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## G = C2×C7⋊Dic7order 392 = 23·72

### Direct product of C2 and C7⋊Dic7

Aliases: C2×C7⋊Dic7, C14⋊Dic7, C14.15D14, C142.2C2, (C7×C14)⋊3C4, C727(C2×C4), C72(C2×Dic7), (C2×C14).4D7, C22.(C7⋊D7), (C7×C14).14C22, C2.2(C2×C7⋊D7), SmallGroup(392,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C2×C7⋊Dic7
 Chief series C1 — C7 — C72 — C7×C14 — C7⋊Dic7 — C2×C7⋊Dic7
 Lower central C72 — C2×C7⋊Dic7
 Upper central C1 — C22

Generators and relations for C2×C7⋊Dic7
G = < a,b,c,d | a2=b7=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 368 in 80 conjugacy classes, 53 normal (7 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, C14, Dic7, C2×C14, C72, C2×Dic7, C7×C14, C7×C14, C7⋊Dic7, C142, C2×C7⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D7, Dic7, D14, C2×Dic7, C7⋊D7, C7⋊Dic7, C2×C7⋊D7, C2×C7⋊Dic7

Smallest permutation representation of C2×C7⋊Dic7
Regular action on 392 points
Generators in S392
(1 199)(2 200)(3 201)(4 202)(5 203)(6 204)(7 205)(8 206)(9 207)(10 208)(11 209)(12 210)(13 197)(14 198)(15 290)(16 291)(17 292)(18 293)(19 294)(20 281)(21 282)(22 283)(23 284)(24 285)(25 286)(26 287)(27 288)(28 289)(29 98)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 385)(44 386)(45 387)(46 388)(47 389)(48 390)(49 391)(50 392)(51 379)(52 380)(53 381)(54 382)(55 383)(56 384)(57 349)(58 350)(59 337)(60 338)(61 339)(62 340)(63 341)(64 342)(65 343)(66 344)(67 345)(68 346)(69 347)(70 348)(71 177)(72 178)(73 179)(74 180)(75 181)(76 182)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(99 193)(100 194)(101 195)(102 196)(103 183)(104 184)(105 185)(106 186)(107 187)(108 188)(109 189)(110 190)(111 191)(112 192)(113 316)(114 317)(115 318)(116 319)(117 320)(118 321)(119 322)(120 309)(121 310)(122 311)(123 312)(124 313)(125 314)(126 315)(127 263)(128 264)(129 265)(130 266)(131 253)(132 254)(133 255)(134 256)(135 257)(136 258)(137 259)(138 260)(139 261)(140 262)(141 333)(142 334)(143 335)(144 336)(145 323)(146 324)(147 325)(148 326)(149 327)(150 328)(151 329)(152 330)(153 331)(154 332)(155 268)(156 269)(157 270)(158 271)(159 272)(160 273)(161 274)(162 275)(163 276)(164 277)(165 278)(166 279)(167 280)(168 267)(211 305)(212 306)(213 307)(214 308)(215 295)(216 296)(217 297)(218 298)(219 299)(220 300)(221 301)(222 302)(223 303)(224 304)(225 353)(226 354)(227 355)(228 356)(229 357)(230 358)(231 359)(232 360)(233 361)(234 362)(235 363)(236 364)(237 351)(238 352)(239 369)(240 370)(241 371)(242 372)(243 373)(244 374)(245 375)(246 376)(247 377)(248 378)(249 365)(250 366)(251 367)(252 368)
