Copied to
clipboard

## G = Dic3×Dic7order 336 = 24·3·7

### Direct product of Dic3 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — Dic3×Dic7
 Chief series C1 — C7 — C21 — C42 — C2×C42 — C6×Dic7 — Dic3×Dic7
 Lower central C21 — Dic3×Dic7
 Upper central C1 — C22

Generators and relations for Dic3×Dic7
G = < a,b,c,d | a6=c14=1, b2=a3, d2=c7, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 220 in 60 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C3, C4, C22, C6, C7, C2×C4, Dic3, Dic3, C12, C2×C6, C14, C42, C21, C2×Dic3, C2×Dic3, C2×C12, Dic7, Dic7, C28, C2×C14, C42, C4×Dic3, C2×Dic7, C2×Dic7, C2×C28, C7×Dic3, C3×Dic7, Dic21, C2×C42, C4×Dic7, C6×Dic7, Dic3×C14, C2×Dic21, Dic3×Dic7
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, D7, C42, C4×S3, C2×Dic3, Dic7, D14, C4×Dic3, C4×D7, C2×Dic7, S3×D7, C4×Dic7, Dic3×D7, S3×Dic7, D21⋊C4, Dic3×Dic7

Smallest permutation representation of Dic3×Dic7
Regular action on 336 points
Generators in S336
(1 305 96 99 167 124)(2 306 97 100 168 125)(3 307 98 101 155 126)(4 308 85 102 156 113)(5 295 86 103 157 114)(6 296 87 104 158 115)(7 297 88 105 159 116)(8 298 89 106 160 117)(9 299 90 107 161 118)(10 300 91 108 162 119)(11 301 92 109 163 120)(12 302 93 110 164 121)(13 303 94 111 165 122)(14 304 95 112 166 123)(15 280 181 314 29 48)(16 267 182 315 30 49)(17 268 169 316 31 50)(18 269 170 317 32 51)(19 270 171 318 33 52)(20 271 172 319 34 53)(21 272 173 320 35 54)(22 273 174 321 36 55)(23 274 175 322 37 56)(24 275 176 309 38 43)(25 276 177 310 39 44)(26 277 178 311 40 45)(27 278 179 312 41 46)(28 279 180 313 42 47)(57 149 326 243 230 185)(58 150 327 244 231 186)(59 151 328 245 232 187)(60 152 329 246 233 188)(61 153 330 247 234 189)(62 154 331 248 235 190)(63 141 332 249 236 191)(64 142 333 250 237 192)(65 143 334 251 238 193)(66 144 335 252 225 194)(67 145 336 239 226 195)(68 146 323 240 227 196)(69 147 324 241 228 183)(70 148 325 242 229 184)(71 132 217 291 258 210)(72 133 218 292 259 197)(73 134 219 293 260 198)(74 135 220 294 261 199)(75 136 221 281 262 200)(76 137 222 282 263 201)(77 138 223 283 264 202)(78 139 224 284 265 203)(79 140 211 285 266 204)(80 127 212 286 253 205)(81 128 213 287 254 206)(82 129 214 288 255 207)(83 130 215 289 256 208)(84 131 216 290 257 209)
(1 57 99 243)(2 58 100 244)(3 59 101 245)(4 60 102 246)(5 61 103 247)(6 62 104 248)(7 63 105 249)(8 64 106 250)(9 65 107 251)(10 66 108 252)(11 67 109 239)(12 68 110 240)(13 69 111 241)(14 70 112 242)(15 289 314 83)(16 290 315 84)(17 291 316 71)(18 292 317 72)(19 293 318 73)(20 294 319 74)(21 281 320 75)(22 282 321 76)(23 283 322 77)(24 284 309 78)(25 285 310 79)(26 286 311 80)(27 287 312 81)(28 288 313 82)(29 208 280 215)(30 209 267 216)(31 210 268 217)(32 197 269 218)(33 198 270 219)(34 199 271 220)(35 200 272 221)(36 201 273 222)(37 202 274 223)(38 203 275 224)(39 204 276 211)(40 205 277 212)(41 206 278 213)(42 207 279 214)(43 265 176 139)(44 266 177 140)(45 253 178 127)(46 254 179 128)(47 255 180 129)(48 256 181 130)(49 257 182 131)(50 258 169 132)(51 259 170 133)(52 260 171 134)(53 261 172 135)(54 262 173 136)(55 263 174 137)(56 264 175 138)(85 233 113 152)(86 234 114 153)(87 235 115 154)(88 236 116 141)(89 237 117 142)(90 238 118 143)(91 225 119 144)(92 226 120 145)(93 227 121 146)(94 228 122 147)(95 229 123 148)(96 230 124 149)(97 231 125 150)(98 232 126 151)(155 328 307 187)(156 329 308 188)(157 330 295 189)(158 331 296 190)(159 332 297 191)(160 333 298 192)(161 334 299 193)(162 335 300 194)(163 336 301 195)(164 323 302 196)(165 324 303 183)(166 325 304 184)(167 326 305 185)(168 327 306 186)
(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)
(1 313 8 320)(2 312 9 319)(3 311 10 318)(4 310 11 317)(5 309 12 316)(6 322 13 315)(7 321 14 314)(15 105 22 112)(16 104 23 111)(17 103 24 110)(18 102 25 109)(19 101 26 108)(20 100 27 107)(21 99 28 106)(29 297 36 304)(30 296 37 303)(31 295 38 302)(32 308 39 301)(33 307 40 300)(34 306 41 299)(35 305 42 298)(43 93 50 86)(44 92 51 85)(45 91 52 98)(46 90 53 97)(47 89 54 96)(48 88 55 95)(49 87 56 94)(57 82 64 75)(58 81 65 74)(59 80 66 73)(60 79 67 72)(61 78 68 71)(62 77 69 84)(63 76 70 83)(113 177 120 170)(114 176 121 169)(115 175 122 182)(116 174 123 181)(117 173 124 180)(118 172 125 179)(119 171 126 178)(127 144 134 151)(128 143 135 150)(129 142 136 149)(130 141 137 148)(131 154 138 147)(132 153 139 146)(133 152 140 145)(155 277 162 270)(156 276 163 269)(157 275 164 268)(158 274 165 267)(159 273 166 280)(160 272 167 279)(161 271 168 278)(183 209 190 202)(184 208 191 201)(185 207 192 200)(186 206 193 199)(187 205 194 198)(188 204 195 197)(189 203 196 210)(211 336 218 329)(212 335 219 328)(213 334 220 327)(214 333 221 326)(215 332 222 325)(216 331 223 324)(217 330 224 323)(225 260 232 253)(226 259 233 266)(227 258 234 265)(228 257 235 264)(229 256 236 263)(230 255 237 262)(231 254 238 261)(239 292 246 285)(240 291 247 284)(241 290 248 283)(242 289 249 282)(243 288 250 281)(244 287 251 294)(245 286 252 293)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 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 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 3 3 3 3 7 7 7 7 21 21 21 21 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 4 4 6 ··· 6 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - + + - - + image C1 C2 C2 C2 C4 C4 C4 S3 Dic3 D6 D7 C4×S3 Dic7 D14 C4×D7 S3×D7 Dic3×D7 S3×Dic7 D21⋊C4 kernel Dic3×Dic7 C6×Dic7 Dic3×C14 C2×Dic21 C7×Dic3 C3×Dic7 Dic21 C2×Dic7 Dic7 C2×C14 C2×Dic3 C14 Dic3 C2×C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 4 4 1 2 1 3 4 6 3 12 3 3 3 3

