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## G = C2×C21⋊Q8order 336 = 24·3·7

### Direct product of C2 and C21⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C2×C21⋊Q8
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — C21⋊Q8 — C2×C21⋊Q8
 Lower central C21 — C42 — C2×C21⋊Q8
 Upper central C1 — C22

Generators and relations for C2×C21⋊Q8
G = < a,b,c,d | a2=b21=c4=1, d2=c2, ab=ba, ac=ca, ad=da, cbc-1=b8, dbd-1=b13, dcd-1=c-1 >

Subgroups: 348 in 76 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C7, C2×C4, Q8, Dic3, Dic3, C12, C2×C6, C14, C14, C2×Q8, C21, Dic6, C2×Dic3, C2×Dic3, C2×C12, Dic7, Dic7, C28, C2×C14, C42, C42, C2×Dic6, Dic14, C2×Dic7, C2×Dic7, C2×C28, C7×Dic3, C3×Dic7, Dic21, C2×C42, C2×Dic14, C21⋊Q8, C6×Dic7, Dic3×C14, C2×Dic21, C2×C21⋊Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, D7, C2×Q8, Dic6, C22×S3, D14, C2×Dic6, Dic14, C22×D7, S3×D7, C2×Dic14, C21⋊Q8, C2×S3×D7, C2×C21⋊Q8

Smallest permutation representation of C2×C21⋊Q8
Regular action on 336 points
Generators in S336
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(85 147)(86 127)(87 128)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 145)(105 146)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 148)(123 149)(124 150)(125 151)(126 152)(169 229)(170 230)(171 231)(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)(190 245)(191 246)(192 247)(193 248)(194 249)(195 250)(196 251)(197 252)(198 232)(199 233)(200 234)(201 235)(202 236)(203 237)(204 238)(205 239)(206 240)(207 241)(208 242)(209 243)(210 244)(253 300)(254 301)(255 302)(256 303)(257 304)(258 305)(259 306)(260 307)(261 308)(262 309)(263 310)(264 311)(265 312)(266 313)(267 314)(268 315)(269 295)(270 296)(271 297)(272 298)(273 299)(274 320)(275 321)(276 322)(277 323)(278 324)(279 325)(280 326)(281 327)(282 328)(283 329)(284 330)(285 331)(286 332)(287 333)(288 334)(289 335)(290 336)(291 316)(292 317)(293 318)(294 319)
(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)
(1 156 39 128)(2 164 40 136)(3 151 41 144)(4 159 42 131)(5 167 22 139)(6 154 23 147)(7 162 24 134)(8 149 25 142)(9 157 26 129)(10 165 27 137)(11 152 28 145)(12 160 29 132)(13 168 30 140)(14 155 31 127)(15 163 32 135)(16 150 33 143)(17 158 34 130)(18 166 35 138)(19 153 36 146)(20 161 37 133)(21 148 38 141)(43 109 81 87)(44 117 82 95)(45 125 83 103)(46 112 84 90)(47 120 64 98)(48 107 65 85)(49 115 66 93)(50 123 67 101)(51 110 68 88)(52 118 69 96)(53 126 70 104)(54 113 71 91)(55 121 72 99)(56 108 73 86)(57 116 74 94)(58 124 75 102)(59 111 76 89)(60 119 77 97)(61 106 78 105)(62 114 79 92)(63 122 80 100)(169 297 199 319)(170 305 200 327)(171 313 201 335)(172 300 202 322)(173 308 203 330)(174 295 204 317)(175 303 205 325)(176 311 206 333)(177 298 207 320)(178 306 208 328)(179 314 209 336)(180 301 210 323)(181 309 190 331)(182 296 191 318)(183 304 192 326)(184 312 193 334)(185 299 194 321)(186 307 195 329)(187 315 196 316)(188 302 197 324)(189 310 198 332)(211 253 236 276)(212 261 237 284)(213 269 238 292)(214 256 239 279)(215 264 240 287)(216 272 241 274)(217 259 242 282)(218 267 243 290)(219 254 244 277)(220 262 245 285)(221 270 246 293)(222 257 247 280)(223 265 248 288)(224 273 249 275)(225 260 250 283)(226 268 251 291)(227 255 252 278)(228 263 232 286)(229 271 233 294)(230 258 234 281)(231 266 235 289)
(1 236 39 211)(2 249 40 224)(3 241 41 216)(4 233 42 229)(5 246 22 221)(6 238 23 213)(7 251 24 226)(8 243 25 218)(9 235 26 231)(10 248 27 223)(11 240 28 215)(12 232 29 228)(13 245 30 220)(14 237 31 212)(15 250 32 225)(16 242 33 217)(17 234 34 230)(18 247 35 222)(19 239 36 214)(20 252 37 227)(21 244 38 219)(43 202 81 172)(44 194 82 185)(45 207 83 177)(46 199 84 169)(47 191 64 182)(48 204 65 174)(49 196 66 187)(50 209 67 179)(51 201 68 171)(52 193 69 184)(53 206 70 176)(54 198 71 189)(55 190 72 181)(56 203 73 173)(57 195 74 186)(58 208 75 178)(59 200 76 170)(60 192 77 183)(61 205 78 175)(62 197 79 188)(63 210 80 180)(85 317 107 295)(86 330 108 308)(87 322 109 300)(88 335 110 313)(89 327 111 305)(90 319 112 297)(91 332 113 310)(92 324 114 302)(93 316 115 315)(94 329 116 307)(95 321 117 299)(96 334 118 312)(97 326 119 304)(98 318 120 296)(99 331 121 309)(100 323 122 301)(101 336 123 314)(102 328 124 306)(103 320 125 298)(104 333 126 311)(105 325 106 303)(127 284 155 261)(128 276 156 253)(129 289 157 266)(130 281 158 258)(131 294 159 271)(132 286 160 263)(133 278 161 255)(134 291 162 268)(135 283 163 260)(136 275 164 273)(137 288 165 265)(138 280 166 257)(139 293 167 270)(140 285 168 262)(141 277 148 254)(142 290 149 267)(143 282 150 259)(144 274 151 272)(145 287 152 264)(146 279 153 256)(147 292 154 269)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 7A 7B 7C 12A 12B 12C 12D 14A ··· 14I 21A 21B 21C 28A ··· 28L 42A ··· 42I order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 7 7 7 12 12 12 12 14 ··· 14 21 21 21 28 ··· 28 42 ··· 42 size 1 1 1 1 2 6 6 14 14 42 42 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 4 4 6 ··· 6 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + - + + - + - + image C1 C2 C2 C2 C2 S3 Q8 D6 D6 D7 Dic6 D14 D14 Dic14 S3×D7 C21⋊Q8 C2×S3×D7 kernel C2×C21⋊Q8 C21⋊Q8 C6×Dic7 Dic3×C14 C2×Dic21 C2×Dic7 C42 Dic7 C2×C14 C2×Dic3 C14 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 2 2 1 3 4 6 3 12 3 6 3

Matrix representation of C2×C21⋊Q8 in GL4(𝔽337) generated by

 336 0 0 0 0 336 0 0 0 0 1 0 0 0 0 1
,
 0 227 0 0 144 303 0 0 0 0 1 141 0 0 43 335
,
 1 0 0 0 0 1 0 0 0 0 33 290 0 0 310 304
,
 251 7 0 0 99 86 0 0 0 0 45 186 0 0 279 292
G:=sub<GL(4,GF(337))| [336,0,0,0,0,336,0,0,0,0,1,0,0,0,0,1],[0,144,0,0,227,303,0,0,0,0,1,43,0,0,141,335],[1,0,0,0,0,1,0,0,0,0,33,310,0,0,290,304],[251,99,0,0,7,86,0,0,0,0,45,279,0,0,186,292] >;

C2×C21⋊Q8 in GAP, Magma, Sage, TeX

C_2\times C_{21}\rtimes Q_8
% in TeX

G:=Group("C2xC21:Q8");
// GroupNames label

G:=SmallGroup(336,160);
// by ID

G=gap.SmallGroup(336,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,121,55,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^21=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^8,d*b*d^-1=b^13,d*c*d^-1=c^-1>;
// generators/relations

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