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G = C23×C42order 336 = 24·3·7

Abelian group of type [2,2,2,42]

Aliases: C23×C42, SmallGroup(336,228)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C42
 Chief series C1 — C7 — C21 — C42 — C2×C42 — C22×C42 — C23×C42
 Lower central C1 — C23×C42
 Upper central 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 [×15], C3, C22 [×35], C6 [×15], C7, C23 [×15], C2×C6 [×35], C14 [×15], C24, C21, C22×C6 [×15], C2×C14 [×35], C42 [×15], C23×C6, C22×C14 [×15], C2×C42 [×35], C23×C14, C22×C42 [×15], C23×C42
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], C7, C23 [×15], C2×C6 [×35], C14 [×15], C24, C21, C22×C6 [×15], C2×C14 [×35], C42 [×15], C23×C6, C22×C14 [×15], C2×C42 [×35], C23×C14, C22×C42 [×15], C23×C42

Smallest permutation representation of C23×C42
Regular action on 336 points
Generators in S336
(1 201)(2 202)(3 203)(4 204)(5 205)(6 206)(7 207)(8 208)(9 209)(10 210)(11 169)(12 170)(13 171)(14 172)(15 173)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 183)(26 184)(27 185)(28 186)(29 187)(30 188)(31 189)(32 190)(33 191)(34 192)(35 193)(36 194)(37 195)(38 196)(39 197)(40 198)(41 199)(42 200)(43 248)(44 249)(45 250)(46 251)(47 252)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 220)(58 221)(59 222)(60 223)(61 224)(62 225)(63 226)(64 227)(65 228)(66 229)(67 230)(68 231)(69 232)(70 233)(71 234)(72 235)(73 236)(74 237)(75 238)(76 239)(77 240)(78 241)(79 242)(80 243)(81 244)(82 245)(83 246)(84 247)(85 256)(86 257)(87 258)(88 259)(89 260)(90 261)(91 262)(92 263)(93 264)(94 265)(95 266)(96 267)(97 268)(98 269)(99 270)(100 271)(101 272)(102 273)(103 274)(104 275)(105 276)(106 277)(107 278)(108 279)(109 280)(110 281)(111 282)(112 283)(113 284)(114 285)(115 286)(116 287)(117 288)(118 289)(119 290)(120 291)(121 292)(122 293)(123 294)(124 253)(125 254)(126 255)(127 320)(128 321)(129 322)(130 323)(131 324)(132 325)(133 326)(134 327)(135 328)(136 329)(137 330)(138 331)(139 332)(140 333)(141 334)(142 335)(143 336)(144 295)(145 296)(146 297)(147 298)(148 299)(149 300)(150 301)(151 302)(152 303)(153 304)(154 305)(155 306)(156 307)(157 308)(158 309)(159 310)(160 311)(161 312)(162 313)(163 314)(164 315)(165 316)(166 317)(167 318)(168 319)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 161)(51 162)(52 163)(53 164)(54 165)(55 166)(56 167)(57 168)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 150)(82 151)(83 152)(84 153)(169 281)(170 282)(171 283)(172 284)(173 285)(174 286)(175 287)(176 288)(177 289)(178 290)(179 291)(180 292)(181 293)(182 294)(183 253)(184 254)(185 255)(186 256)(187 257)(188 258)(189 259)(190 260)(191 261)(192 262)(193 263)(194 264)(195 265)(196 266)(197 267)(198 268)(199 269)(200 270)(201 271)(202 272)(203 273)(204 274)(205 275)(206 276)(207 277)(208 278)(209 279)(210 280)(211 310)(212 311)(213 312)(214 313)(215 314)(216 315)(217 316)(218 317)(219 318)(220 319)(221 320)(222 321)(223 322)(224 323)(225 324)(226 325)(227 326)(228 327)(229 328)(230 329)(231 330)(232 331)(233 332)(234 333)(235 334)(236 335)(237 336)(238 295)(239 296)(240 297)(241 298)(242 299)(243 300)(244 301)(245 302)(246 303)(247 304)(248 305)(249 306)(250 307)(251 308)(252 309)
