C3^3:7C16 - GroupNames

G = C₃³⋊₇C₁₆ order 432 = 2⁴·3³

3^rd semidirect product of C₃³ and C₁₆ acting via C₁₆/C₈=C₂

metabelian, supersoluble, monomial, A-group

Aliases: C₃³⋊₇C₁₆, C₃⋊(C₂₄.S₃), C₃²⋊₆(C₃⋊C₁₆), (C₃×C₂₄).₂₆S₃, C₂.(C₃³⋊₇C₈), (C₃²×C₆).₇C₈, C₂₄.₁₁(C₃⋊S₃), (C₃²×C₂₄).₇C₂, (C₃²×C₁₂).₁₂C₄, C₁₂.₇(C₃⋊Dic₃), C₄.₂(C₃³⋊₅C₄), C₆.₃(C₃²⋊₄C₈), C₈.₂(C₃³⋊C₂), (C₃×C₁₂).₂₈Dic₃, (C₃×C₆).₁₂(C₃⋊C₈), SmallGroup(432,231)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₃³⋊₇C₁₆

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×C₁₂ — C₃²×C₂₄ — C₃³⋊₇C₁₆

Lower central

C₃³ — C₃³⋊₇C₁₆

Upper central

C₁ — C₈

Generators and relations for C₃³⋊₇C₁₆
G = < a,b,c,d | a³=b³=c³=d¹⁶=1, ab=ba, ac=ca, dad^-1=a^-1, bc=cb, dbd^-1=b^-1, dcd^-1=c^-1 >

Subgroups: 296 in 140 conjugacy classes, 113 normal (9 characteristic)
C₁, C₂, C₃, C₄, C₆, C₈, C₃², C₁₂, C₁₆, C₃×C₆, C₂₄, C₃³, C₃×C₁₂, C₃⋊C₁₆, C₃²×C₆, C₃×C₂₄, C₃²×C₁₂, C₂₄.S₃, C₃²×C₂₄, C₃³⋊₇C₁₆
Quotients: C₁, C₂, C₄, S₃, C₈, Dic₃, C₁₆, C₃⋊S₃, C₃⋊C₈, C₃⋊Dic₃, C₃⋊C₁₆, C₃³⋊C₂, C₃²⋊₄C₈, C₃³⋊₅C₄, C₂₄.S₃, C₃³⋊₇C₈, C₃³⋊₇C₁₆

Smallest permutation representation of C₃³⋊₇C₁₆
►Regular action on 432 points

Generators in S₄₃₂

(1 193 181)(2 182 194)(3 195 183)(4 184 196)(5 197 185)(6 186 198)(7 199 187)(8 188 200)(9 201 189)(10 190 202)(11 203 191)(12 192 204)(13 205 177)(14 178 206)(15 207 179)(16 180 208)(17 37 236)(18 237 38)(19 39 238)(20 239 40)(21 41 240)(22 225 42)(23 43 226)(24 227 44)(25 45 228)(26 229 46)(27 47 230)(28 231 48)(29 33 232)(30 233 34)(31 35 234)(32 235 36)(49 65 271)(50 272 66)(51 67 257)(52 258 68)(53 69 259)(54 260 70)(55 71 261)(56 262 72)(57 73 263)(58 264 74)(59 75 265)(60 266 76)(61 77 267)(62 268 78)(63 79 269)(64 270 80)(81 306 111)(82 112 307)(83 308 97)(84 98 309)(85 310 99)(86 100 311)(87 312 101)(88 102 313)(89 314 103)(90 104 315)(91 316 105)(92 106 317)(93 318 107)(94 108 319)(95 320 109)(96 110 305)(113 336 133)(114 134 321)(115 322 135)(116 136 323)(117 324 137)(118 138 325)(119 326 139)(120 140 327)(121 328 141)(122 142 329)(123 330 143)(124 144 331)(125 332 129)(126 130 333)(127 334 131)(128 132 335)(145 294 212)(146 213 295)(147 296 214)(148 215 297)(149 298 216)(150 217 299)(151 300 218)(152 219 301)(153 302 220)(154 221 303)(155 304 222)(156 223 289)(157 290 224)(158 209 291)(159 292 210)(160 211 293)(161 411 380)(162 381 412)(163 413 382)(164 383 414)(165 415 384)(166 369 416)(167 401 370)(168 371 402)(169 403 372)(170 373 404)(171 405 374)(172 375 406)(173 407 376)(174 377 408)(175 409 378)(176 379 410)(241 281 351)(242 352 282)(243 283 337)(244 338 284)(245 285 339)(246 340 286)(247 287 341)(248 342 288)(249 273 343)(250 344 274)(251 275 345)(252 346 276)(253 277 347)(254 348 278)(255 279 349)(256 350 280)(353 400 421)(354 422 385)(355 386 423)(356 424 387)(357 388 425)(358 426 389)(359 390 427)(360 428 391)(361 392 429)(362 430 393)(363 394 431)(364 432 395)(365 396 417)(366 418 397)(367 398 419)(368 420 399)