Matrix representation of Dic3×Dic7 in GL6(𝔽337)

 336 0 0 0 0 0 0 336 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 336 1 0 0 0 0 336 0
,
 148 0 0 0 0 0 0 148 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 54 271 0 0 0 0 325 283
,
 143 336 0 0 0 0 86 228 0 0 0 0 0 0 0 336 0 0 0 0 1 194 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 54 205 0 0 0 0 234 283 0 0 0 0 0 0 300 264 0 0 0 0 28 37 0 0 0 0 0 0 336 0 0 0 0 0 0 336

G:=sub<GL(6,GF(337))| [336,0,0,0,0,0,0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,336,336,0,0,0,0,1,0],[148,0,0,0,0,0,0,148,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,54,325,0,0,0,0,271,283],[143,86,0,0,0,0,336,228,0,0,0,0,0,0,0,1,0,0,0,0,336,194,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[54,234,0,0,0,0,205,283,0,0,0,0,0,0,300,28,0,0,0,0,264,37,0,0,0,0,0,0,336,0,0,0,0,0,0,336] >;

Dic3×Dic7 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm Dic}_7
% in TeX

G:=Group("Dic3xDic7");
// GroupNames label

G:=SmallGroup(336,41);
// by ID

G=gap.SmallGroup(336,41);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,55,490,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^14=1,b^2=a^3,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