(1 217 292 86 370 267 134)(2 218 293 87 371 268 135)(3 219 294 88 372 269 136)(4 220 281 89 373 270 137)(5 221 282 90 374 271 138)(6 222 283 91 375 272 139)(7 223 284 92 376 273 140)(8 224 285 93 377 274 127)(9 211 286 94 378 275 128)(10 212 287 95 365 276 129)(11 213 288 96 366 277 130)(12 214 289 97 367 278 131)(13 215 290 98 368 279 132)(14 216 291 85 369 280 133)(15 29 252 166 254 197 295)(16 30 239 167 255 198 296)(17 31 240 168 256 199 297)(18 32 241 155 257 200 298)(19 33 242 156 258 201 299)(20 34 243 157 259 202 300)(21 35 244 158 260 203 301)(22 36 245 159 261 204 302)(23 37 246 160 262 205 303)(24 38 247 161 263 206 304)(25 39 248 162 264 207 305)(26 40 249 163 265 208 306)(27 41 250 164 266 209 307)(28 42 251 165 253 210 308)(43 121 360 77 152 100 62)(44 122 361 78 153 101 63)(45 123 362 79 154 102 64)(46 124 363 80 141 103 65)(47 125 364 81 142 104 66)(48 126 351 82 143 105 67)(49 113 352 83 144 106 68)(50 114 353 84 145 107 69)(51 115 354 71 146 108 70)(52 116 355 72 147 109 57)(53 117 356 73 148 110 58)(54 118 357 74 149 111 59)(55 119 358 75 150 112 60)(56 120 359 76 151 99 61)(169 330 194 340 385 310 232)(170 331 195 341 386 311 233)(171 332 196 342 387 312 234)(172 333 183 343 388 313 235)(173 334 184 344 389 314 236)(174 335 185 345 390 315 237)(175 336 186 346 391 316 238)(176 323 187 347 392 317 225)(177 324 188 348 379 318 226)(178 325 189 349 380 319 227)(179 326 190 350 381 320 228)(180 327 191 337 382 321 229)(181 328 192 338 383 322 230)(182 329 193 339 384 309 231)
(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)(337 338 339 340 341 342 343 344 345 346 347 348 349 350)(351 352 353 354 355 356 357 358 359 360 361 362 363 364)(365 366 367 368 369 370 371 372 373 374 375 376 377 378)(379 380 381 382 383 384 385 386 387 388 389 390 391 392)
(1 320 8 313)(2 319 9 312)(3 318 10 311)(4 317 11 310)(5 316 12 309)(6 315 13 322)(7 314 14 321)(15 60 22 67)(16 59 23 66)(17 58 24 65)(18 57 25 64)(19 70 26 63)(20 69 27 62)(21 68 28 61)(29 112 36 105)(30 111 37 104)(31 110 38 103)(32 109 39 102)(33 108 40 101)(34 107 41 100)(35 106 42 99)(43 300 50 307)(44 299 51 306)(45 298 52 305)(46 297 53 304)(47 296 54 303)(48 295 55 302)(49 308 56 301)(71 163 78 156)(72 162 79 155)(73 161 80 168)(74 160 81 167)(75 159 82 166)(76 158 83 165)(77 157 84 164)(85 191 92 184)(86 190 93 183)(87 189 94 196)(88 188 95 195)(89 187 96 194)(90 186 97 193)(91 185 98 192)(113 210 120 203)(114 209 121 202)(115 208 122 201)(116 207 123 200)(117 206 124 199)(118 205 125 198)(119 204 126 197)(127 235 134 228)(128 234 135 227)(129 233 136 226)(130 232 137 225)(131 231 138 238)(132 230 139 237)(133 229 140 236)(141 240 148 247)(142 239 149 246)(143 252 150 245)(144 251 151 244)(145 250 152 243)(146 249 153 242)(147 248 154 241)(169 270 176 277)(170 269 177 276)(171 268 178 275)(172 267 179 274)(173 280 180 273)(174 279 181 272)(175 278 182 271)(211 387 218 380)(212 386 219 379)(213 385 220 392)(214 384 221 391)(215 383 222 390)(216 382 223 389)(217 381 224 388)(253 359 260 352)(254 358 261 351)(255 357 262 364)(256 356 263 363)(257 355 264 362)(258 354 265 361)(259 353 266 360)(281 347 288 340)(282 346 289 339)(283 345 290 338)(284 344 291 337)(285 343 292 350)(286 342 293 349)(287 341 294 348)(323 366 330 373)(324 365 331 372)(325 378 332 371)(326 377 333 370)(327 376 334 369)(328 375 335 368)(329 374 336 367)

