direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C7⋊Q16, C21⋊8Q16, C42.43D4, C12.39D14, C84.39C22, Dic14.4C6, C7⋊C8.2C6, C7⋊5(C3×Q16), C4.4(C6×D7), Q8.2(C3×D7), (C3×Q8).2D7, (C7×Q8).7C6, C28.25(C2×C6), C14.26(C3×D4), (Q8×C21).2C2, C6.26(C7⋊D4), (C3×Dic14).4C2, (C3×C7⋊C8).2C2, C2.7(C3×C7⋊D4), SmallGroup(336,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C7⋊Q16
G = < a,b,c,d | a3=b7=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Smallest permutation representation of C3×C7⋊Q16
►Regular action on 336 points
Generators in S336
(1 196 333)(2 197 334)(3 198 335)(4 199 336)(5 200 329)(6 193 330)(7 194 331)(8 195 332)(9 102 317)(10 103 318)(11 104 319)(12 97 320)(13 98 313)(14 99 314)(15 100 315)(16 101 316)(17 173 205)(18 174 206)(19 175 207)(20 176 208)(21 169 201)(22 170 202)(23 171 203)(24 172 204)(25 116 140)(26 117 141)(27 118 142)(28 119 143)(29 120 144)(30 113 137)(31 114 138)(32 115 139)(33 307 148)(34 308 149)(35 309 150)(36 310 151)(37 311 152)(38 312 145)(39 305 146)(40 306 147)(41 326 163)(42 327 164)(43 328 165)(44 321 166)(45 322 167)(46 323 168)(47 324 161)(48 325 162)(49 133 157)(50 134 158)(51 135 159)(52 136 160)(53 129 153)(54 130 154)(55 131 155)(56 132 156)(57 262 179)(58 263 180)(59 264 181)(60 257 182)(61 258 183)(62 259 184)(63 260 177)(64 261 178)(65 225 293)(66 226 294)(67 227 295)(68 228 296)(69 229 289)(70 230 290)(71 231 291)(72 232 292)(73 278 111)(74 279 112)(75 280 105)(76 273 106)(77 274 107)(78 275 108)(79 276 109)(80 277 110)(81 272 188)(82 265 189)(83 266 190)(84 267 191)(85 268 192)(86 269 185)(87 270 186)(88 271 187)(89 121 249)(90 122 250)(91 123 251)(92 124 252)(93 125 253)(94 126 254)(95 127 255)(96 128 256)(209 233 287)(210 234 288)(211 235 281)(212 236 282)(213 237 283)(214 238 284)(215 239 285)(216 240 286)(217 241 302)(218 242 303)(219 243 304)(220 244 297)(221 245 298)(222 246 299)(223 247 300)(224 248 301)
(1 92 173 78 60 314 87)(2 88 315 61 79 174 93)(3 94 175 80 62 316 81)(4 82 317 63 73 176 95)(5 96 169 74 64 318 83)(6 84 319 57 75 170 89)(7 90 171 76 58 320 85)(8 86 313 59 77 172 91)(9 260 278 208 127 199 265)(10 266 200 128 201 279 261)(11 262 280 202 121 193 267)(12 268 194 122 203 273 263)(13 264 274 204 123 195 269)(14 270 196 124 205 275 257)(15 258 276 206 125 197 271)(16 272 198 126 207 277 259)(17 108 182 99 186 333 252)(18 253 334 187 100 183 109)(19 110 184 101 188 335 254)(20 255 336 189 102 177 111)(21 112 178 103 190 329 256)(22 249 330 191 104 179 105)(23 106 180 97 192 331 250)(24 251 332 185 98 181 107)(25 167 226 151 304 56 281)(26 282 49 297 152 227 168)(27 161 228 145 298 50 283)(28 284 51 299 146 229 162)(29 163 230 147 300 52 285)(30 286 53 301 148 231 164)(31 165 232 149 302 54 287)(32 288 55 303 150 225 166)(33 291 42 113 216 129 224)(34 217 130 209 114 43 292)(35 293 44 115 210 131 218)(36 219 132 211 116 45 294)(37 295 46 117 212 133 220)(38 221 134 213 118 47 296)(39 289 48 119 214 135 222)(40 223 136 215 120 41 290)(65 321 139 234 155 242 309)(66 310 243 156 235 140 322)(67 323 141 236 157 244 311)(68 312 245 158 237 142 324)(69 325 143 238 159 246 305)(70 306 247 160 239 144 326)(71 327 137 240 153 248 307)(72 308 241 154 233 138 328)
