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G = C7×C3⋊Q16order 336 = 24·3·7

Direct product of C7 and C3⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C7×C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — C84 — C7×Dic6 — C7×C3⋊Q16
 Lower central C3 — C6 — C12 — C7×C3⋊Q16
 Upper central C1 — C14 — C28 — C7×Q8

Generators and relations for C7×C3⋊Q16
G = < a,b,c,d | a7=b3=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Smallest permutation representation of C7×C3⋊Q16
Regular action on 336 points
Generators in S336
(1 253 115 81 191 317 91)(2 254 116 82 192 318 92)(3 255 117 83 185 319 93)(4 256 118 84 186 320 94)(5 249 119 85 187 313 95)(6 250 120 86 188 314 96)(7 251 113 87 189 315 89)(8 252 114 88 190 316 90)(9 108 326 181 67 98 205)(10 109 327 182 68 99 206)(11 110 328 183 69 100 207)(12 111 321 184 70 101 208)(13 112 322 177 71 102 201)(14 105 323 178 72 103 202)(15 106 324 179 65 104 203)(16 107 325 180 66 97 204)(17 300 158 41 231 276 134)(18 301 159 42 232 277 135)(19 302 160 43 225 278 136)(20 303 153 44 226 279 129)(21 304 154 45 227 280 130)(22 297 155 46 228 273 131)(23 298 156 47 229 274 132)(24 299 157 48 230 275 133)(25 209 241 49 233 36 217)(26 210 242 50 234 37 218)(27 211 243 51 235 38 219)(28 212 244 52 236 39 220)(29 213 245 53 237 40 221)(30 214 246 54 238 33 222)(31 215 247 55 239 34 223)(32 216 248 56 240 35 224)(57 262 307 331 197 78 170)(58 263 308 332 198 79 171)(59 264 309 333 199 80 172)(60 257 310 334 200 73 173)(61 258 311 335 193 74 174)(62 259 312 336 194 75 175)(63 260 305 329 195 76 176)(64 261 306 330 196 77 169)(121 165 291 145 283 141 267)(122 166 292 146 284 142 268)(123 167 293 147 285 143 269)(124 168 294 148 286 144 270)(125 161 295 149 287 137 271)(126 162 296 150 288 138 272)(127 163 289 151 281 139 265)(128 164 290 152 282 140 266)
(1 98 57)(2 58 99)(3 100 59)(4 60 101)(5 102 61)(6 62 103)(7 104 63)(8 64 97)(9 307 115)(10 116 308)(11 309 117)(12 118 310)(13 311 119)(14 120 312)(15 305 113)(16 114 306)(17 212 125)(18 126 213)(19 214 127)(20 128 215)(21 216 121)(22 122 209)(23 210 123)(24 124 211)(25 131 268)(26 269 132)(27 133 270)(28 271 134)(29 135 272)(30 265 136)(31 129 266)(32 267 130)(33 281 225)(34 226 282)(35 283 227)(36 228 284)(37 285 229)(38 230 286)(39 287 231)(40 232 288)(41 236 149)(42 150 237)(43 238 151)(44 152 239)(45 240 145)(46 146 233)(47 234 147)(48 148 235)(49 155 292)(50 293 156)(51 157 294)(52 295 158)(53 159 296)(54 289 160)(55 153 290)(56 291 154)(65 176 89)(66 90 169)(67 170 91)(68 92 171)(69 172 93)(70 94 173)(71 174 95)(72 96 175)(73 184 320)(74 313 177)(75 178 314)(76 315 179)(77 180 316)(78 317 181)(79 182 318)(80 319 183)(81 108 331)(82 332 109)(83 110 333)(84 334 111)(85 112 335)(86 336 105)(87 106 329)(88 330 107)(137 276 220)(138 221 277)(139 278 222)(140 223 279)(141 280 224)(142 217 273)(143 274 218)(144 219 275)(161 300 244)(162 245 301)(163 302 246)(164 247 303)(165 304 248)(166 241 297)(167 298 242)(168 243 299)(185 328 199)(186 200 321)(187 322 193)(188 194 323)(189 324 195)(190 196 325)(191 326 197)(192 198 327)(201 258 249)(202 250 259)(203 260 251)(204 252 261)(205 262 253)(206 254 263)(207 264 255)(208 256 257)
(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)
(1 123 5 127)(2 122 6 126)(3 121 7 125)(4 128 8 124)(9 156 13 160)(10 155 14 159)(11 154 15 158)(12 153 16 157)(17 100 21 104)(18 99 22 103)(19 98 23 102)(20 97 24 101)(25 175 29 171)(26 174 30 170)(27 173 31 169)(28 172 32 176)(33 197 37 193)(34 196 38 200)(35 195 39 199)(36 194 40 198)(41 110 45 106)(42 109 46 105)(43 108 47 112)(44 107 48 111)(49 312 53 308)(50 311 54 307)(51 310 55 306)(52 309 56 305)(57 210 61 214)(58 209 62 213)(59 216 63 212)(60 215 64 211)(65 134 69 130)(66 133 70 129)(67 132 71 136)(68 131 72 135)(73 223 77 219)(74 222 78 218)(75 221 79 217)(76 220 80 224)(81 147 85 151)(82 146 86 150)(83 145 87 149)(84 152 88 148)(89 271 93 267)(90 270 94 266)(91 269 95 265)(92 268 96 272)(113 295 117 291)(114 294 118 290)(115 293 119 289)(116 292 120 296)(137 319 141 315)(138 318 142 314)(139 317 143 313)(140 316 144 320)(161 255 165 251)(162 254 166 250)(163 253 167 249)(164 252 168 256)(177 278 181 274)(178 277 182 273)(179 276 183 280)(180 275 184 279)(185 283 189 287)(186 282 190 286)(187 281 191 285)(188 288 192 284)(201 302 205 298)(202 301 206 297)(203 300 207 304)(204 299 208 303)(225 326 229 322)(226 325 230 321)(227 324 231 328)(228 323 232 327)(233 336 237 332)(234 335 238 331)(235 334 239 330)(236 333 240 329)(241 259 245 263)(242 258 246 262)(243 257 247 261)(244 264 248 260)