G:=sub<Sym(392)| (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,197)(14,198)(15,290)(16,291)(17,292)(18,293)(19,294)(20,281)(21,282)(22,283)(23,284)(24,285)(25,286)(26,287)(27,288)(28,289)(29,98)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,385)(44,386)(45,387)(46,388)(47,389)(48,390)(49,391)(50,392)(51,379)(52,380)(53,381)(54,382)(55,383)(56,384)(57,349)(58,350)(59,337)(60,338)(61,339)(62,340)(63,341)(64,342)(65,343)(66,344)(67,345)(68,346)(69,347)(70,348)(71,177)(72,178)(73,179)(74,180)(75,181)(76,182)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(99,193)(100,194)(101,195)(102,196)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,316)(114,317)(115,318)(116,319)(117,320)(118,321)(119,322)(120,309)(121,310)(122,311)(123,312)(124,313)(125,314)(126,315)(127,263)(128,264)(129,265)(130,266)(131,253)(132,254)(133,255)(134,256)(135,257)(136,258)(137,259)(138,260)(139,261)(140,262)(141,333)(142,334)(143,335)(144,336)(145,323)(146,324)(147,325)(148,326)(149,327)(150,328)(151,329)(152,330)(153,331)(154,332)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,267)(211,305)(212,306)(213,307)(214,308)(215,295)(216,296)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,353)(226,354)(227,355)(228,356)(229,357)(230,358)(231,359)(232,360)(233,361)(234,362)(235,363)(236,364)(237,351)(238,352)(239,369)(240,370)(241,371)(242,372)(243,373)(244,374)(245,375)(246,376)(247,377)(248,378)(249,365)(250,366)(251,367)(252,368), (1,217,292,86,370,267,134)(2,218,293,87,371,268,135)(3,219,294,88,372,269,136)(4,220,281,89,373,270,137)(5,221,282,90,374,271,138)(6,222,283,91,375,272,139)(7,223,284,92,376,273,140)(8,224,285,93,377,274,127)(9,211,286,94,378,275,128)(10,212,287,95,365,276,129)(11,213,288,96,366,277,130)(12,214,289,97,367,278,131)(13,215,290,98,368,279,132)(14,216,291,85,369,280,133)(15,29,252,166,254,197,295)(16,30,239,167,255,198,296)(17,31,240,168,256,199,297)(18,32,241,155,257,200,298)(19,33,242,156,258,201,299)(20,34,243,157,259,202,300)(21,35,244,158,260,203,301)(22,36,245,159,261,204,302)(23,37,246,160,262,205,303)(24,38,247,161,263,206,304)(25,39,248,162,264,207,305)(26,40,249,163,265,208,306)(27,41,250,164,266,209,307)(28,42,251,165,253,210,308)(43,121,360,77,152,100,62)(44,122,361,78,153,101,63)(45,123,362,79,154,102,64)(46,124,363,80,141,103,65)(47,125,364,81,142,104,66)(48,126,351,82,143,105,67)(49,113,352,83,144,106,68)(50,114,353,84,145,107,69)(51,115,354,71,146,108,70)(52,116,355,72,147,109,57)(53,117,356,73,148,110,58)(54,118,357,74,149,111,59)(55,119,358,75,150,112,60)(56,120,359,76,151,99,61)(169,330,194,340,385,310,232)(170,331,195,341,386,311,233)(171,332,196,342,387,312,234)(172,333,183,343,388,313,235)(173,334,184,344,389,314,236)(174,335,185,345,390,315,237)(175,336,186,346,391,316,238)(176,323,187,347,392,317,225)(177,324,188,348,379,318,226)(178,325,189,349,380,319,227)(179,326,190,350,381,320,228)(180,327,191,337,382,321,229)(181,328,192,338,383,322,230)(182,329,193,339,384,309,231), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,320,8,313)(2,319,9,312)(3,318,10,311)(4,317,11,310)(5,316,12,309)(6,315,13,322)(7,314,14,321)(15,60,22,67)(16,59,23,66)(17,58,24,65)(18,57,25,64)(19,70,26,63)(20,69,27,62)(21,68,28,61)(29,112,36,105)(30,111,37,104)(31,110,38,103)(32,109,39,102)(33,108,40,101)(34,107,41,100)(35,106,42,99)(43,300,50,307)(44,299,51,306)(45,298,52,305)(46,297,53,304)(47,296,54,303)(48,295,55,302)(49,308,56,301)(71,163,78,156)(72,162,79,155)(73,161,80,168)(74,160,81,167)(75,159,82,166)(76,158,83,165)(77,157,84,164)(85,191,92,184)(86,190,93,183)(87,189,94,196)(88,188,95,195)(89,187,96,194)(90,186,97,193)(91,185,98,192)(113,210,120,203)(114,209,121,202)(115,208,122,201)(116,207,123,200)(117,206,124,199)(118,205,125,198)(119,204,126,197)(127,235,134,228)(128,234,135,227)(129,233,136,226)(130,232,137,225)(131,231,138,238)(132,230,139,237)(133,229,140,236)(141,240,148,247)(142,239,149,246)(143,252,150,245)(144,251,151,244)(145,250,152,243)(146,249,153,242)(147,248,154,241)(169,270,176,277)(170,269,177,276)(171,268,178,275)(172,267,179,274)(173,280,180,273)(174,279,181,272)(175,278,182,271)(211,387,218,380)(212,386,219,379)(213,385,220,392)(214,384,221,391)(215,383,222,390)(216,382,223,389)(217,381,224,388)(253,359,260,352)(254,358,261,351)(255,357,262,364)(256,356,263,363)(257,355,264,362)(258,354,265,361)(259,353,266,360)(281,347,288,340)(282,346,289,339)(283,345,290,338)(284,344,291,337)(285,343,292,350)(286,342,293,349)(287,341,294,348)(323,366,330,373)(324,365,331,372)(325,378,332,371)(326,377,333,370)(327,376,334,369)(328,375,335,368)(329,374,336,367)>;