(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 35 5 39)(2 34 6 38)(3 33 7 37)(4 40 8 36)(9 160 13 156)(10 159 14 155)(11 158 15 154)(12 157 16 153)(17 166 21 162)(18 165 22 161)(19 164 23 168)(20 163 24 167)(25 111 29 107)(26 110 30 106)(27 109 31 105)(28 108 32 112)(41 172 45 176)(42 171 46 175)(43 170 47 174)(44 169 48 173)(49 101 53 97)(50 100 54 104)(51 99 55 103)(52 98 56 102)(57 213 61 209)(58 212 62 216)(59 211 63 215)(60 210 64 214)(65 128 69 124)(66 127 70 123)(67 126 71 122)(68 125 72 121)(73 120 77 116)(74 119 78 115)(75 118 79 114)(76 117 80 113)(81 224 85 220)(82 223 86 219)(83 222 87 218)(84 221 88 217)(89 296 93 292)(90 295 94 291)(91 294 95 290)(92 293 96 289)(129 320 133 316)(130 319 134 315)(131 318 135 314)(132 317 136 313)(137 273 141 277)(138 280 142 276)(139 279 143 275)(140 278 144 274)(145 334 149 330)(146 333 150 329)(147 332 151 336)(148 331 152 335)(177 285 181 281)(178 284 182 288)(179 283 183 287)(180 282 184 286)(185 304 189 300)(186 303 190 299)(187 302 191 298)(188 301 192 297)(193 312 197 308)(194 311 198 307)(195 310 199 306)(196 309 200 305)(201 325 205 321)(202 324 206 328)(203 323 207 327)(204 322 208 326)(225 256 229 252)(226 255 230 251)(227 254 231 250)(228 253 232 249)(233 262 237 258)(234 261 238 257)(235 260 239 264)(236 259 240 263)(241 267 245 271)(242 266 246 270)(243 265 247 269)(244 272 248 268)
G:=sub<Sym(336)| (1,196,333)(2,197,334)(3,198,335)(4,199,336)(5,200,329)(6,193,330)(7,194,331)(8,195,332)(9,102,317)(10,103,318)(11,104,319)(12,97,320)(13,98,313)(14,99,314)(15,100,315)(16,101,316)(17,173,205)(18,174,206)(19,175,207)(20,176,208)(21,169,201)(22,170,202)(23,171,203)(24,172,204)(25,116,140)(26,117,141)(27,118,142)(28,119,143)(29,120,144)(30,113,137)(31,114,138)(32,115,139)(33,307,148)(34,308,149)(35,309,150)(36,310,151)(37,311,152)(38,312,145)(39,305,146)(40,306,147)(41,326,163)(42,327,164)(43,328,165)(44,321,166)(45,322,167)(46,323,168)(47,324,161)(48,325,162)(49,133,157)(50,134,158)(51,135,159)(52,136,160)(53,129,153)(54,130,154)(55,131,155)(56,132,156)(57,262,179)(58,263,180)(59,264,181)(60,257,182)(61,258,183)(62,259,184)(63,260,177)(64,261,178)(65,225,293)(66,226,294)(67,227,295)(68,228,296)(69,229,289)(70,230,290)(71,231,291)(72,232,292)(73,278,111)(74,279,112)(75,280,105)(76,273,106)(77,274,107)(78,275,108)(79,276,109)(80,277,110)(81,272,188)(82,265,189)(83,266,190)(84,267,191)(85,268,192)(86,269,185)(87,270,186)(88,271,187)(89,121,249)(90,122,250)(91,123,251)(92,124,252)(93,125,253)(94,126,254)(95,127,255)(96,128,256)(209,233,287)(210,234,288)(211,235,281)(212,236,282)(213,237,283)(214,238,284)(215,239,285)(216,240,286)(217,241,302)(218,242,303)(219,243,304)(220,244,297)(221,245,298)(222,246,299)(223,247,300)(224,248,301), (1,92,173,78,60,314,87)(2,88,315,61,79,174,93)(3,94,175,80,62,316,81)(4,82,317,63,73,176,95)(5,96,169,74,64,318,83)(6,84,319,57,75,170,89)(7,90,171,76,58,320,85)(8,86,313,59,77,172,91)(9,260,278,208,127,199,265)(10,266,200,128,201,279,261)(11,262,280,202,121,193,267)(12,268,194,122,203,273,263)(13,264,274,204,123,195,269)(14,270,196,124,205,275,257)(15