direct product, abelian, monomial, 2-elementary
Aliases: C23×C42, SmallGroup(336,228)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C23×C42 |
| C1 — C23×C42 |
| C1 — C23×C42 |
Generators and relations for C23×C42
G = < a,b,c,d | a2=b2=c2=d42=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
Subgroups: 268, all normal (8 characteristic)
C1, C2, C3, C22, C6, C7, C23, C2×C6, C14, C24, C21, C22×C6, C2×C14, C42, C23×C6, C22×C14, C2×C42, C23×C14, C22×C42, C23×C42
Quotients: C1, C2, C3, C22, C6, C7, C23, C2×C6, C14, C24, C21, C22×C6, C2×C14, C42, C23×C6, C22×C14, C2×C42, C23×C14, C22×C42, C23×C42
►Regular action on 336 points
Generators in S336
(1 209)(2 210)(3 169)(4 170)(5 171)(6 172)(7 173)(8 174)(9 175)(10 176)(11 177)(12 178)(13 179)(14 180)(15 181)(16 182)(17 183)(18 184)(19 185)(20 186)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 236)(44 237)(45 238)(46 239)(47 240)(48 241)(49 242)(50 243)(51 244)(52 245)(53 246)(54 247)(55 248)(56 249)(57 250)(58 251)(59 252)(60 211)(61 212)(62 213)(63 214)(64 215)(65 216)(66 217)(67 218)(68 219)(69 220)(70 221)(71 222)(72 223)(73 224)(74 225)(75 226)(76 227)(77 228)(78 229)(79 230)(80 231)(81 232)(82 233)(83 234)(84 235)(85 261)(86 262)(87 263)(88 264)(89 265)(90 266)(91 267)(92 268)(93 269)(94 270)(95 271)(96 272)(97 273)(98 274)(99 275)(100 276)(101 277)(102 278)(103 279)(104 280)(105 281)(106 282)(107 283)(108 284)(109 285)(110 286)(111 287)(112 288)(113 289)(114 290)(115 291)(116 292)(117 293)(118 294)(119 253)(120 254)(121 255)(122 256)(123 257)(124 258)(125 259)(126 260)(127 298)(128 299)(129 300)(130 301)(131 302)(132 303)(133 304)(134 305)(135 306)(136 307)(137 308)(138 309)(139 310)(140 311)(141 312)(142 313)(143 314)(144 315)(145 316)(146 317)(147 318)(148 319)(149 320)(150 321)(151 322)(152 323)(153 324)(154 325)(155 326)(156 327)(157 328)(158 329)(159 330)(160 331)(161 332)(162 333)(163 334)(164 335)(165 336)(166 295)(167 296)(168 297)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 113)(25 114)(26 115)(27 116)(28 117)(29 118)(30 119)(31 120)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 85)(39 86)(40 87)(41 88)(42 89)(43 140)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(169 268)(170 269)(171 270)(172 271)(173 272)(174 273)(175 274)(176 275)(177 276)(178 277)(179 278)(180 279)(181 280)(182 281)(183 282)(184 283)(185 284)(186 285)(187 286)(188 287)(189 288)(190 289)(191 290)(192 291)(193 292)(194 293)(195 294)(196 253)(197 254)(198 255)(199 256)(200 257)(201 258)(202 259)(203 260)(204 261)(205 262)(206 263)(207 264)(208 265)(209 266)(210 267)(211 328)(212 329)(213 330)(214 331)(215 332)(216 333)(217 334)(218 335)(219 336)(220 295)(221 296)(222 297)(223 298)(224 299)(225 300)(226 301)(227 302)(228 303)(229 304)(230 305)(231 306)(232 307)(233 308)(234 309)(235 310)(236 311)(237 312)(238 313)(239 314)(240 315)(241 316)(242 317)(243 318)(244 319)(245 320)(246 321)(247 322)(248 323)(249 324)(250 325)(251 326)(252 327)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 155)(94 156)(95 157)(96 158)(97 159)(98 160)(99 161)(100 162)(101 163)(102 164)(103 165)(104 166)(105 167)(106 168)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(169 250)(170 251)(171 252)(172 211)(173 212)(174 213)(175 214)(176 215)(177 216)(178 217)(179 218)(180 219)(181 220)(182 221)(183 222)(184 223)(185 224)(186 225)(187 226)(188 227)(189 228)(190 229)(191 230)(192 231)(193 232)(194 233)(195 234)(196 235)(197 236)(198 237)(199 238)(200 239)(201 240)(202 241)(203 242)(204 243)(205 244)(206 245)(207 246)(208 247)(209 248)(210 249)(253 310)(254 311)(255 312)(256 313)(257 314)(258 315)(259 316)(260 317)(261 318)(262 319)(263 320)(264 321)(265 322)(266 323)(267 324)(268 325)(269 326)(270 327)(271 328)(272 329)(273 330)(274 331)(275 332)(276 333)(277 334)(278 335)(279 336)(280 295)(281 296)(282 297)(283 298)(284 299)(285 300)(286 301)(287 302)(288 303)(289 304)(290 305)(291 306)(292 307)(293 308)(294 309)