(1 83)(2 84)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 81)(42 82)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)(109 161)(110 162)(111 163)(112 164)(113 165)(114 166)(115 167)(116 168)(117 127)(118 128)(119 129)(120 130)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(169 214)(170 215)(171 216)(172 217)(173 218)(174 219)(175 220)(176 221)(177 222)(178 223)(179 224)(180 225)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 235)(191 236)(192 237)(193 238)(194 239)(195 240)(196 241)(197 242)(198 243)(199 244)(200 245)(201 246)(202 247)(203 248)(204 249)(205 250)(206 251)(207 252)(208 211)(209 212)(210 213)(253 327)(254 328)(255 329)(256 330)(257 331)(258 332)(259 333)(260 334)(261 335)(262 336)(263 295)(264 296)(265 297)(266 298)(267 299)(268 300)(269 301)(270 302)(271 303)(272 304)(273 305)(274 306)(275 307)(276 308)(277 309)(278 310)(279 311)(280 312)(281 313)(282 314)(283 315)(284 316)(285 317)(286 318)(287 319)(288 320)(289 321)(290 322)(291 323)(292 324)(293 325)(294 326)
(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)| (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,169)(12,170)(13,171)(14,172)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,199)(42,200)(43,248)(44,249)(45,250)(46,251)(47,252)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,220)(58,221)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,233)(71,234)(72,235)(73,236)(74,237)(75,238)(76,239)(77,240)(78,241)(79,242)(80,243)(81,244)(82,245)(83,246)(84,247)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,272)(102,273)(103,274)(104,275)(105,276)(106,277)(107,278)(108,279)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,289)(119,290)(120,291)(121,292)(122,293)(123,294)(124,253)(125,254)(126,255)(127,320)(128,321)(129,322)(130,323)(131,324)(132,325)(133,326)(134,327)(135,328)(136,329)(137,330)(138,331)(139,332)(140,333)(141,334)(142,335)(143,336)(144,295)(145,296)(146,297)(147,298)(148,299)(149,300)(150,301)(151,302)(152,303)(153,304)(154,305)(155,306)(156,307)(157,308)(158,309)(159,310)(160,311)(161,312)(162,313)(163,314)(164,315)(165,316)(166,317)(167,318)(168,319), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(169,281)(170,282)(171,283)(172,284)(173,285)(174,286)(175,287)(176,288)(177,289)(178,290)(179,291)(180,292)(181,293)(182,294)(183,253)(184,254)(185,255)(186,256)(187,257)(188,258)(189,259)(190,260)(191,261)(192,262)(193,263)(194,264)(195,265)(196,266)(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)(210,280)(211,310)(212,311)(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)(221,320)(222,321)(223,322)(224,323)(225,324)(226,325)(227,326)(228,327)(229,328)(230,329)(231,330)(232,331)(233,332)(234,333)(235,334)(236,335)(237,336)(238,295)(239,296)(240,297)(241,298)(242,299)(243,300)(244,301)(245,302)(246,303)(247,304)(248,305)(249,306)(250,307)(251,308)(252,309), (1,83)(2,84)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(169,214)(170,215)(171,216)(172,217)(173,218)(174,219)(175,220)(176,221)(177,222)(178,223)(179,224)(180,225)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,241)(197,242)(198,243)(199,244)(200,245)(201,246)(202,247)(203,248)(204,249)(205,250)(206,251)(207,252)(208,211)(209,212)(210,213)(253,327)(254,328)(255,329)(256,330)(257,331)(258,332)(259,333)(260,334)(261,335)(262,336)(263,295)(264,296)(265,297)(266,298)(267,299)(268,300)(269,301)(270,302)(271,303)(272,304)(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)(280,312)(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)(288,320)(289,321)(290,322)(291,323)(292,324)(293,325)(294,326), (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( (1,201)(2,202)(3,203)(4,204)(5,205)(6,206)(7,207)(8,208)(9,209)(10,210)(11,169)(12,170)(13,171)(14,172)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,199)(42,200)(43,248)(44,249)(45,250)(46,251)(47,252)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,220)(58,221)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,233)(71,234)(72,235)(73,236)(74,237)(75,238)(76,239)(77,240)(78,241)(79,242)(80,243)(81,244)(82,245)(83,246)(84,247)(85,256)(86,257)(87,258)(88,259)(89,260)(90,261