G:=Group( (1,199)(2,200)(3,201)(4,202)(5,203)(6,204)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,197)(14,198)(15,290)(16,291)(17,292)(18,293)(19,294)(20,281)(21,282)(22,283)(23,284)(24,285)(25,286)(26,287)(27,288)(28,289)(29,98)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,385)(44,386)(45,387)(46,388)(47,389)(48,390)(49,391)(50,392)(51,379)(52,380)(53,381)(54,382)(55,383)(56,384)(57,349)(58,350)(59,337)(60,338)(61,339)(62,340)(63,341)(64,342)(65,343)(66,344)(67,345)(68,346)(69,347)(70,348)(71,177)(72,178)(73,179)(74,180)(75,181)(76,182)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(99,193)(100,194)(101,195)(102,196)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(113,316)(114,317)(115,318)(116,319)(117,320)(118,321)(119,322)(120,309)(121,310)(122,311)(123,312)(124,313)(125,314)(126,315)(127,263)(128,264)(129,265)(130,266)(131,253)(132,254)(133,255)(134,256)(135,257)(136,258)(137,259)(138,260)(139,261)(140,262)(141,333)(142,334)(143,335)(144,336)(145,323)(146,324)(147,325)(148,326)(149,327)(150,328)(151,329)(152,330)(153,331)(154,332)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,267)(211,305)(212,306)(213,307)(214,308)(215,295)(216,296)(217,297)(218,298)(219,299)(220,300)(221,301)(222,302)(223,303)(224,304)(225,353)(226,354)(227,355)(228,356)(229,357)(230,358)(231,359)(232,360)(233,361)(234,362)(235,363)(236,364)(237,351)(238,352)(239,369)(240,370)(241,371)(242,372)(243,373)(244,374)(245,375)(246,376)(247,377)(248,378)(249,365)(250,366)(251,367)(252,368), (1,217,292,86,370,267,134)(2,218,293,87,371,268,135)(3,219,294,88,372,269,136)(4,220,281,89,373,270,137)(5,221,282,90,374,271,138)(6,222,283,91,375,272,139)(7,223,284,92,376,273,140)(8,224,285,93,377,274,127)(9,211,286,94,378,275,128)(10,212,287,95,365,276,129)(11,213,288,96,366,277,130)(12,214,289,97,367,278,131)(13,215,290,98,368,279,132)(14,216,291,85,369,280,133)(15,29,252,166,254,197,295)(16,30,239,167,255,198,296)(17,31,240,168,256,199,297)(18,32,241,155,257,200,298)(19,33,242,156,258,201,299)(20,34,243,157,259,202,300)(21,35,244,158,260,203,301)(22,36,245,159,261,204,302)(23,37,246,160,262,205,303)(24,38,247,161,263,206,304)(25,39,248,162,264,207,305)(26,40,249,163,265,208,306)(27,41,250,164,266,209,307)(28,42,251,165,253,210,308)(43,121,360,77,152,100,62)(44,122,361,78,153,101,63)(45,123,362,79,154,102,64)(46,124,363,80,141,103,65)(47,125,364,81,142,104,66)(48,126,351,82,143,105,67)(49,113,352,83,144,106,68)(50,114,353,84,145,107,69)(51,115,354,71,146,108,70)(52,116,355,72,147,109,57)(53,117,356,73,148,110,58)(54,118,357,74,149,111,59)(55,119,358,75,150,112,60)(56,120,359,76,151,99,61)(169,330,194,340,385,310,232)(170,331,195,341,386,311,233)(171,332,196,342,387,312,234)(172,333,183,343,388,313,235)(173,334,184,344,389,314,236)(174,335,185,345,390,315,237)(175,336,186,346,391,316,238)(176,323,187,347,392,317,225)(177,324,188,348,379,318,226)(178,325,189,349,380,319,227)(179,326,190,350,381,320,228)(180,327,191,337,382,321,229)(181,328,192,338,383,322,230)(182,329,193,339,384,309,231), (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350)(351,352,353,354,355,356,357,358,359,360,361,362,363,364)(365,366,367,368,369,370,371,372,373,374,375,376,377,378)(379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,320,8,313)(2,319,9,312)(3,318,10,311)(4,317,11,310)(5,316,12,309)(6,315,13,322)(7,314,14,321)(15,60,22,67)(16,59,23,66)(17,58,24,65)(18,57,25,64)(19,70,26,63)(20,69,27,62)(21,68,28,61)(29,112,36,105)(30,111,37,104)(31,110,38,103)(32,109,39,102)(33,108,40,101)(34,107,41,100)(35,106,42,99)(43,300,50,307)(44,299,51,306)(45,298,52,305)(46,297,53,304)(47,296,54,303)(48,295,55,302)(49,308,56,301)(71,163,78,156)(72,162,79,155)(73,161,80,168)(74,160,81,167)(75,159,82,166)(76,158,83,165)(77,157,84,164)(85,191,92,184)(86,190,93,183)(87,189,94,196)(88,188,95,195)(89,187,96,194)(90,186,97,193)(91,185,98,192)(113,210,120,203)(114,209,121,202)(115,208,122,201)(116,207,123,200)(117,206,124,199)(118,205,125,198)(119,204,126,197)(127,235,134,228)(128,234,135,227)(129,233,136,226)(130,232,137,225)(131,231,138,238)(132,230,139,237)(133,229,140,236)(141,240,148,247)(142,239,149,246)(143,252,150,245)(144,251,151,244)(145,250,152,243)(146,249,153,242)(147,248,154,241)(169,270,176,277)(170,269,177,276)(171,268,178,275)(172,267,179,274)(173,280,180,273)(174,279,181,272)(175,278,182,271)(211,387,218,380)(212,386,219,379)(213,385,220,392)(214,384,221,391)(215,383,222,390)(216,382,223,389)(217,381,224,388)(253,359,260,352)(254,358,261,351)(255,357,262,364)(256,356,263,363)(257,355,264,362)(258,354,265,361)(259,353,266,360)(281,347,288,340)(282,346,289,339)(283,345,290,338)(284,344,291,337)(285,343,292,350)(286,342,293,349)(287,341,294,348)(323,366,330,373)(324,365,331,372)(325,378,332,371)(326,377,333,370)(327,376,334,369)(328,375,335,368)(329,374,336,367) );