(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)
G:=sub<Sym(336)| (1,209)(2,210)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,236)(44,237)(45,238)(46,239)(47,240)(48,241)(49,242)(50,243)(51,244)(52,245)(53,246)(54,247)(55,248)(56,249)(57,250)(58,251)(59,252)(60,211)(61,212)(62,213)(63,214)(64,215)(65,216)(66,217)(67,218)(68,219)(69,220)(70,221)(71,222)(72,223)(73,224)(74,225)(75,226)(76,227)(77,228)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,261)(86,262)(87,263)(88,264)(89,265)(90,266)(91,267)(92,268)(93,269)(94,270)(95,271)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,289)(114,290)(115,291)(116,292)(117,293)(118,294)(119,253)(120,254)(121,255)(122,256)(123,257)(124,258)(125,259)(126,260)(127,298)(128,299)(129,300)(130,301)(131,302)(132,303)(133,304)(134,305)(135,306)(136,307)(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)(144,315)(145,316)(146,317)(147,318)(148,319)(149,320)(150,321)(151,322)(152,323)(153,324)(154,325)(155,326)(156,327)(157,328)(158,329)(159,330)(160,331)(161,332)(162,333)(163,334)(164,335)(165,336)(166,295)(167,296)(168,297), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,85)(39,86)(40,87)(41,88)(42,89)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(169,268)(170,269)(171,270)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,280)(182,281)(183,282)(184,283)(185,284)(186,285)(187,286)(188,287)(189,288)(190,289)(191,290)(192,291)(193,292)(194,293)(195,294)(196,253)(197,254)(198,255)(199,256)(200,257)(201,258)(202,259)(203,260)(204,261)(205,262)(206,263)(207,264)(208,265)(209,266)(210,267)(211,328)(212,329)(213,330)(214,331)(215,332)(216,333)(217,334)(218,335)(219,336)(220,295)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,321)(247,322)(248,323)(249,324)(250,325)(251,326)(252,327), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(169,250)(170,251)(171,252)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(196,235)(197,236)(198,237)(199,238)(200,239)(201,240)(202,241)(203,242)(204,243)(205,244)(206,245)(207,246)(208,247)(209,248)(210,249)(253,310)(254,311)(255,312)(256,313)(257,314)(258,315)(259,316)(260,317)(261,318)(262,319)(263,320)(264,321)(265,322)(266,323)(267,324)(268,325)(269,326)(270,327)(271,328)(272,329)(273,330)(274,331)(275,332)(276,333)(277,334)(278,335)(279,336)(280,295)(281,296)(282,297)(283,298)(284,299)(285,300)(286,301)(287,302)(288,303)(289,304)(290,305)(291,306)(292,307)(293,308)(294,309), (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)>;
G:=Group( (1,209)(2,210)(3,169)(4,170)(5,171)(6,172)(7,173)(8,174)(9,175)(10,176)(11,177)(12,178)(13,179)(14,180)(15,181)(16,182)(17,183)(18,184)(19,185)(20,186)(21,187)(22,188)(23,189)(24,190)(25,191)(26,192)(27,193)(28,194)(29,195)(30,196)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,236)(44,237)(45,238)(46,239)(47,240)(48,241)(49,242)(50,243)(51,244)(52,245)(53,246)(54,247)(55,248)(56,249)(57,250)(58,251)(59,252)(60,211)(61,212)(62,213)(63,214)(64,215)(65,216)(66,217)(67,218)(68,219)(69,220)(70,221)(71,222)(72,223)(73,224)(74,225)(75,226)(76,227)(77,228)(78,229)(79,230)(80,231)(81,232)(82,233)(83,234)(84,235)(85,261)(86,262)(87,263)(88,264)(89,265)(90,266)(91,267)(92,268)(93,269)(94,270)(95,271)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,289)(114,290)(115,291)(116,292)(117,293)(118,294)(119,253)(120,254)(121,255)(122,256)(123,257)(124,258)(125,259)(126,260)(127,298)(128,299)(129,300)(130,301)(131,302)(132,303)(133,304)(134,305)(135,306)(136,307)(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)(144,315)(145,316)(146,317)(147,318)(148,319)(149,320)(150,321)(151,322)(152,323)(153,324)(154