,0,0,0,88,88],[52,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,64,0,0,0,34,36] >;

C₂²×C₁₁⋊C₈ in GAP, Magma, Sage, TeX

C_2^2\times C_{11}\rtimes C_8

% in TeX

G:=Group("C2^2xC11:C8");

// GroupNames label

G:=SmallGroup(352,115);

// by ID

G=gap.SmallGroup(352,115);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,48,69,11525]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^11=d^8=1,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

G = C22×C11⋊C8 order 352 = 25·11

Direct product of C22 and C11⋊C8

G = C₂²×C₁₁⋊C₈ order 352 = 2⁵·11

Direct product of C₂² and C₁₁⋊C₈