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

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

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

84 conjugacy classes

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

84 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 C7 C14 C14 C14 S3 D4 D6 Q16 C3⋊D4 S3×C7 C7×D4 S3×C14 C7×Q16 C7×C3⋊D4 C3⋊Q16 C7×C3⋊Q16 kernel C7×C3⋊Q16 C7×C3⋊C8 C7×Dic6 Q8×C21 C3⋊Q16 C3⋊C8 Dic6 C3×Q8 C7×Q8 C42 C28 C21 C14 Q8 C6 C4 C3 C2 C7 C1 # reps 1 1 1 1 6 6 6 6 1 1 1 2 2 6 6 6 12 12 1 6

Matrix representation of C7×C3⋊Q16 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 52 0 0 0 0 52
,
 336 336 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 148 0 0 0 189 189 0 0 0 0 13 324 0 0 13 13
,
 139 278 0 0 59 198 0 0 0 0 314 12 0 0 12 23
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,52,0,0,0,0,52],[336,1,0,0,336,0,0,0,0,0,1,0,0,0,0,1],[148,189,0,0,0,189,0,0,0,0,13,13,0,0,324,13],[139,59,0,0,278,198,0,0,0,0,314,12,0,0,12,23] >;

C7×C3⋊Q16 in GAP, Magma, Sage, TeX

C_7\times C_3\rtimes Q_{16}
% in TeX

G:=Group("C7xC3:Q16");
// GroupNames label

G:=SmallGroup(336,88);
// by ID

G=gap.SmallGroup(336,88);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-3,336,361,343,2019,1017,69,8069]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^3=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

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