,325)(155,326)(156,327)(157,328)(158,329)(159,330)(160,331)(161,332)(162,333)(163,334)(164,335)(165,336)(166,295)(167,296)(168,297), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,113)(25,114)(26,115)(27,116)(28,117)(29,118)(30,119)(31,120)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,85)(39,86)(40,87)(41,88)(42,89)(43,140)(44,141)(45,142)(46,143)(47,144)(48,145)(49,146)(50,147)(51,148)(52,149)(53,150)(54,151)(55,152)(56,153)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(169,268)(170,269)(171,270)(172,271)(173,272)(174,273)(175,274)(176,275)(177,276)(178,277)(179,278)(180,279)(181,280)(182,281)(183,282)(184,283)(185,284)(186,285)(187,286)(188,287)(189,288)(190,289)(191,290)(192,291)(193,292)(194,293)(195,294)(196,253)(197,254)(198,255)(199,256)(200,257)(201,258)(202,259)(203,260)(204,261)(205,262)(206,263)(207,264)(208,265)(209,266)(210,267)(211,328)(212,329)(213,330)(214,331)(215,332)(216,333)(217,334)(218,335)(219,336)(220,295)(221,296)(222,297)(223,298)(224,299)(225,300)(226,301)(227,302)(228,303)(229,304)(230,305)(231,306)(232,307)(233,308)(234,309)(235,310)(236,311)(237,312)(238,313)(239,314)(240,315)(241,316)(242,317)(243,318)(244,319)(245,320)(246,321)(247,322)(248,323)(249,324)(250,325)(251,326)(252,327), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,81)(28,82)(29,83)(30,84)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(85,147)(86,148)(87,149)(88,150)(89,151)(90,152)(91,153)(92,154)(93,155)(94,156)(95,157)(96,158)(97,159)(98,160)(99,161)(100,162)(101,163)(102,164)(103,165)(104,166)(105,167)(106,168)(107,127)(108,128)(109,129)(110,130)(111,131)(112,132)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(169,250)(170,251)(171,252)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(196,235)(197,236)(198,237)(199,238)(200,239)(201,240)(202,241)(203,242)(204,243)(205,244)(206,245)(207,246)(208,247)(209,248)(210,249)(253,310)(254,311)(255,312)(256,313)(257,314)(258,315)(259,316)(260,317)(261,318)(262,319)(263,320)(264,321)(265,322)(266,323)(267,324)(268,325)(269,326)(270,327)(271,328)(272,329)(273,330)(274,331)(275,332)(276,333)(277,334)(278,335)(279,336)(280,295)(281,296)(282,297)(283,298)(284,299)(285,300)(286,301)(287,302)(288,303)(289,304)(290,305)(291,306)(292,307)(293,308)(294,309), (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) );
G=PermutationGroup([[(1,209),(2,210),(3,169),(4,170),(5,171),(6,172),(7,173),(8,174),(9,175),(10,176),(11,177),(12,178),(13,179),(14,180),(15,181),(16,182),(17,183),(18,184),(19,185),(20,186),(21,187),(22,188),(23,189),(24,190),(25,191),(26,192),(27,193),(28,194),(29,195),(30,196),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,236),(44,237),(45,238),(46,239),(47,240),(48,241),(49,242),(50,243),(51,244),(52,245),(53,246),(54,247),(55,248),(56,249),(57,250),(58,251),(59,252),(60,211),(61,212),(62,213),(63,214),(64,215),(65,216),(66,217),(67,218),(68,219),(69,220),(70,221),(71,222),(72,223),(73,224),(74,225),(75,226),(76,227),(77,228),(78,229),(79,230),(80,231),(81,232),(82,233),(83,234),(84,235),(85,261),(86,262),(87,263),(88,264),(89,265),(90,266),(91,267),(92,268),(93,269),(94,270),(95,271),(96,272),(97,273),(98,274),(99,275),(100,276),(101,277),(102,278),(103,279),(104,280),(105,281),(106,282),(107,283),(108,284),(109,285),(110,286),(111,287),(112,288),(113,289),(114,290),(115,291),(116,292),(117,293),(118,294),(119,253),(120,254),(121,255),(122,256),(123,257),(124,258),(125,259),(126,260),(127,298),(128,299),(129,300),(130,301),(131,302),(132,303),(133,304),(134,305),(135,306),(136,307),(137,308),(138,309),(139,310),(140,311),(141,312),(142,313),(143,314),(144,315),(145,316),(146,317),(147,318),(148,319),(149,320),(150,321),(151,322),(152,323),(153,324),(154,325),(155,326),(156,327),(157,328),(158,329),(159,330),(160