(1 349 56)(2 57 350)(3 351 58)(4 59 352)(5 337 60)(6 61 338)(7 339 62)(8 63 340)(9 341 64)(10 49 342)(11 343 50)(12 51 344)(13 345 52)(14 53 346)(15 347 54)(16 55 348)(17 367 110)(18 111 368)(19 353 112)(20 97 354)(21 355 98)(22 99 356)(23 357 100)(24 101 358)(25 359 102)(26 103 360)(27 361 104)(28 105 362)(29 363 106)(30 107 364)(31 365 108)(32 109 366)(33 394 317)(34 318 395)(35 396 319)(36 320 397)(37 398 305)(38 306 399)(39 400 307)(40 308 385)(41 386 309)(42 310 387)(43 388 311)(44 312 389)(45 390 313)(46 314 391)(47 392 315)(48 316 393)(65 288 190)(66 191 273)(67 274 192)(68 177 275)(69 276 178)(70 179 277)(71 278 180)(72 181 279)(73 280 182)(74 183 281)(75 282 184)(76 185 283)(77 284 186)(78 187 285)(79 286 188)(80 189 287)(81 420 237)(82 238 421)(83 422 239)(84 240 423)(85 424 225)(86 226 425)(87 426 227)(88 228 427)(89 428 229)(90 230 429)(91 430 231)(92 232 431)(93 432 233)(94 234 417)(95 418 235)(96 236 419)(113 210 405)(114 406 211)(115 212 407)(116 408 213)(117 214 409)(118 410 215)(119 216 411)(120 412 217)(121 218 413)(122 414 219)(123 220 415)(124 416 221)(125 222 401)(126 402 223)(127 224 403)(128 404 209)(129 304 167)(130 168 289)(131 290 169)(132 170 291)(133 292 171)(134 172 293)(135 294 173)(136 174 295)(137 296 175)(138 176 297)(139 298 161)(140 162 299)(141 300 163)(142 164 301)(143 302 165)(144 166 303)(145 376 322)(146 323 377)(147 378 324)(148 325 379)(149 380 326)(150 327 381)(151 382 328)(152 329 383)(153 384 330)(154 331 369)(155 370 332)(156 333 371)(157 372 334)(158 335 373)(159 374 336)(160 321 375)(193 255 262)(194 263 256)(195 241 264)(196 265 242)(197 243 266)(198 267 244)(199 245 268)(200 269 246)(201 247 270)(202 271 248)(203 249 272)(204 257 250)(205 251 258)(206 259 252)(207 253 260)(208 261 254)

(1 335 40)(2 41 336)(3 321 42)(4 43 322)(5 323 44)(6 45 324)(7 325 46)(8 47 326)(9 327 48)(10 33 328)(11 329 34)(12 35 330)(13 331 36)(14 37 332)(15 333 38)(16 39 334)(17 125 206)(18 207 126)(19 127 208)(20 193 128)(21 113 194)(22 195 114)(23 115 196)(24 197 116)(25 117 198)(26 199 118)(27 119 200)(28 201 120)(29 121 202)(30 203 122)(31 123 204)(32 205 124)(49 394 151)(50 152 395)(51 396 153)(52 154 397)(53 398 155)(54 156 399)(55 400 157)(56 158 385)(57 386 159)(58 160 387)(59 388 145)(60 146 389)(61 390 147)(62 148 391)(63 392 149)(64 150 393)(65 431 300)(66 301 432)(67 417 302)(68 303 418)(69 419 304)(70 289 420)(71 421 290)(72 291 422)(73 423 292)(74 293 424)(75 425 294)(76 295 426)(77 427 296)(78 297 428)(79 429 298)(80 299 430)(81 277 168)(82 169 278)(83 279 170)(84 171 280)(85 281 172)(86 173 282)(87 283 174)(88 175 284)(89 285 176)(90 161 286)(91 287 162)(92 163 288)(93 273 164)(94 165 274)(95 275 166)(96 167 276)(97 255 404)(98 405 256)(99 241 406)(100 407 242)(101 243 408)(102 409 244)(103 245 410)(104 411 246)(105 247 412)(106 413 248)(107 249 414)(108 415 250)(109 251 416)(110 401 252)(111 253 402)(112 403 254)(129 178 236)(130 237 179)(131 180 238)(132 239 181)(133 182 240)(134 225 183)(135 184 226)(136 227 185)(137 186 228)(138 229 187)(139 188 230)(140 231 189)(141 190 232)(142 233 191)(143 192 234)(144 235 177)(209 354 262)(210 263 355)(211 356 264)(212 265 357)(213 358 266)(214 267 359)(215 360 268)(216 269 361)(217 362 270)(218 271 363)(219 364 272)(220 257 365)(221 366 258)(222 259 367)(223 368 260)(224 261 353)(305 370 346)(306 347 371)(307 372 348)(308 349 373)(309 374 350)(310 351 375)(311 376 352)(312 337 377)(313 378 338)(314 339 379)(315 380 340)(316 341 381)(317 382 342)(318 343 383)(319 384 344)(320 345 369)

