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## G = C6×C72order 432 = 24·33

### Abelian group of type [6,72]

Aliases: C6×C72, SmallGroup(432,209)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C72
 Chief series C1 — C2 — C6 — C12 — C3×C12 — C3×C36 — C3×C72 — C6×C72
 Lower central C1 — C6×C72
 Upper central C1 — C6×C72

Generators and relations for C6×C72
G = < a,b | a6=b72=1, ab=ba >

Subgroups: 110, all normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C3×C6, C3×C6, C24, C2×C12, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C2×C24, C2×C24, C3×C18, C3×C18, C72, C2×C36, C3×C24, C6×C12, C3×C36, C6×C18, C2×C72, C6×C24, C3×C72, C6×C36, C6×C72
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C9, C32, C12, C2×C6, C2×C8, C18, C3×C6, C24, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C2×C24, C3×C18, C72, C2×C36, C3×C24, C6×C12, C3×C36, C6×C18, C2×C72, C6×C24, C3×C72, C6×C36, C6×C72

Smallest permutation representation of C6×C72
Regular action on 432 points
Generators in S432
(1 355 90 288 418 199)(2 356 91 217 419 200)(3 357 92 218 420 201)(4 358 93 219 421 202)(5 359 94 220 422 203)(6 360 95 221 423 204)(7 289 96 222 424 205)(8 290 97 223 425 206)(9 291 98 224 426 207)(10 292 99 225 427 208)(11 293 100 226 428 209)(12 294 101 227 429 210)(13 295 102 228 430 211)(14 296 103 229 431 212)(15 297 104 230 432 213)(16 298 105 231 361 214)(17 299 106 232 362 215)(18 300 107 233 363 216)(19 301 108 234 364 145)(20 302 109 235 365 146)(21 303 110 236 366 147)(22 304 111 237 367 148)(23 305 112 238 368 149)(24 306 113 239 369 150)(25 307 114 240 370 151)(26 308 115 241 371 152)(27 309 116 242 372 153)(28 310 117 243 373 154)(29 311 118 244 374 155)(30 312 119 245 375 156)(31 313 120 246 376 157)(32 314 121 247 377 158)(33 315 122 248 378 159)(34 316 123 249 379 160)(35 317 124 250 380 161)(36 318 125 251 381 162)(37 319 126 252 382 163)(38 320 127 253 383 164)(39 321 128 254 384 165)(40 322 129 255 385 166)(41 323 130 256 386 167)(42 324 131 257 387 168)(43 325 132 258 388 169)(44 326 133 259 389 170)(45 327 134 260 390 171)(46 328 135 261 391 172)(47 329 136 262 392 173)(48 330 137 263 393 174)(49 331 138 264 394 175)(50 332 139 265 395 176)(51 333 140 266 396 177)(52 334 141 267 397 178)(53 335 142 268 398 179)(54 336 143 269 399 180)(55 337 144 270 400 181)(56 338 73 271 401 182)(57 339 74 272 402 183)(58 340 75 273 403 184)(59 341 76 274 404 185)(60 342 77 275 405 186)(61 343 78 276 406 187)(62 344 79 277 407 188)(63 345 80 278 408 189)(64 346 81 279 409 190)(65 347 82 280 410 191)(66 348 83 281 411 192)(67 349 84 282 412 193)(68 350 85 283 413 194)(69 351 86 284 414 195)(70 352 87 285 415 196)(71 353 88 286 416 197)(72 354 89 287 417 198)
(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)| (1,355,90,288,418,199)(2,356,91,217,419,200)(3,357,92,218,420,201)(4,358,93,219,421,202)(5,359,94,220,422,203)(6,360,95,221,423,204)(7,289,96,222,424,205)(8,290,97,223,425,206)(9,291,98,224,426,207)(10,292,99,225,427,208)(11,293,100,226,428,209)(12,294,101,227,429,210)(13,295,102,228,430,211)(14,296,103,229,431,212)(15,297,104,230,432,213)(16,298,105,231,361,214)(17,299,106,232,362,215)(18,300,107,233,363,216)(19,301,108,234,364,145)(20,302,109,235,365,146)(21,303,110,236,366,147)(22,304,111,237,367,148)(23,305,112,238,368,149)(24,306,113,239,369,150)(25,307,114,240,370,151)(26,308,115,241,371,152)(27,309,116,242,372,153)(28,310,117,243,373,154)(29,311,118,244,374,155)(30,312,119,245,375,156)(31,313,120,246,376,157)(32,314,121,247,377,158)(33,315,122,248,378,159)(34,316,123,249,379,160)(35,317,124,250,380,161)(36,318,125,251,381,162)(37,319,126,252,382,163)(38,320,127,253,383,164)(39,321,128,254,384,165)(40,322,129,255,385,166)(41,323,130,256,386,167)(42,324,131,257,387,168)(43,325,132,258,388,169)(44,326,133,259,389,170)(45,327,134,260,390,171)(46,328,135,261,391,172)(47,329,136,262,392,173)(48,330,137,263,393,174)(49,331,138,264,394,175)(50,332,139,265,395,176)(51,333,140,266,396,177)(52,334,141,267,397,178)(53,335,142,268,398,179)(54,336,143,269,399,180)(55,337,144,270,400,181)(56,338,73,271,401,182)(57,339,74,272,402,183)(58,340,75,273,403,184)(59,341,76,274,404,185)(60,342,77,275,405,186)(61,343,78,276,406,187)(62,344,79,277,407,188)(63,345,80,278,408,189)(64,346,81,279,409,190)(65,347,82,280,410,191)(66,348,83,281,411,192)(67,349,84,282,412,193)(68,350,85,283,413