,258,276,206,125,197,271)(16,272,198,126,207,277,259)(17,108,182,99,186,333,252)(18,253,334,187,100,183,109)(19,110,184,101,188,335,254)(20,255,336,189,102,177,111)(21,112,178,103,190,329,256)(22,249,330,191,104,179,105)(23,106,180,97,192,331,250)(24,251,332,185,98,181,107)(25,167,226,151,304,56,281)(26,282,49,297,152,227,168)(27,161,228,145,298,50,283)(28,284,51,299,146,229,162)(29,163,230,147,300,52,285)(30,286,53,301,148,231,164)(31,165,232,149,302,54,287)(32,288,55,303,150,225,166)(33,291,42,113,216,129,224)(34,217,130,209,114,43,292)(35,293,44,115,210,131,218)(36,219,132,211,116,45,294)(37,295,46,117,212,133,220)(38,221,134,213,118,47,296)(39,289,48,119,214,135,222)(40,223,136,215,120,41,290)(65,321,139,234,155,242,309)(66,310,243,156,235,140,322)(67,323,141,236,157,244,311)(68,312,245,158,237,142,324)(69,325,143,238,159,246,305)(70,306,247,160,239,144,326)(71,327,137,240,153,248,307)(72,308,241,154,233,138,328), (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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,160,13,156)(10,159,14,155)(11,158,15,154)(12,157,16,153)(17,166,21,162)(18,165,22,161)(19,164,23,168)(20,163,24,167)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(41,172,45,176)(42,171,46,175)(43,170,47,174)(44,169,48,173)(49,101,53,97)(50,100,54,104)(51,99,55,103)(52,98,56,102)(57,213,61,209)(58,212,62,216)(59,211,63,215)(60,210,64,214)(65,128,69,124)(66,127,70,123)(67,126,71,122)(68,125,72,121)(73,120,77,116)(74,119,78,115)(75,118,79,114)(76,117,80,113)(81,224,85,220)(82,223,86,219)(83,222,87,218)(84,221,88,217)(89,296,93,292)(90,295,94,291)(91,294,95,290)(92,293,96,289)(129,320,133,316)(130,319,134,315)(131,318,135,314)(132,317,136,313)(137,273,141,277)(138,280,142,276)(139,279,143,275)(140,278,144,274)(145,334,149,330)(146,333,150,329)(147,332,151,336)(148,331,152,335)(177,285,181,281)(178,284,182,288)(179,283,183,287)(180,282,184,286)(185,304,189,300)(186,303,190,299)(187,302,191,298)(188,301,192,297)(193,312,197,308)(194,311,198,307)(195,310,199,306)(196,309,200,305)(201,325,205,321)(202,324,206,328)(203,323,207,327)(204,322,208,326)(225,256,229,252)(226,255,230,251)(227,254,231,250)(228,253,232,249)(233,262,237,258)(234,261,238,257)(235,260,239,264)(236,259,240,263)(241,267,245,271)(242,266,246,270)(243,265,247,269)(244,272,248,268)>;
G:=Group( (1,196,333)(2,197,334)(3,198,335)(4,199,336)(5,200,329)(6,193,330)(7,194,331)(8,195,332)(9,102,317)(10,103,318)(11,104,319)(12,97,320)(13,98,313)(14,99,314)(15,100,315)(16,101,316)(17,173,205)(18,174,206)(19,175,207)(20,176,208)(21,169,201)(22,170,202)(23,171,203)(24,172,204)(25,116,140)(26,117,141)(27,118,142)(28,119,143)(29,120,144)(30,113,137)(31,114,138)(32,115,139)(33,307,148)(34,308,149)(35,309,150)(36,310,151)(37,311,152)(38,312,145)(39,305,146)(40,306,147)(41,326,163)(42,327,164)(43,328,165)(44,321,166)(45,322,167)(46,323,168)(47,324,161)(48,325,162)(49,133,157)(50,134,158)(51,135,159)(52,136,160)(53,129,153)(54,130,154)(55,131,155)(56,132,156)(57,262,179)(58,263,180)(59,264,181)(60,257,182)(61,258,183)(62,259,184)(63,260,177)(64,261,178)(65,225,293)(66,226,294)(67,227,295)(68,228,296)(69,229,289)(70,