(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)(385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400)(401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416)(417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)

G:=sub<Sym(432)| (1,193,181)(2,182,194)(3,195,183)(4,184,196)(5,197,185)(6,186,198)(7,199,187)(8,188,200)(9,201,189)(10,190,202)(11,203,191)(12,192,204)(13,205,177)(14,178,206)(15,207,179)(16,180,208)(17,37,236)(18,237,38)(19,39,238)(20,239,40)(21,41,240)(22,225,42)(23,43,226)(24,227,44)(25,45,228)(26,229,46)(27,47,230)(28,231,48)(29,33,232)(30,233,34)(31,35,234)(32,235,36)(49,65,271)(50,272,66)(51,67,257)(52,258,68)(53,69,259)(54,260,70)(55,71,261)(56,262,72)(57,73,263)(58,264,74)(59,75,265)(60,266,76)(61,77,267)(62,268,78)(63,79,269)(64,270,80)(81,306,111)(82,112,307)(83,308,97)(84,98,309)(85,310,99)(86,100,311)(87,312,101)(88,102,313)(89,314,103)(90,104,315)(91,316,105)(92,106,317)(93,318,107)(94,108,319)(95,320,109)(96,110,305)(113,336,133)(114,134,321)(115,322,135)(116,136,323)(117,324,137)(118,138,325)(119,326,139)(120,140,327)(121,328,141)(122,142,329)(123,330,143)(124,144,331)(125,332,129)(126,130,333)(127,334,131)(128,132,335)(145,294,212)(146,213,295)(147,296,214)(148,215,297)(149,298,216)(150,217,299)(151,300,218)(152,219,301)(153,302,220)(154,221,303)(155,304,222)(156,223,289)(157,290,224)(158,209,291)(159,292,210)(160,211,293)(161,411,380)(162,381,412)(163,413,382)(164,383,414)(165,415,384)(166,369,416)(167,401,370)(168,371,402)(169,403,372)(170,373,404)(171,405,374)(172,375,406)(173,407,376)(174,377,408)(175,409,378)(176,379,410)(241,281,351)(242,352,282)(243,283,337)(244,338,284)(245,285,339)(246,340,286)(247,287,341)(248,342,288)(249,273,343)(250,344,274)(251,275,345)(252,346,276)(253,277,347)(254,348,278)(255,279,349)(256,350,280)(353,400,421)(354,422,385)(355,386,423)(356,424,387)(357,388,425)(358,426,389)(359,390,427)(360,428,391)(361,392,429)(362,430,393)(363,394,431)(364,432,395)(365,396,417)(366,418,397)(367,398,419)(368,420,399), (1,349,56)(2,57,350)(3,351,58)(4,59,352)(5,337,60)(6,61,338)(7,339,62)(8,63,340)(9,341,64)(10,49,342)(11,343,50)(12,51,344)(13,345,52)(14,53,346)(15,347,54)(16,55,348)(17,367,110)(18,111,368)(19,353,112)(20,97,354)(21,355,98)(22,99,356)(23,357,100)(24,101,358)(25,359,102)(26,103,360)(27,361,104)(28,105,362)(29,363,106)(30,107,364)(31,365,108)(32,109,366)(33,394,317)(34,318,395)(35,396,319)(36,320,397)(37,398,305)(38,306,399)(39,400,307)(40,308,385)(41,386,309)(42,310,387)(43,388,311)(44,312,389)(45,390,313)(46,314,391)(47,392,315)(48,316,393)(65,288,190)(66,191,273)(67,274,192)(68,177,275)(69,276,178)(70,179,277)(71,278,180)(72,181,279)(73,280,182)(74,183,281)(75,282,184)(76,185,283)(77,284,186)(78,187,285)(79,286,188)(80,189,287)(81,420,237)(82,238,421)(83,422,239)(84,240,423)(85,424,225)(86,226,425)(87,426,227)(88,228,427)(89,428,229)(90,230,429)(91,430,231)(92,232,431)(93,432,233)(94,234,417)(95,418,235)(96,236,419)(113,210,405)(114,406,211)(115,212,407)(116,408,213)(117,214,409)(118,410,215)(119,216,411)(120,412,217)(121,218,413)(122,414,219)(123,220,415)(124,416,221)(125,222,401)(126,402,223)(127,224,403)(128,404,209)(129,304,167)(130,168,289)(131,290,169)(132,170,291)(133,292,171)(134,172,293)(135,294,173)(136,174,295)(137,296,175)(138,176,297)(139,298,161)(140,162,299)(141,300,163)(142,164,301)(143,302,165)(144,166,303)(145,376,322)(146,323,377)(147,378,324)(148,325,379)(149,380,326)(150,327,381)(151,382,328)(152,329,383)(153,384,330)(154,331,369)(155,370,332)(156,333,371)(157,372,334)(158,335,373)(159,374,336)(160,321,375)(193,255,262)(194,263,256)(195,241,264)(196,265,242)(197,243,266)(198,267,244)(199,245,268)(200,269,246)(201,247,270)(202,271,248)(203,249,272)(204,257,250)(205,251,258)(206,259,252)(207,253,260)(208,261,254), (1,335,40)(2,41,336)(3,321,42)(4,43,322)(5,323,44)(6,45,324)(7,325,46)(8,47,326)(9,327,48)(10,33,328)(11,329,34)(12,35,330)(13,331,36)(14,37,332)(15,333,38)(16,39,334)(17,125,206)(18,207,126)(19,127,208)(20,193,128)(21,113,194)(22,195,114)(23,115,196)(24,197,116)(25,117,198)(26,199,118)(27,119,200)(28,201,120)(29,121,202)(30,203,122)(31,123,204)(32,205,124)(49,394,151)(50,152,395)(51,396,153)(52,154,397)(53,398,155)(54,156,399)(55,400,157)(56,158,385)(57,386,159)(58,160,387)(59,388,145)(60,146,389)(61,390,147)(62,148,391)(63,392,149)(64,150,393)(65,431,300)(66,301,432)(67,417,302)(68,303,418)(69,419,304)(70,289,420)(71,421,290)(72,291,422)(73,423,292)(74,293,424)(75,425,294)(76,295,426)(77,427,296)(78,297,428)(79,429,298)(80,299,430)(81,277,168)(82,169,278)(83,279,170)(84,171,280)(85,281,172)(86,173,282)(87,283,174)(88,175,284)(89,285,176)(90,161,286)(91,287,162)(92,163,288)(93,273,164)(94,165,274)(95,275,166)(96,167,276)(97,255,404)(98,405,256)(99,241,406)(100,407,242)(101,243,408)(102,409,244)(103,245,410)(104,411,246)(105,247,412)(106,413,248)(107,249,414)(108,415,250)(109,251,416)(110,401,252)(111,253,402)(112,403,254)(129,178,236)(130,237,179)(131,180,238)(132,239,181)(133,182,240)(134,225,183)(135,184,226)(136,227,185)(137,186,228)(138,229,187)(139,188,230)(140,231,189)(141,190,232)(142,233,191)(143,192,234)(144,235,177)(209,354,262)(210,263,355)(211,356,264)(212,265,357)(213,358,266)(214,267,359)(215,360,268)(216,269,361)(217,362,270)(218,271,363)(219,364,272)(220,257,365)(221,366,258)(222,259,367)(223,368,260)(224,261,353)(305,370,346)(306,347,371)(307,372,348)(308,349,373)(309,374,350)(310,351,375)(311,376,352)(312,337,377)(313,378,338)(314,339,379)(315,380,340)(316,341,381)(317,382,342)(318,343,383)(319,384,344)(320,345,369), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)>;

G:=Group( (1,193,181)(2,182,194)(3,195,183)(4,184,196)(5,197,185)(6,186,198)(7,199,187)(8,188,200)(9,201,189)(10,190,202)(11,203,191)(12,192,204)(13,205,177)(14,178,206)(15,207,179)(16,180,208)(17,37,236)(18,237,38)(19,39,238)(20,239,40)(21,41,240)(22,225,42)(23,43,226)(24,227,44)(25,45,228)(26,229,46)(27,47,230)(28,231,48)(29,33,232)(30,233,34)(31,35,234)(32,235,36)(49,65,271)(50,272,66)(51,67,257)(52,258,68)(53,69,259)(54,260,70)(55,71,261)(56,262,72)(57,73,263)(58,264,74)(59,75,265)(60,266,76)(61,77,267)(62,268,78)(63,79,269)(64,270,80)(81,306,111)(82,112,307)(83,308,97)(84,98,309)(85,310,99)(86,100,311)(87,312,101)(88,102,313)(89,314,103)(90,104,315)(91,316,105)(92,106,317)(93,318,107)(94,108,319)(95,320,109)(96,110,305)(113,336,133)(114,134,321)(115,322,135)(116,136,323)(117,324,137)(118,138,325)(119,326,139)(120,140,327)(121,328,141)(122,142,329)(123,330,143)(124,144,331)(125,332,129)(126,130,333)(127,334,131)(128,132,335)(145,294,212)(146,213,295)(147,296,214)(148,215,297)(149,298,216)(150,217,299)(151,300,218)(152,219,301)(153,302,220)(154,221,303)(155,304,222)(156,223,289)(157,290,224)(158,209,291)(159,292,210)(160,211,293)(161,411,380)(162,381,412)(163,413,382)(164,383,414)(165,415,384)(166,369,416)(167,401,370)(168,371,402)(169,403,372)(170,373,404)(171,405,374)(172,375,406)(173,407,376)(174,377,408)(175,409,378)(176,379,410)(241,281,351)(242,352,282)(243,283,337)(244,338,284)(245,285,339)(246,340,286)(247,287,341)(248,342,288)(249,273,343)(250,344,274)(251,275,345)(252,346,276)(253,277,347)(254,348,278)(255,279,349)(256,350,280)(353,400,421)(354,422,385)(355,386,423)(356,424,387)(357,388,425)(358,426,389)(359,390,427)(360,428,391)(361,392,429)(362,430,393)(363,394,431)(364,432,395)(365,396,417)(366,418,397)(367,398,419)(368,420,399), (1,349,56)(2,57,350)(3,351,58)(4,59,352)(5,337,60)(6,61,338)(7,339,62)(8,63,340)(9,341,64)(10,49,342)(11,343,50)(12,51,344)(13,345,52)(14,53,346)(15,347,54)(16,55,348)(17,367,110)(18,111,368)(19,353,112)(20,97,354)(21,355,98)(22,99,356)(23,357,100)(24,101,358)(25,359,102)(26,103,360)(27,361,104)(28,105,362)(29,363,106)(30,107,364)(31,365,108)(32,109,366)(33,394,317)(34,318,395)(35,396,319)(36,320,397)(37,398,305)(38,306,399)(39,400,307)(40,308,385)(41,386,309)(42,310,387)(43,388,311)(44,312,389)(45,390,313)(46,314,391)(47,392,315)(48,316,393)(65,288,190)(66,191,273)(67,274,192)(68,177,275)(69,276,178)(70,179,277)(71,278,180)(72,181,279)(73,280,182)(74,183,281)(75,282,184)(76,185,283)(77,284,186)(78,187,285)(79,286,188)(80,189,287)(81,420,237)(82,238,421)(83,422,239)(84,240,423)(85,424,225)(86,226,425)(87,426,227)(88,228,427)(89,428,229)(90,230,429)(91,430,231)(92,232,431)(93,432,233)(94,234,417)(95,418,235)(96,236,419)(113,210,405)(114,406,211)(115,212,407)(116,408,213)(117,214,409)(118,410,215)(119,216,411)(120,412,217)(121,218,413)(122,414,219)(123,220,415)(124,416,221)(125,222,401)(126,402,223)(127,224,403)(128,404,209)(129,304,167)(130,168,289)(131,290,169)(132,170,291)(133,292,171)(134,172,293)(135,294,173)(136,174,295)(137,296,175)(138,176,297)(139,298,161)(140,162,299)(141,300,163)(142,164,301)(143,302,165)(144,166,303)(145,376,322)(146,323,377)(147,378,324)(148,325,379)(149,380,326)(150,327,381)(151,382,328)(152,329,383)(153,384,330)(154,331,369)(155,370,332)(156,333,371)(157,372,334)(158,335,373)(159,374,336)(160,321,375)(193,255,262)(194,263,256)(195,241,264)(196,265,242)(197,243,266)(198,267,244)(199,245,268)(200,269,246)(201,247,270)(202,271,248)(203,249,272)(204,257,250)(205,251,258)(206,259,252)(207,253,260)(208,261,254), (1,335,40)(2,41,336)(3,321,42)(4,43,322)(5,323,44)(6,45,324)(7,325,46)(8,47,326)(9,327,48)(10,33,328)(11,329,34)(12,35,330)(13,331,36)(14,37,332)(15,333,38)(16,39,334)(17,125,206)(18,207,126)(19,127,208)(20,193,128)(21,113,194)(22,195,114)(23,115,196)(24,197,116)(25,117,198)(26,199,118)(27,119,200)(28,201,120)(29,121,202)(30,203,122)(31,123,204)(32,205,124)(49,394,151)(50,152,395)(51,396,153)(52,154,397)(53,398,155)(54,156,399)(55,400,157)(56,158,385)(57,386,159)(58,160,387)(59,388,145)(60,146,389)(61,390,147)(62,148,391)(63,392,149)(64,150,393)(65,431,300)(66,301,432)(67,417,302)(68,303,418)(69,419,304)(70,289,420)(71,421,290)(72,291,422)(73,423,292)(74,293,424)(75,425,294)(76,295,426)(77,427,296)(78,297,428)(79,429,298)(80,299,430)(81,277,168)(82,169,278)(83,279,170)(84,171,280)(85,281,172)(86,173,282)(87,283,174)(88,175,284)(89,285,176)(90,161,286)(91,287,162)(92,163,288)(93,273,164)(94,165,274)(95,275,166)(96,167,276)(97,255,404)(98,405,256)(99,241,406)(100,407,242)(101,243,408)(102,409,244)(103,245,410)(104,411,246)(105,247,412)(106,413,248)(107,249,414)(108,415,250)(109,251,416)(110,401,252)(111,253,402)(112,403,254)(129,178,236)(130,237,179)(131,180,238)(132,239,181)(133,182,240)(134,225,183)(135,184,226)(136,227,185)(137,186,228)(138,229,187)(139,188,230)(140,231,189)(141,190,232)(142,233,191)(143,192,234)(144,235,177)(209,354,262)(210,263,355)(211,356,264)(212,265,357)(213,358,266)(214,267,359)(215,360,268)(216,269,361)(217,362,270)(218,271,363)(219,364,272)(220,257,365)(221,366,258)(222,259,367)(223,368,260)(224,261,353)(305,370,346)(306,347,371)(307,372,348)(308,349,373)(309,374,350)(310,351,375)(311,376,352)(312,337,377)(313,378,338)(314,339,379)(315,380,340)(316,341,381)(317,382,342)(318,343,383)(319,384,344)(320,345,369), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416)(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432) );

G=PermutationGroup([[(1,193,181),(2,182,194),(3,195,183),(4,184,196),(5,197,185),(6,186,198),(7,199,187),(8,188,200),(9,201,189),(10,190,202),(11,203,191),(12,192,204),(13,205,177),(14,178,206),(15,207,179),(16,180,208),(17,37,236),(18,237,38),(19,39,238),(20,239,40),(21,41,240),(22,225,42),(23,43,226),(24,227,44),(25,45,228),(26,229,46),(27,47,230),(28,231,48),(29,33,232),(30,233,34),(31,35,234),(32,235,36),(49,65,271),(50,272,66),(51,67,257),(52,258,68),(53,69,259),(54,260,70),(55,71,261),(56,262,72),(57,73,263),(58,264,74),(59,75,265),(60,266,76),(61,77,267),(62,268,78),(63,79,269),(64,270,80),(81,306,111),(82,112,307),(83,308,97),(84,98,309),(85,310,99),(86,100,311),(87,312,101),(88,102,313),(89,314,103),(90,104,315),(91,316,105),(92,106,317),(93,318,107),(94,108,319),(95,320,109),(96,110,305),(113,336,133),(114,134,321),(115,322,135),(116,136,323),(117,324,137),(118,138,325),(119,326,139),(120,140,327),(121,328,141),(122,142,329),(123,330,143),(124,144,331),(125,332,129),(126,130,333),(127,334,131),(128,132,335),(145,294,212),(146,213,295),(147,296,214),(148,215,297),(149,298,216),(150,217,299),(151,300,218),(152,219,301),(153,302,220),(154,221,303),(155,304,222),(156,223,289),(157,290,224),(158,209,291),(159,292,210),(160,211,293),(161,411,380),(162,381,412),(163,413,382),(164,383,414),(165,415,384),(166,369,416),(167,401,370),(168,371,402),(169,403,372),(170,373,404),(171,405,374),(172,375,406),(173,407,376),(174,377,408),(175,409,378),(176,379,410),(241,281,351),(242,352,282),(243,283,337),(244,338,284),(245,285,339),(246,340,286),(247,287,341),(248,342,288),(249,273,343),(250,344,274),(251,275,345),(252,346,276),(253,277,347),(254,348,278),(255,279,349),(256,350,280),(353,400,421),(354,422,385),(355,386,423),(356,424,387),(357,388,425),(358,426,389),(359,390,427),(360,428,391),(361,392,429),(362,430,393),(363,394,431),(364,432,395),(365,396,417),(366,418,397),(367,398,419),(368,420,399)], [(1,349,56),(2,57,350),(3,351,58),(4,59,352),(5,337,60),(6,61,338),(7,339,62),(8,63,340),(9,341,64),(10,49,342),(11,343,50),(12,51,344),(13,345,52),(14,53,346),(15,347,54),(16,55,348),(17,367,110),(18,111,368),(19,353,112),(20,97,354),(21,355,98),(22,99,356),(23,357,100),(24,101,358),(25,359,102),(26,103,360),(27,361,104),(28,105,362),(29,363,106),(30,107,364),(31,365,108),(32,109,366),(33,394,317),(34,318,395),(35,396,319),(36,320,397),(37,398,305),(38,306,399),(39,400,307),(40,308,385),(41,386,309),(42,310,387),(43,388,311),(44,312,389),(45,390,313),(46,314,391),(47,392,315),(48,316,393),(65,288,190),(66,191,273),(67,274,192),(68,177,275),(69,276,178),(70,179,277),(71,278,180),(72,181,279),(73,280,182),(74,183,281),(75,282,184),(76,185,283),(77,284,186),(78,187,285),(79,286,188),(80,189,287),(81,420,237),(82,238,421),(83,422,239),(84,240,423),(85,424,225),(86,226,425),(87,426,227),(88,228,427),(89,428,229),(90,230,429),(91,430,231),(92,232,431),(93,432,233),(94,234,417),(95,418,235),(96,236,419),(113,210,405),(114,406,211),(115,212,407),(116,408,213),(117,214,409),(118,410,215),(119,216,411),(120,412,217),(121,218,413),(122,414,219),(123,220,415),(124,416,221),(125,222,401),(126,402,223),(127,224,403),(128,404,209),(129,304,167),(130,168,289),(131,290,169),(132,170,291),(133,292,171),(134,172,293),(135,294,173),(136,174,295),(137,296,175),(138,176,297),(139,298,161),(140,162,299),(141,300,163),(142,164,301),(143,302,165),(144,166,303),(145,376,322),(146,323,377),(147,378,324),(148,325,379),(149,380,326),(150,327,381),(151,382,328),(152,329,383),(153,384,330),(154,331,369),(155,370,332),(156,333,371),(157,372,334),(158,335,373),(159,374,336),(160,321,375),(193,255,262),(194,263,256),(195,241,264),(196,265,242),(197,243,266),(198,267,244),(199,245,268),(200,269,246),(201,247,270),(202,271,248),(203,249,272),(204,257,250),(205,251,258),(206,259,252),(207,253,260),(208,261,254)], [(1,335,40),(2,41,336),(3,321,42),(4,43,322),(5,323,44),(6,45,324),(7,325,46),(8,47,326),(9,327,48),(10,33,328),(11,329,34),(12,35,330),(13,331,36),(14,37,332),(15,333,38),(16,39,334),(17,125,206),(18,207,126),(19,127,208),(20,193,128),(21,113,194),(22,195,114),(23,115,196),(24,197,116),(25,117,198),(26,199,118),(27,119,200),(28,201,120),(29,121,202),(30,203,122),(31,123,204),(32,205,124),(49,394,151),(50,152,395),(51,396,153),(52,154,397),(53,398,155),(54,156,399),(55,400,157),(56,158,385),(57,386,159),(58,160,387),(59,388,145),(60,146,389),(61,390,147),(62,148,391),(63,392,149),(64,150,393),(65,431,300),(66,301,432),(67,417,302),(68,303,418),(69,419,304),(70,289,420),(71,421,290),(72,291,422),(73,423,292),(74,293,424),(75,425,294),(76,295,426),(77,427,296),(78,297,428),(79,429,298),(80,299,430),(81,277,168),(82,169,278),(83,279,170),(84,171,280),(85,281,172),(86,173,282),(87,283,174),(88,175,284),(89,285,176),(90,161,286),(91,287,162),(92,163,288),(93,273,164),(94,165,274),(95,275,166),(96,167,276),(97,255,404),(98,405,256),(99,241,406),(100,407,242),(101,243,408),(102,409,244),(103,245,410),(104,411,246),(105,247,412),(106,413,248),(107,249,414),(108,415,250),(109,251,416),(110,401,252),(111,253,402),(112,403,254),(129,178,236),(130,237,179),(131,180,238),(132,239,181),(133,182,240),(134,225,183),(135,184,226),(136,227,185),(137,186,228),(138,229,187),(139,188,230),(140,231,189),(141,190,232),(142,233,191),(143,192,234),(144,235,177),(209,354,262),(210,263,355),(211,356,264),(212,265,357),(213,358,266),(214,267,359),(215,360,268),(216,269,361),(217,362,270),(218,271,363),(219,364,272),(220,257,365),(221,366,258),(222,259,367),(223,368,260),(224,261,353),(305,370,346),(306,347,371),(307,372,348),(308,349,373),(309,374,350),(310,351,375),(311,376,352),(312,337,377),(313,378,338),(314,339,379),(315,380,340),(316,341,381),(317,382,342),(318,343,383),(319,384,344),(320,345,369)], [(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),(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400),(401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416),(417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)]])