G=PermutationGroup([[(1,199),(2,200),(3,201),(4,202),(5,203),(6,204),(7,205),(8,206),(9,207),(10,208),(11,209),(12,210),(13,197),(14,198),(15,290),(16,291),(17,292),(18,293),(19,294),(20,281),(21,282),(22,283),(23,284),(24,285),(25,286),(26,287),(27,288),(28,289),(29,98),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,385),(44,386),(45,387),(46,388),(47,389),(48,390),(49,391),(50,392),(51,379),(52,380),(53,381),(54,382),(55,383),(56,384),(57,349),(58,350),(59,337),(60,338),(61,339),(62,340),(63,341),(64,342),(65,343),(66,344),(67,345),(68,346),(69,347),(70,348),(71,177),(72,178),(73,179),(74,180),(75,181),(76,182),(77,169),(78,170),(79,171),(80,172),(81,173),(82,174),(83,175),(84,176),(99,193),(100,194),(101,195),(102,196),(103,183),(104,184),(105,185),(106,186),(107,187),(108,188),(109,189),(110,190),(111,191),(112,192),(113,316),(114,317),(115,318),(116,319),(117,320),(118,321),(119,322),(120,309),(121,310),(122,311),(123,312),(124,313),(125,314),(126,315),(127,263),(128,264),(129,265),(130,266),(131,253),(132,254),(133,255),(134,256),(135,257),(136,258),(137,259),(138,260),(139,261),(140,262),(141,333),(142,334),(143,335),(144,336),(145,323),(146,324),(147,325),(148,326),(149,327),(150,328),(151,329),(152,330),(153,331),(154,332),(155,268),(156,269),(157,270),(158,271),(159,272),(160,273),(161,274),(162,275),(163,276),(164,277),(165,278),(166,279),(167,280),(168,267),(211,305),(212,306),(213,307),(214,308),(215,295),(216,296),(217,297),(218,298),(219,299),(220,300),(221,301),(222,302),(223,303),(224,304),(225,353),(226,354),(227,355),(228,356),(229,357),(230,358),(231,359),(232,360),(233,361),(234,362),(235,363),(236,364),(237,351),(238,352),(239,369),(240,370),(241,371),(242,372),(243,373),(244,374),(245,375),(246,376),(247,377),(248,378),(249,365),(250,366),(251,367),(252,368)], [(1,217,292,86,370,267,134),(2,218,293,87,371,268,135),(3,219,294,88,372,269,136),(4,220,281,89,373,270,137),(5,221,282,90,374,271,138),(6,222,283,91,375,272,139),(7,223,284,92,376,273,140),(8,224,285,93,377,274,127),(9,211,286,94,378,275,128),(10,212,287,95,365,276,129),(11,213,288,96,366,277,130),(12,214,289,97,367,278,131),(13,215,290,98,368,279,132),(14,216,291,85,369,280,133),(15,29,252,166,254,197,295),(16,30,239,167,255,198,296),(17,31,240,168,256,199,297),(18,32,241,155,257,200,298),(19,33,242,156,258,201,299),(20,34,243,157,259,202,300),(21,35,244,158,260,203,301),(22,36,245,159,261,204,302),(23,37,246,160,262,205,303),(24,38,247,161,263,206,304),(25,39,248,162,264,207,305),(26,40,249,163,265,208,306),(27,41,250,164,266,209,307),(28,42,251,165,253,210,308),(43,121,360,77,152,100,62),(44,122,361,78,153,101,63),(45,123,362,79,154,102,64),(46,124,363,80,141,103,65),(47,125,364,81,142,104,66),(48,126,351,82,143,105,67),(49,113,352,83,144,106,68),(50,114,353,84,145,107,69),(51,115,354,71,146,108,70),(52,116,355,72,147,109,57),(53,117,356,73,148,110,58),(54,118,357,74,149,111,59),(55,119,358,75,150,112,60),(56,120,359,76,151,99,61),(169,330,194,340,385,310,232),(170,331,195,341,386,311,233),(171,332,196,342,387,312,234),(172,333,183,343,388,313,235),(173,334,184,344,389,314,236),(174,335,185,345,390,315,237),(175,336,186,346,391,316,238),(176,323,187,347,392,317,225),(177,324,188,348,379,318,226),(178,325,189,349,380,319,227),(179,326,190,350,381,320,228),(180,327,191,337,382,321,229),(181,328,192,338,383,322,230),(182,329,193,339,384,309,231)], [(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),(337,338,339,340,341,342,343,344,345,346,347,348,349,350),(351,352,353,354,355,356,357,358,359,360,361,362,363,364),(365,366,367,368,369,370,371,372,373,374,375,376,377,378),(379,380,381,382,383,384,385,386,387,388,389,390,391,392)], [(1,320,8,313),(2,319,9,312),(3,318,10,311),(4,317,11,310),(5,316,12,309),(6,315,13,322),(7,314,14,321),(15,60,22,67),(16,59,23,66),(17,58,24,65),(18,57,25,64),(19,70,26,63),(20,69,27,62),(21,68,28,61),(29,112,36,105),(30,111,37,104),(31,110,38,103),(32,109,39,102),(33,108,40,101),(34,107,41,100),(35,106,42,99),(43,300,50,307),(44,299,51,306),(45,298,52,305),(46,297,53,304),(47,296,54,303),(48,295,55,302),(49,308,56,301),(71,163,78,156),(72,162,79,155),(73,161,80,168),(74,160,81,167),(75,159,82,166),(76,158,83,165),(77,157,84,164),(85,191,92,184),(86,190,93,183),(87,189,94,196),(88,188,95,195),(89,187,96,194),(90,186,97,193),(91,185,98,192),(113,210,120,203),(114,209,121,202),(115,208,122,201),(116,207,123,200),(117,206,124,199),(118,205,125,198),(119,204,126,197),(127,235,134,228),(128,234,135,227),(129,233,136,226),(130,232,137,225),(131,231,138,238),(132,230,139,237),(133,229,140,236),(141,240,148,247),(142,239,149,246),(143,252,150,245),(144,251,151,244),(145,250,152,243),(146,249,153,242),(147,248,154,241),(169,270,176,277),(170,269,177,276),(171,268,178,275),(172,267,179,274),(173,280,180,273),(174,279,181,272),(175,278,182,271),(211,387,218,380),(212,386,219,379),(213,385,220,392),(214,384,221,391),(215,383,222,390),(216,382,223,389),(217,381,224,388),(253,359,260,352),(254,358,261,351),(255,357,262,364),(256,356,263,363),(257,355,264,362),(258,354,265,361),(259,353,266,360),(281,347,288,340),(282,346,289,339),(283,345,290,338),(284,344,291,337),(285,343,292,350),(286,342,293,349),(287,341,294,348),(323,366,330,373),(324,365,331,372),(325,378,332,371),(326,377,333,370),(327,376,334,369),(328,375,335,368),(329,374,336,367)]])

104 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A ··· 7X 14A ··· 14BT order 1 2 2 2 4 4 4 4 7 ··· 7 14 ··· 14 size 1 1 1 1 49 49 49 49 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 2 2 2 type + + + + - + image C1 C2 C2 C4 D7 Dic7 D14 kernel C2×C7⋊Dic7 C7⋊Dic7 C142 C7×C14 C2×C14 C14 C14 # reps 1 2 1 4 24 48 24

Matrix representation of C2×C7⋊Dic7 in GL5(𝔽29)

 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28
,
 1 0 0 0 0 0 16 0 0 0 0 0 20 0 0 0 0 0 23 0 0 0 0 0 24
,
 28 0 0 0 0 0 24 0 0 0 0 0 23 0 0 0 0 0 20 0 0 0 0 0 16
,
 17 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,16,0,0,0,0,0,20,0,0,0,0,0,23,0,0,0,0,0,24],[28,0,0,0,0,0,24,0,0,0,0,0,23,0,0,0,0,0,20,0,0,0,0,0,16],[17,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×C7⋊Dic7 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes {\rm Dic}_7
% in TeX

G:=Group("C2xC7:Dic7");
// GroupNames label

G:=SmallGroup(392,31);
// by ID

G=gap.SmallGroup(392,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-7,-7,20,963,8404]);
// Polycyclic

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

׿
×
𝔽