230,290)(71,231,291)(72,232,292)(73,278,111)(74,279,112)(75,280,105)(76,273,106)(77,274,107)(78,275,108)(79,276,109)(80,277,110)(81,272,188)(82,265,189)(83,266,190)(84,267,191)(85,268,192)(86,269,185)(87,270,186)(88,271,187)(89,121,249)(90,122,250)(91,123,251)(92,124,252)(93,125,253)(94,126,254)(95,127,255)(96,128,256)(209,233,287)(210,234,288)(211,235,281)(212,236,282)(213,237,283)(214,238,284)(215,239,285)(216,240,286)(217,241,302)(218,242,303)(219,243,304)(220,244,297)(221,245,298)(222,246,299)(223,247,300)(224,248,301), (1,92,173,78,60,314,87)(2,88,315,61,79,174,93)(3,94,175,80,62,316,81)(4,82,317,63,73,176,95)(5,96,169,74,64,318,83)(6,84,319,57,75,170,89)(7,90,171,76,58,320,85)(8,86,313,59,77,172,91)(9,260,278,208,127,199,265)(10,266,200,128,201,279,261)(11,262,280,202,121,193,267)(12,268,194,122,203,273,263)(13,264,274,204,123,195,269)(14,270,196,124,205,275,257)(15,258,276,206,125,197,271)(16,272,198,126,207,277,259)(17,108,182,99,186,333,252)(18,253,334,187,100,183,109)(19,110,184,101,188,335,254)(20,255,336,189,102,177,111)(21,112,178,103,190,329,256)(22,249,330,191,104,179,105)(23,106,180,97,192,331,250)(24,251,332,185,98,181,107)(25,167,226,151,304,56,281)(26,282,49,297,152,227,168)(27,161,228,145,298,50,283)(28,284,51,299,146,229,162)(29,163,230,147,300,52,285)(30,286,53,301,148,231,164)(31,165,232,149,302,54,287)(32,288,55,303,150,225,166)(33,291,42,113,216,129,224)(34,217,130,209,114,43,292)(35,293,44,115,210,131,218)(36,219,132,211,116,45,294)(37,295,46,117,212,133,220)(38,221,134,213,118,47,296)(39,289,48,119,214,135,222)(40,223,136,215,120,41,290)(65,321,139,234,155,242,309)(66,310,243,156,235,140,322)(67,323,141,236,157,244,311)(68,312,245,158,237,142,324)(69,325,143,238,159,246,305)(70,306,247,160,239,144,326)(71,327,137,240,153,248,307)(72,308,241,154,233,138,328), (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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,160,13,156)(10,159,14,155)(11,158,15,154)(12,157,16,153)(17,166,21,162)(18,165,22,161)(19,164,23,168)(20,163,24,167)(25,111,29,107)(26,110,30,106)(27,109,31,105)(28,108,32,112)(41,172,45,176)(42,171,46,175)(43,170,47,174)(44,169,48,173)(49,101,53,97)(50,100,54,104)(51,99,55,103)(52,98,56,102)(57,213,61,209)(58,212,62,216)(59,211,63,215)(60,210,64,214)(65,128,69,124)(66,127,70,123)(67,126,71,122)(68,125,72,121)(73,120,77,116)(74,119,78,115)(75,118,79,114)(76,117,80,113)(81,224,85,220)(82,223,86,219)(83,222,87,218)(84,221,88,217)(89,296,93,292)(90,295,94,291)(91,294,95,290)(92,293,96,289)(129,320,133,316)(130,319,134,315)(131,318,135,314)(132,317,136,313)(137,273,141,277)(138,280,142,276)(139,279,143,275)(140,278,144,274)(145,334,149,330)(146,333,150,329)(147,332,151,336)(148,331,152,335)(177,285,181,281)(178,284,182,288)(179,283,183,287)(180,282,184,286)(185,304,189,300)(186,303,190,299)(187,302,191,298)(188,301,192,297)(193,312,197,308)(194,311,198,307)(195,310,199,306)(196,309,200,305)(201,325,205,321)(202,324,206,328)(203,323,207,327)(204,322,208,326)(225,256,229,252)(226,255,230,251)(227,254,231,250)(228,253,232,249)(233,262,237,258)(234,261,238,257)(235,260,239,264)(236,259,240,263)(241,267,245,271)(242,266,246,270)(243,265,247,269)(244,272,248,268) );