,331),(161,332),(162,333),(163,334),(164,335),(165,336),(166,295),(167,296),(168,297)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,113),(25,114),(26,115),(27,116),(28,117),(29,118),(30,119),(31,120),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,85),(39,86),(40,87),(41,88),(42,89),(43,140),(44,141),(45,142),(46,143),(47,144),(48,145),(49,146),(50,147),(51,148),(52,149),(53,150),(54,151),(55,152),(56,153),(57,154),(58,155),(59,156),(60,157),(61,158),(62,159),(63,160),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(169,268),(170,269),(171,270),(172,271),(173,272),(174,273),(175,274),(176,275),(177,276),(178,277),(179,278),(180,279),(181,280),(182,281),(183,282),(184,283),(185,284),(186,285),(187,286),(188,287),(189,288),(190,289),(191,290),(192,291),(193,292),(194,293),(195,294),(196,253),(197,254),(198,255),(199,256),(200,257),(201,258),(202,259),(203,260),(204,261),(205,262),(206,263),(207,264),(208,265),(209,266),(210,267),(211,328),(212,329),(213,330),(214,331),(215,332),(216,333),(217,334),(218,335),(219,336),(220,295),(221,296),(222,297),(223,298),(224,299),(225,300),(226,301),(227,302),(228,303),(229,304),(230,305),(231,306),(232,307),(233,308),(234,309),(235,310),(236,311),(237,312),(238,313),(239,314),(240,315),(241,316),(242,317),(243,318),(244,319),(245,320),(246,321),(247,322),(248,323),(249,324),(250,325),(251,326),(252,327)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,81),(28,82),(29,83),(30,84),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(85,147),(86,148),(87,149),(88,150),(89,151),(90,152),(91,153),(92,154),(93,155),(94,156),(95,157),(96,158),(97,159),(98,160),(99,161),(100,162),(101,163),(102,164),(103,165),(104,166),(105,167),(106,168),(107,127),(108,128),(109,129),(110,130),(111,131),(112,132),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(169,250),(170,251),(171,252),(172,211),(173,212),(174,213),(175,214),(176,215),(177,216),(178,217),(179,218),(180,219),(181,220),(182,221),(183,222),(184,223),(185,224),(186,225),(187,226),(188,227),(189,228),(190,229),(191,230),(192,231),(193,232),(194,233),(195,234),(196,235),(197,236),(198,237),(199,238),(200,239),(201,240),(202,241),(203,242),(204,243),(205,244),(206,245),(207,246),(208,247),(209,248),(210,249),(253,310),(254,311),(255,312),(256,313),(257,314),(258,315),(259,316),(260,317),(261,318),(262,319),(263,320),(264,321),(265,322),(266,323),(267,324),(268,325),(269,326),(270,327),(271,328),(272,329),(273,330),(274,331),(275,332),(276,333),(277,334),(278,335),(279,336),(280,295),(281,296),(282,297),(283,298),(284,299),(285,300),(286,301),(287,302),(288,303),(289,304),(290,305),(291,306),(292,307),(293,308),(294,309)], [(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)]])
336 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3A | 3B | 6A | ··· | 6AD | 7A | ··· | 7F | 14A | ··· | 14CL | 21A | ··· | 21L | 42A | ··· | 42FX |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
336 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 |
| kernel | C23×C42 | C22×C42 | C23×C14 | C22×C14 | C23×C6 | C22×C6 | C24 | C23 |
| # reps | 1 | 15 | 2 | 30 | 6 | 90 | 12 | 180 |
Matrix representation of C23×C42 ►in GL4(𝔽43) generated by
| 42 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 42 |
| 38 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(43))| [42,0,0,0,0,1,0,0,0,0,42,0,0,0,0,42],[42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,1],[42,0,0,0,0,42,0,0,0,0,1,0,0,0,0,42],[38,0,0,0,0,31,0,0,0,0,31,0,0,0,0,8] >;
C23×C42 in GAP, Magma, Sage, TeX
C_2^3\times C_{42} % in TeX
G:=Group("C2^3xC42"); // GroupNames label
G:=SmallGroup(336,228);
// by ID
G=gap.SmallGroup(336,228);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^2=d^42=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations