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## G = C7×Dic12order 336 = 24·3·7

### Direct product of C7 and Dic12

Aliases: C7×Dic12, C216Q16, C56.3S3, C168.4C2, C24.1C14, C28.54D6, C42.30D4, C14.15D12, C84.70C22, Dic6.1C14, C8.(S3×C7), C31(C7×Q16), C6.3(C7×D4), C2.5(C7×D12), C4.10(S3×C14), C12.10(C2×C14), (C7×Dic6).3C2, SmallGroup(336,78)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C7×Dic12
 Chief series C1 — C3 — C6 — C12 — C84 — C7×Dic6 — C7×Dic12
 Lower central C3 — C6 — C12 — C7×Dic12
 Upper central C1 — C14 — C28 — C56

Generators and relations for C7×Dic12
G = < a,b,c | a7=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Dic12
Regular action on 336 points
Generators in S336
(1 335 113 180 232 278 121)(2 336 114 181 233 279 122)(3 313 115 182 234 280 123)(4 314 116 183 235 281 124)(5 315 117 184 236 282 125)(6 316 118 185 237 283 126)(7 317 119 186 238 284 127)(8 318 120 187 239 285 128)(9 319 97 188 240 286 129)(10 320 98 189 217 287 130)(11 321 99 190 218 288 131)(12 322 100 191 219 265 132)(13 323 101 192 220 266 133)(14 324 102 169 221 267 134)(15 325 103 170 222 268 135)(16 326 104 171 223 269 136)(17 327 105 172 224 270 137)(18 328 106 173 225 271 138)(19 329 107 174 226 272 139)(20 330 108 175 227 273 140)(21 331 109 176 228 274 141)(22 332 110 177 229 275 142)(23 333 111 178 230 276 143)(24 334 112 179 231 277 144)(25 54 298 146 255 86 210)(26 55 299 147 256 87 211)(27 56 300 148 257 88 212)(28 57 301 149 258 89 213)(29 58 302 150 259 90 214)(30 59 303 151 260 91 215)(31 60 304 152 261 92 216)(32 61 305 153 262 93 193)(33 62 306 154 263 94 194)(34 63 307 155 264 95 195)(35 64 308 156 241 96 196)(36 65 309 157 242 73 197)(37 66 310 158 243 74 198)(38 67 311 159 244 75 199)(39 68 312 160 245 76 200)(40 69 289 161 246 77 201)(41 70 290 162 247 78 202)(42 71 291 163 248 79 203)(43 72 292 164 249 80 204)(44 49 293 165 250 81 205)(45 50 294 166 251 82 206)(46 51 295 167 252 83 207)(47 52 296 168 253 84 208)(48 53 297 145 254 85 209)
(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)
(1 166 13 154)(2 165 14 153)(3 164 15 152)(4 163 16 151)(5 162 17 150)(6 161 18 149)(7 160 19 148)(8 159 20 147)(9 158 21 146)(10 157 22 145)(11 156 23 168)(12 155 24 167)(25 240 37 228)(26 239 38 227)(27 238 39 226)(28 237 40 225)(29 236 41 224)(30 235 42 223)(31 234 43 222)(32 233 44 221)(33 232 45 220)(34 231 46 219)(35 230 47 218)(36 229 48 217)(49 267 61 279)(50 266 62 278)(51 265 63 277)(52 288 64 276)(53 287 65 275)(54 286 66 274)(55 285 67 273)(56 284 68 272)(57 283 69 271)(58 282 70 270)(59 281 71 269)(60 280 72 268)(73 110 85 98)(74 109 86 97)(75 108 87 120)(76 107 88 119)(77 106 89 118)(78 105 90 117)(79 104 91 116)(80 103 92 115)(81 102 93 114)(82 101 94 113)(83 100 95 112)(84 99 96 111)(121 294 133 306)(122 293 134 305)(123 292 135 304)(124 291 136 303)(125 290 137 302)(126 289 138 301)(127 312 139 300)(128 311 140 299)(129 310 141 298)(130 309 142 297)(131 308 143 296)(132 307 144 295)(169 193 181 205)(170 216 182 204)(171 215 183 203)(172 214 184 202)(173 213 185 201)(174 212 186 200)(175 211 187 199)(176 210 188 198)(177 209 189 197)(178 208 190 196)(179 207 191 195)(180 206 192 194)(241 333 253 321)(242 332 254 320)(243 331 255 319)(244 330 256 318)(245 329 257 317)(246 328 258 316)(247 327 259 315)(248 326 260 314)(249 325 261 313)(250 324 262 336)(251 323 263 335)(252 322 264 334)

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

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

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

105 conjugacy classes

 class 1 2 3 4A 4B 4C 6 7A ··· 7F 8A 8B 12A 12B 14A ··· 14F 21A ··· 21F 24A 24B 24C 24D 28A ··· 28F 28G ··· 28R 42A ··· 42F 56A ··· 56L 84A ··· 84L 168A ··· 168X order 1 2 3 4 4 4 6 7 ··· 7 8 8 12 12 14 ··· 14 21 ··· 21 24 24 24 24 28 ··· 28 28 ··· 28 42 ··· 42 56 ··· 56 84 ··· 84 168 ··· 168 size 1 1 2 2 12 12 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C7 C14 C14 S3 D4 D6 Q16 D12 S3×C7 Dic12 C7×D4 S3×C14 C7×Q16 C7×D12 C7×Dic12 kernel C7×Dic12 C168 C7×Dic6 Dic12 C24 Dic6 C56 C42 C28 C21 C14 C8 C7 C6 C4 C3 C2 C1 # reps 1 1 2 6 6 12 1 1 1 2 2 6 4 6 6 12 12 24

Matrix representation of C7×Dic12 in GL2(𝔽337) generated by

 8 0 0 8
,
 129 155 182 284
,
 241 91 332 96
G:=sub<GL(2,GF(337))| [8,0,0,8],[129,182,155,284],[241,332,91,96] >;

C7×Dic12 in GAP, Magma, Sage, TeX

C_7\times {\rm Dic}_{12}
% in TeX

G:=Group("C7xDic12");
// GroupNames label

G:=SmallGroup(336,78);
// by ID

G=gap.SmallGroup(336,78);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,336,361,511,2019,69,8069]);
// Polycyclic

G:=Group<a,b,c|a^7=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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