G=PermutationGroup([[(1,196,333),(2,197,334),(3,198,335),(4,199,336),(5,200,329),(6,193,330),(7,194,331),(8,195,332),(9,102,317),(10,103,318),(11,104,319),(12,97,320),(13,98,313),(14,99,314),(15,100,315),(16,101,316),(17,173,205),(18,174,206),(19,175,207),(20,176,208),(21,169,201),(22,170,202),(23,171,203),(24,172,204),(25,116,140),(26,117,141),(27,118,142),(28,119,143),(29,120,144),(30,113,137),(31,114,138),(32,115,139),(33,307,148),(34,308,149),(35,309,150),(36,310,151),(37,311,152),(38,312,145),(39,305,146),(40,306,147),(41,326,163),(42,327,164),(43,328,165),(44,321,166),(45,322,167),(46,323,168),(47,324,161),(48,325,162),(49,133,157),(50,134,158),(51,135,159),(52,136,160),(53,129,153),(54,130,154),(55,131,155),(56,132,156),(57,262,179),(58,263,180),(59,264,181),(60,257,182),(61,258,183),(62,259,184),(63,260,177),(64,261,178),(65,225,293),(66,226,294),(67,227,295),(68,228,296),(69,229,289),(70,230,290),(71,231,291),(72,232,292),(73,278,111),(74,279,112),(75,280,105),(76,273,106),(77,274,107),(78,275,108),(79,276,109),(80,277,110),(81,272,188),(82,265,189),(83,266,190),(84,267,191),(85,268,192),(86,269,185),(87,270,186),(88,271,187),(89,121,249),(90,122,250),(91,123,251),(92,124,252),(93,125,253),(94,126,254),(95,127,255),(96,128,256),(209,233,287),(210,234,288),(211,235,281),(212,236,282),(213,237,283),(214,238,284),(215,239,285),(216,240,286),(217,241,302),(218,242,303),(219,243,304),(220,244,297),(221,245,298),(222,246,299),(223,247,300),(224,248,301)], [(1,92,173,78,60,314,87),(2,88,315,61,79,174,93),(3,94,175,80,62,316,81),(4,82,317,63,73,176,95),(5,96,169,74,64,318,83),(6,84,319,57,75,170,89),(7,90,171,76,58,320,85),(8,86,313,59,77,172,91),(9,260,278,208,127,199,265),(10,266,200,128,201,279,261),(11,262,280,202,121,193,267),(12,268,194,122,203,273,263),(13,264,274,204,123,195,269),(14,270,196,124,205,275,257),(15,258,276,206,125,197,271),(16,272,198,126,207,277,259),(17,108,182,99,186,333,252),(18,253,334,187,100,183,109),(19,110,184,101,188,335,254),(20,255,336,189,102,177,111),(21,112,178,103,190,329,256),(22,249,330,191,104,179,105),(23,106,180,97,192,331,250),(24,251,332,185,98,181,107),(25,167,226,151,304,56,281),(26,282,49,297,152,227,168),(27,161,228,145,298,50,283),(28,284,51,299,146,229,162),(29,163,230,147,300,52,285),(30,286,53,301,148,231,164),(31,165,232,149,302,54,287),(32,288,55,303,150,225,166),(33,291,42,113,216,129,224),(34,217,130,209,114,43,292),(35,293,44,115,210,131,218),(36,219,132,211,116,45,294),(37,295,46,117,212,133,220),(38,221,134,213,118,47,296),(39,289,48,119,214,135,222),(40,223,136,215,120,41,290),(65,321,139,234,155,242,309),(66,310,243,156,235,140,322),(67,323,141,236,157,244,311),(68,312,245,158,237,142,324),(69,325,143,238,159,246,305),(70,306,247,160,239,144,326),(71,327,137,240,153,248,307),(72,308,241,154,233,138,328)], [(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)], [(1,35,5,39),(2,34,6,38),(3,33,7,37),(4,40,8,36),(9,160,13,156),(10,159,14,155),(11,158,15,154),(12,157,16,153),(17,166,21,162),(18,165,22,161),(19,164,23,168),(20,163,24,167),(25,111,29,107),(26,110,30,106),(27,109,31,105),(28,108,32,112),(41,172,45,176),(42,171,46,175),(43,170,47,174),(44,169,48,173),(49,101,53,97),(50,100,54,104),(51,99,55,103),(52,98,56,102),(57,213,61,209),(58,212,62,216),(59,211,63,215),(60,210,64,214),(65,128,69,124),(66,127,70,123),(67,126,71,122),(68,125,72,121),(73,120,77,116),(74,119,78,115),(75,118,79,114),(76,117,80,113),(81,224,85,220),(82,223,86,219),(83,222,87,218),(84,221,88,217),(89,296,93,292),(90,295,94,291),(91,294,95,290),(92,293,96,289),(129,320,133,316),(130,319,134,315),(131,318,135,314),(132,317,136,313),(137,273,141,277),(138,280,142,276),(139,279,143,275),(140,278,144,274),(145,334,149,330),(146,333,150,329),(147,332,151,336),(148,331,152,335),(177,285,181,281),(178,284,182,288),(179,283,183,287),(180,282,184,286),(185,304,189,300),(186,303,190,299),(187,302,191,298),(188,301,192,297),(193,312,197,308),(194,311,198,307),(195,310,199,306),(196,309,200,305),(201,325,205,321),(202,324,206,328),(203,323,207,327),(204,322,208,326),(225,256,229,252),(226,255,230,251),(227,254,231,250),(228,253,232,249),(233,262,237,258),(234,261,238,257),(235,260,239,264),(236,259,240,263),(241,267,245,271),(242,266,246,270),(243,265,247,269),(244,272,248,268)]])
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28I | 42A | ··· | 42F | 84A | ··· | 84R |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 2 | 4 | 28 | 1 | 1 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D7 | Q16 | C3×D4 | D14 | C3×D7 | C3×Q16 | C7⋊D4 | C6×D7 | C3×C7⋊D4 | C7⋊Q16 | C3×C7⋊Q16 |
| kernel | C3×C7⋊Q16 | C3×C7⋊C8 | C3×Dic14 | Q8×C21 | C7⋊Q16 | C7⋊C8 | Dic14 | C7×Q8 | C42 | C3×Q8 | C21 | C14 | C12 | Q8 | C7 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 3 | 6 |
Matrix representation of C3×C7⋊Q16 ►in GL6(𝔽337)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 208 | 0 | 0 | 0 |
| 0 | 0 | 0 | 208 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 228 | 336 | 0 | 0 | 0 | 0 |
| 229 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 143 | 227 | 0 | 0 | 0 | 0 |
| 143 | 194 | 0 | 0 | 0 | 0 |
| 0 | 0 | 231 | 172 | 0 | 0 |
| 0 | 0 | 15 | 106 | 0 | 0 |
| 0 | 0 | 0 | 0 | 311 | 131 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 336 | 0 | 0 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 7 | 0 | 0 |
| 0 | 0 | 96 | 292 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 304 |
| 0 | 0 | 0 | 0 | 18 | 321 |
G:=sub<GL(6,GF(337))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,208,0,0,0,0,0,0,208,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[228,229,0,0,0,0,336,336,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[143,143,0,0,0,0,227,194,0,0,0,0,0,0,231,15,0,0,0,0,172,106,0,0,0,0,0,0,311,18,0,0,0,0,131,0],[336,0,0,0,0,0,0,336,0,0,0,0,0,0,45,96,0,0,0,0,7,292,0,0,0,0,0,0,16,18,0,0,0,0,304,321] >;
C3×C7⋊Q16 in GAP, Magma, Sage, TeX
C_3\times C_7\rtimes Q_{16} % in TeX
G:=Group("C3xC7:Q16"); // GroupNames label
G:=SmallGroup(336,72);
// by ID
G=gap.SmallGroup(336,72);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,151,867,441,69,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^7=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export