)(91,262)(92,263)(93,264)(94,265)(95,266)(96,267)(97,268)(98,269)(99,270)(100,271)(101,272)(102,273)(103,274)(104,275)(105,276)(106,277)(107,278)(108,279)(109,280)(110,281)(111,282)(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,289)(119,290)(120,291)(121,292)(122,293)(123,294)(124,253)(125,254)(126,255)(127,320)(128,321)(129,322)(130,323)(131,324)(132,325)(133,326)(134,327)(135,328)(136,329)(137,330)(138,331)(139,332)(140,333)(141,334)(142,335)(143,336)(144,295)(145,296)(146,297)(147,298)(148,299)(149,300)(150,301)(151,302)(152,303)(153,304)(154,305)(155,306)(156,307)(157,308)(158,309)(159,310)(160,311)(161,312)(162,313)(163,314)(164,315)(165,316)(166,317)(167,318)(168,319), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,120)(22,121)(23,122)(24,123)(25,124)(26,125)(27,126)(28,85)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,154)(44,155)(45,156)(46,157)(47,158)(48,159)(49,160)(50,161)(51,162)(52,163)(53,164)(54,165)(55,166)(56,167)(57,168)(58,127)(59,128)(60,129)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,150)(82,151)(83,152)(84,153)(169,281)(170,282)(171,283)(172,284)(173,285)(174,286)(175,287)(176,288)(177,289)(178,290)(179,291)(180,292)(181,293)(182,294)(183,253)(184,254)(185,255)(186,256)(187,257)(188,258)(189,259)(190,260)(191,261)(192,262)(193,263)(194,264)(195,265)(196,266)(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)(210,280)(211,310)(212,311)(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)(221,320)(222,321)(223,322)(224,323)(225,324)(226,325)(227,326)(228,327)(229,328)(230,329)(231,330)(232,331)(233,332)(234,333)(235,334)(236,335)(237,336)(238,295)(239,296)(240,297)(241,298)(242,299)(243,300)(244,301)(245,302)(246,303)(247,304)(248,305)(249,306)(250,307)(251,308)(252,309), (1,83)(2,84)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,81)(42,82)(85,137)(86,138)(87,139)(88,140)(89,141)(90,142)(91,143)(92,144)(93,145)(94,146)(95,147)(96,148)(97,149)(98,150)(99,151)(100,152)(101,153)(102,154)(103,155)(104,156)(105,157)(106,158)(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)(116,168)(117,127)(118,128)(119,129)(120,130)(121,131)(122,132)(123,133)(124,134)(125,135)(126,136)(169,214)(170,215)(171,216)(172,217)(173,218)(174,219)(175,220)(176,221)(177,222)(178,223)(179,224)(180,225)(181,226)(182,227)(183,228)(184,229)(185,230)(186,231)(187,232)(188,233)(189,234)(190,235)(191,236)(192,237)(193,238)(194,239)(195,240)(196,241)(197,242)(198,243)(199,244)(200,245)(201,246)(202,247)(203,248)(204,249)(205,250)(206,251)(207,252)(208,211)(209,212)(210,213)(253,327)(254,328)(255,329)(256,330)(257,331)(258,332)(259,333)(260,334)(261,335)(262,336)(263,295)(264,296)(265,297)(266,298)(267,299)(268,300)(269,301)(270,302)(271,303)(272,304)(273,305)(274,306)(275,307)(276,308)(277,309)(278,310)(279,311)(280,312)(281,313)(282,314)(283,315)(284,316)(285,317)(286,318)(287,319)(288,320)(289,321)(290,322)(291,323)(292,324)(293,325)(294,326), (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([(1,201),(2,202),(3,203),(4,204),(5,205),(6,206),(7,207),(8,208),(9,209),(10,210),(11,169),(12,170),(13,171),(14,172),(15,173),(16,174),(17,175),(18,176),(19,177),(20,178),(21,179),(22,180),(23,181),(24,182),(25,183),(26,184),(27,185),(28,186),(29,187),(30,188),(31,189),(32,190),(33,191),(34,192),(35,193),(36,194),(37,195),(38,196),(39,197),(40,198),(41,199),(42,200),(43,248),(44,249),(45,250),(46,251),(47,252),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(57,220),(58,221),(59,222),(60,223),(61,224),(62,225),(63,226),(64,227),(65,228),(66,229),(67,230),(68,231),(69,232),(70,233),(71,234),(72,235),(73,236),(74,237),(75,238),(76,239),(77,240),(78,241),(79,