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## G = C22×Dic21order 336 = 24·3·7

### Direct product of C22 and Dic21

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C22×Dic21
 Chief series C1 — C7 — C21 — C42 — Dic21 — C2×Dic21 — C22×Dic21
 Lower central C21 — C22×Dic21
 Upper central C1 — C23

Generators and relations for C22×Dic21
G = < a,b,c,d | a2=b2=c42=1, d2=c21, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 416 in 108 conjugacy classes, 75 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C7, C2×C4, C23, Dic3, C2×C6, C14, C14, C22×C4, C21, C2×Dic3, C22×C6, Dic7, C2×C14, C42, C42, C22×Dic3, C2×Dic7, C22×C14, Dic21, C2×C42, C22×Dic7, C2×Dic21, C22×C42, C22×Dic21
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, D7, C22×C4, C2×Dic3, C22×S3, Dic7, D14, D21, C22×Dic3, C2×Dic7, C22×D7, Dic21, D42, C22×Dic7, C2×Dic21, C22×D21, C22×Dic21

Smallest permutation representation of C22×Dic21
Regular action on 336 points
Generators in S336
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 141)(8 142)(9 143)(10 144)(11 145)(12 146)(13 147)(14 148)(15 149)(16 150)(17 151)(18 152)(19 153)(20 154)(21 155)(22 156)(23 157)(24 158)(25 159)(26 160)(27 161)(28 162)(29 163)(30 164)(31 165)(32 166)(33 167)(34 168)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(169 309)(170 310)(171 311)(172 312)(173 313)(174 314)(175 315)(176 316)(177 317)(178 318)(179 319)(180 320)(181 321)(182 322)(183 323)(184 324)(185 325)(186 326)(187 327)(188 328)(189 329)(190 330)(191 331)(192 332)(193 333)(194 334)(195 335)(196 336)(197 295)(198 296)(199 297)(200 298)(201 299)(202 300)(203 301)(204 302)(205 303)(206 304)(207 305)(208 306)(209 307)(210 308)(211 253)(212 254)(213 255)(214 256)(215 257)(216 258)(217 259)(218 260)(219 261)(220 262)(221 263)(222 264)(223 265)(224 266)(225 267)(226 268)(227 269)(228 270)(229 271)(230 272)(231 273)(232 274)(233 275)(234 276)(235 277)(236 278)(237 279)(238 280)(239 281)(240 282)(241 283)(242 284)(243 285)(244 286)(245 287)(246 288)(247 289)(248 290)(249 291)(250 292)(251 293)(252 294)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 137)(114 138)(115 139)(116 140)(117 141)(118 142)(119 143)(120 144)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(169 232)(170 233)(171 234)(172 235)(173 236)(174 237)(175 238)(176 239)(177 240)(178 241)(179 242)(180 243)(181 244)(182 245)(183 246)(184 247)(185 248)(186 249)(187 250)(188 251)(189 252)(190 211)(191 212)(192 213)(193 214)(194 215)(195 216)(196 217)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 225)(205 226)(206 227)(207 228)(208 229)(209 230)(210 231)(253 330)(254 331)(255 332)(256 333)(257 334)(258 335)(259 336)(260 295)(261 296)(262 297)(263 298)(264 299)(265 300)(266 301)(267 302)(268 303)(269 304)(270 305)(271 306)(272 307)(273 308)(274 309)(275 310)(276 311)(277 312)(278 313)(279 314)(280 315)(281 316)(282 317)(283 318)(284 319)(285 320)(286 321)(287 322)(288 323)(289 324)(290 325)(291 326)(292 327)(293 328)(294 329)
(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)
(1 232 22 211)(2 231 23 252)(3 230 24 251)(4 229 25 250)(5 228 26 249)(6 227 27 248)(7 226 28 247)(8 225 29 246)(9 224 30 245)(10 223 31 244)(11 222 32 243)(12 221 33 242)(13 220 34 241)(14 219 35 240)(15 218 36 239)(16 217 37 238)(17 216 38 237)(18 215 39 236)(19 214 40 235)(20 213 41 234)(21 212 42 233)(43 176 64 197)(44 175 65 196)(45 174 66 195)(46 173 67 194)(47 172 68 193)(48 171 69 192)(49 170 70 191)(50 169 71 190)(51 210 72 189)(52 209 73 188)(53 208 74 187)(54 207 75 186)(55 206 76 185)(56 205 77 184)(57 204 78 183)(58 203 79 182)(59 202 80 181)(60 201 81 180)(61 200 82 179)(62 199 83 178)(63 198 84 177)(85 335 106 314)(86 334 107 313)(87 333 108 312)(88 332 109 311)(89 331 110 310)(90 330 111 309)(91 329 112 308)(92 328 113 307)(93 327 114 306)(94 326 115 305)(95 325 116 304)(96 324 117 303)(97 323 118 302)(98 322 119 301)(99 321 120 300)(100 320 121 299)(101 319 122 298)(102 318 123 297)(103 317 124 296)(104 316 125 295)(105 315 126 336)(127 282 148 261)(128 281 149 260)(129 280 150 259)(130 279 151 258)(131 278 152 257)(132 277 153 256)(133 276 154 255)(134 275 155 254)(135 274 156 253)(136 273 157 294)(137 272 158 293)(138 271 159 292)(139 270 160 291)(140 269 161 290)(141 268 162 289)(142 267 163 288)(143 266 164 287)(144 265 165 286)(145 264 166 285)(146 263 167 284)(147 262 168 283)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 6A ··· 6G 7A 7B 7C 14A ··· 14U 21A ··· 21F 42A ··· 42AP order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 7 7 7 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 ··· 1 2 21 ··· 21 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + - + + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 D7 Dic7 D14 D21 Dic21 D42 kernel C22×Dic21 C2×Dic21 C22×C42 C2×C42 C22×C14 C2×C14 C2×C14 C22×C6 C2×C6 C2×C6 C23 C22 C22 # reps 1 6 1 8 1 4 3 3 12 9 6 24 18

Matrix representation of C22×Dic21 in GL4(𝔽337) generated by

 336 0 0 0 0 336 0 0 0 0 336 0 0 0 0 336
,
 1 0 0 0 0 336 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 107 101 0 0 236 59
,
 336 0 0 0 0 336 0 0 0 0 228 180 0 0 95 109
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,107,236,0,0,101,59],[336,0,0,0,0,336,0,0,0,0,228,95,0,0,180,109] >;

C22×Dic21 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{21}
% in TeX

G:=Group("C2^2xDic21");
// GroupNames label

G:=SmallGroup(336,202);
// by ID

G=gap.SmallGroup(336,202);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,964,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^42=1,d^2=c^21,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

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