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G = C24×C22  order 352 = 25·11

Abelian group of type [2,2,2,2,22]

direct product, abelian, monomial, 2-elementary

Aliases: C24×C22, SmallGroup(352,195)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C24×C22
 G = < a,b,c,d,e | a2=b2=c2=d2=e22=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 748, all normal (4 characteristic)
C1, C2, C22, C23, C11, C24, C22, C25, C2×C22, C22×C22, C23×C22, C24×C22
Quotients: C1, C2, C22, C23, C11, C24, C22, C25, C2×C22, C22×C22, C23×C22, C24×C22

Smallest permutation representation of C24×C22
►Regular action on 352 points
Generators in S352
(1 192)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 177)(9 178)(10 179)(11 180)(12 181)(13 182)(14 183)(15 184)(16 185)(17 186)(18 187)(19 188)(20 189)(21 190)(22 191)(23 211)(24 212)(25 213)(26 214)(27 215)(28 216)(29 217)(30 218)(31 219)(32 220)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 229)(46 230)(47 231)(48 232)(49 233)(50 234)(51 235)(52 236)(53 237)(54 238)(55 239)(56 240)(57 241)(58 242)(59 221)(60 222)(61 223)(62 224)(63 225)(64 226)(65 227)(66 228)(67 262)(68 263)(69 264)(70 243)(71 244)(72 245)(73 246)(74 247)(75 248)(76 249)(77 250)(78 251)(79 252)(80 253)(81 254)(82 255)(83 256)(84 257)(85 258)(86 259)(87 260)(88 261)(89 268)(90 269)(91 270)(92 271)(93 272)(94 273)(95 274)(96 275)(97 276)(98 277)(99 278)(100 279)(101 280)(102 281)(103 282)(104 283)(105 284)(106 285)(107 286)(108 265)(109 266)(110 267)(111 287)(112 288)(113 289)(114 290)(115 291)(116 292)(117 293)(118 294)(119 295)(120 296)(121 297)(122 298)(123 299)(124 300)(125 301)(126 302)(127 303)(128 304)(129 305)(130 306)(131 307)(132 308)(133 310)(134 311)(135 312)(136 313)(137 314)(138 315)(139 316)(140 317)(141 318)(142 319)(143 320)(144 321)(145 322)(146 323)(147 324)(148 325)(149 326)(150 327)(151 328)(152 329)(153 330)(154 309)(155 346)(156 347)(157 348)(158 349)(159 350)(160 351)(161 352)(162 331)(163 332)(164 333)(165 334)(166 335)(167 336)(168 337)(169 338)(170 339)(171 340)(172 341)(173 342)(174 343)(175 344)(176 345)

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

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

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

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

352 conjugacy classes

class 1 2A···2AE11A···11J22A···22KX
order12···211···1122···22
size11···11···11···1

352 irreducible representations

dim1111
type++
imageC1C2C11C22
kernelC24×C22C23×C22C25C24
# reps13110310

Matrix representation of C24×C22 ►in GL5(𝔽23)

220000
022000
00100
00010
000022
,
10000
01000
002200
000220
000022
,
10000
022000
002200
00010
000022
,
220000
01000
002200
000220
00001
,
190000
019000
00700
000110
00007

G:=sub<GL(5,GF(23))| [22,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,22],[1,0,0,0,0,0,1,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,22],[1,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,1,0,0,0,0,0,22],[22,0,0,0,0,0,1,0,0,0,0,0,22,0,0,0,0,0,22,0,0,0,0,0,1],[19,0,0,0,0,0,19,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,7] >;
 

C24×C22 in GAP, Magma, Sage, TeX 

C_2^4\times C_{22}
 
% in TeX
 
G:=Group("C2^4xC22");
 
// GroupNames label
 
G:=SmallGroup(352,195);
 
// by ID
 
G=gap.SmallGroup(352,195);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-11]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^22=1,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,d*e=e*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