242),(80,243),(81,244),(82,245),(83,246),(84,247),(85,256),(86,257),(87,258),(88,259),(89,260),(90,261),(91,262),(92,263),(93,264),(94,265),(95,266),(96,267),(97,268),(98,269),(99,270),(100,271),(101,272),(102,273),(103,274),(104,275),(105,276),(106,277),(107,278),(108,279),(109,280),(110,281),(111,282),(112,283),(113,284),(114,285),(115,286),(116,287),(117,288),(118,289),(119,290),(120,291),(121,292),(122,293),(123,294),(124,253),(125,254),(126,255),(127,320),(128,321),(129,322),(130,323),(131,324),(132,325),(133,326),(134,327),(135,328),(136,329),(137,330),(138,331),(139,332),(140,333),(141,334),(142,335),(143,336),(144,295),(145,296),(146,297),(147,298),(148,299),(149,300),(150,301),(151,302),(152,303),(153,304),(154,305),(155,306),(156,307),(157,308),(158,309),(159,310),(160,311),(161,312),(162,313),(163,314),(164,315),(165,316),(166,317),(167,318),(168,319)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,113),(15,114),(16,115),(17,116),(18,117),(19,118),(20,119),(21,120),(22,121),(23,122),(24,123),(25,124),(26,125),(27,126),(28,85),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,99),(43,154),(44,155),(45,156),(46,157),(47,158),(48,159),(49,160),(50,161),(51,162),(52,163),(53,164),(54,165),(55,166),(56,167),(57,168),(58,127),(59,128),(60,129),(61,130),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(81,150),(82,151),(83,152),(84,153),(169,281),(170,282),(171,283),(172,284),(173,285),(174,286),(175,287),(176,288),(177,289),(178,290),(179,291),(180,292),(181,293),(182,294),(183,253),(184,254),(185,255),(186,256),(187,257),(188,258),(189,259),(190,260),(191,261),(192,262),(193,263),(194,264),(195,265),(196,266),(197,267),(198,268),(199,269),(200,270),(201,271),(202,272),(203,273),(204,274),(205,275),(206,276),(207,277),(208,278),(209,279),(210,280),(211,310),(212,311),(213,312),(214,313),(215,314),(216,315),(217,316),(218,317),(219,318),(220,319),(221,320),(222,321),(223,322),(224,323),(225,324),(226,325),(227,326),(228,327),(229,328),(230,329),(231,330),(232,331),(233,332),(234,333),(235,334),(236,335),(237,336),(238,295),(239,296),(240,297),(241,298),(242,299),(243,300),(244,301),(245,302),(246,303),(247,304),(248,305),(249,306),(250,307),(251,308),(252,309)], [(1,83),(2,84),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,81),(42,82),(85,137),(86,138),(87,139),(88,140),(89,141),(90,142),(91,143),(92,144),(93,145),(94,146),(95,147),(96,148),(97,149),(98,150),(99,151),(100,152),(101,153),(102,154),(103,155),(104,156),(105,157),(106,158),(107,159),(108,160),(109,161),(110,162),(111,163),(112,164),(113,165),(114,166),(115,167),(116,168),(117,127),(118,128),(119,129),(120,130),(121,131),(122,132),(123,133),(124,134),(125,135),(126,136),(169,214),(170,215),(171,216),(172,217),(173,218),(174,219),(175,220),(176,221),(177,222),(178,223),(179,224),(180,225),(181,226),(182,227),(183,228),(184,229),(185,230),(186,231),(187,232),(188,233),(189,234),(190,235),(191,236),(192,237),(193,238),(194,239),(195,240),(196,241),(197,242),(198,243),(199,244),(200,245),(201,246),(202,247),(203,248),(204,249),(205,250),(206,251),(207,252),(208,211),(209,212),(210,213),(253,327),(254,328),(255,329),(256,330),(257,331),(258,332),(259,333),(260,334),(261,335),(262,336),(263,295),(264,296),(265,297),(266,298),(267,299),(268,300),(269,301),(270,302),(271,303),(272,304),(273,305),(274,306),(275,307),(276,308),(277,309),(278,310),(279,311),(280,312),(281,313),(282,314),(283,315),(284,316),(285,317),(286,318),(287,319),(288,320),(289,321),(290,322),(291,323),(292,324),(293,325),(294,326)], [(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

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