120 conjugacy classes

class	1	2	3A	···	3M	4A	4B	6A	···	6M	8A	8B	8C	8D	12A	···	12Z	16A	···	16H	24A	···	24AZ
order	1	2	3	···	3	4	4	6	···	6	8	8	8	8	12	···	12	16	···	16	24	···	24
size	1	1	2	···	2	1	1	2	···	2	1	1	1	1	2	···	2	27	···	27	2	···	2

120 irreducible representations

dim	1	1	1	1	1	2	2	2	2
type	+	+				+	-
image	C₁	C₂	C₄	C₈	C₁₆	S₃	Dic₃	C₃⋊C₈	C₃⋊C₁₆
kernel	C₃³⋊₇C₁₆	C₃²×C₂₄	C₃²×C₁₂	C₃²×C₆	C₃³	C₃×C₂₄	C₃×C₁₂	C₃×C₆	C₃²
# reps	1	1	2	4	8	13	13	26	52

Matrix representation of C₃³⋊₇C₁₆ ►in GL₆(𝔽₉₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	96	1	0	0
0	0	96	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
96	96	0	0	0	0
0	0	0	96	0	0
0	0	1	96	0	0
0	0	0	0	96	96
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	96	1	0	0
0	0	96	0	0	0
0	0	0	0	0	1
0	0	0	0	96	96

92	5	0	0	0	0
10	5	0	0	0	0
0	0	47	33	0	0
0	0	80	50	0	0
0	0	0	0	96	43
0	0	0	0	44	1

G:=sub<GL(6,GF(97))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,96,0,0,0,0,1,96,0,0,0,0,0,0,0,1,0,0,0,0,96,96,0,0,0,0,0,0,96,1,0,0,0,0,96,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,96,0,0,0,0,1,0,0,0,0,0,0,0,0,96,0,0,0,0,1,96],[92,10,0,0,0,0,5,5,0,0,0,0,0,0,47,80,0,0,0,0,33,50,0,0,0,0,0,0,96,44,0,0,0,0,43,1] >;

C₃³⋊₇C₁₆ in GAP, Magma, Sage, TeX

C_3^3\rtimes_7C_{16}

% in TeX

G:=Group("C3^3:7C16");

// GroupNames label

G:=SmallGroup(432,231);

// by ID

G=gap.SmallGroup(432,231);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,14,36,58,1124,4037,14118]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^16=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;

// generators/relations

G = C33⋊7C16 order 432 = 24·33

3rd semidirect product of C33 and C16 acting via C16/C8=C2

G = C₃³⋊₇C₁₆ order 432 = 2⁴·3³

3^rd semidirect product of C₃³ and C₁₆ acting via C₁₆/C₈=C₂