,194)(69,351,86,284,414,195)(70,352,87,285,415,196)(71,353,88,286,416,197)(72,354,89,287,417,198), (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( (1,355,90,288,418,199)(2,356,91,217,419,200)(3,357,92,218,420,201)(4,358,93,219,421,202)(5,359,94,220,422,203)(6,360,95,221,423,204)(7,289,96,222,424,205)(8,290,97,223,425,206)(9,291,98,224,426,207)(10,292,99,225,427,208)(11,293,100,226,428,209)(12,294,101,227,429,210)(13,295,102,228,430,211)(14,296,103,229,431,212)(15,297,104,230,432,213)(16,298,105,231,361,214)(17,299,106,232,362,215)(18,300,107,233,363,216)(19,301,108,234,364,145)(20,302,109,235,365,146)(21,303,110,236,366,147)(22,304,111,237,367,148)(23,305,112,238,368,149)(24,306,113,239,369,150)(25,307,114,240,370,151)(26,308,115,241,371,152)(27,309,116,242,372,153)(28,310,117,243,373,154)(29,311,118,244,374,155)(30,312,119,245,375,156)(31,313,120,246,376,157)(32,314,121,247,377,158)(33,315,122,248,378,159)(34,316,123,249,379,160)(35,317,124,250,380,161)(36,318,125,251,381,162)(37,319,126,252,382,163)(38,320,127,253,383,164)(39,321,128,254,384,165)(40,322,129,255,385,166)(41,323,130,256,386,167)(42,324,131,257,387,168)(43,325,132,258,388,169)(44,326,133,259,389,170)(45,327,134,260,390,171)(46,328,135,261,391,172)(47,329,136,262,392,173)(48,330,137,263,393,174)(49,331,138,264,394,175)(50,332,139,265,395,176)(51,333,140,266,396,177)(52,334,141,267,397,178)(53,335,142,268,398,179)(54,336,143,269,399,180)(55,337,144,270,400,181)(56,338,73,271,401,182)(57,339,74,272,402,183)(58,340,75,273,403,184)(59,341,76,274,404,185)(60,342,77,275,405,186)(61,343,78,276,406,187)(62,344,79,277,407,188)(63,345,80,278,408,189)(64,346,81,279,409,190)(65,347,82,280,410,191)(66,348,83,281,411,192)(67,349,84,282,412,193)(68,350,85,283,413,194)(69,351,86,284,414,195)(70,352,87,285,415,196)(71,353,88,286,416,197)(72,354,89,287,417,198), (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([[(1,355,90,288,418,199),(2,356,91,217,419,200),(3,357,92,218,420,201),(4,358,93,219,421,202),(5,359,94,220,422,203),(6,360,95,221,423,204),(7,289,96,222,424,205),(8,290,97,223,425,206),(9,291,98,224,426,207),(10,292,99,225,427,208),(11,293,100,226,428,209),(12,294,101,227,429,210),(13,295,102,228,430,211),(14,296,103,229,431,212),(15,297,104,230,432,213),(16,298,105,231,361,214),(17,299,106,232,362,215),(18,300,107,233,363,216),(19,301,108,234,364,145),(20,302,109,235,365,146),(21,303,110,236,366,147),(22,304,111,237,367,148),(23,305,112,238,368,149),(24,306,113,239,369,150),(25,307,114,240,370,151),(26,308,115,241,371,152),(27,309,116,242,372,153),(28,310,117,243,373,154),(29,311,118,244,374,155),(30,312,119,245,375,156),(31,313,120,246,376,157),(32,314,121,247,377,158),(33,315,122,248,378,159),(34,316,123,249,379,160),(35,317,124,250,380,161),(36,318,125,251,381,162),(37,319,126,252,382,163),(38,320,127,253,383,164),(39,321,128,254,384,165),(40,322,129,255,385,166),(41,323,130,256,386,167),(42,324,131,257,387,168),(43,325,132,258,388,169),(44,326,133,259,389,170),(45,327,134,260,390,171),(46,328,135,261,391,172),(47,329,136,262,392,173),(48,330,137,263,393,174),(49,331,138,264,394,175),(50,332,139,265,395,176),(51,333,140,266,396,177),(52,334,141,267,397,178),(53,335,142,268,398,179),(54,336,143,269,399,180),(55,337,144,270,400,181),(56,338,73,271,401,182),(57,339,74,272,402,183),(58,340,75,273,403,184),(59,341,76,274,404,185),(60,342,77,275,405,186),(61,343,78,276,406,187),(62,344,79,277,407,188),(63,345,80,278,408,189),(64,346,81,279,409,190),(65,347,82,280,410,191),(66,348,83,281,411,192),(67,349,84,282,412,193),(68,350,85,283,413,194),(69,351,86,284,414,195),(70,352,87,285,415,196),(71,353,88,286,416,197),(72,354,89,287,417,198)], [(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)]])

432 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 6A ··· 6X 8A ··· 8H 9A ··· 9R 12A ··· 12AF 18A ··· 18BB 24A ··· 24BL 36A ··· 36BT 72A ··· 72EN order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 8 ··· 8 9 ··· 9 12 ··· 12 18 ··· 18 24 ··· 24 36 ··· 36 72 ··· 72 size 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C3 C4 C4 C6 C6 C6 C6 C8 C9 C12 C12 C12 C12 C18 C18 C24 C24 C36 C36 C72 kernel C6×C72 C3×C72 C6×C36 C2×C72 C6×C24 C3×C36 C6×C18 C72 C2×C36 C3×C24 C6×C12 C3×C18 C2×C24 C36 C2×C18 C3×C12 C62 C24 C2×C12 C18 C3×C6 C12 C2×C6 C6 # reps 1 2 1 6 2 2 2 12 6 4 2 8 18 12 12 4 4 36 18 48 16 36 36 144

Matrix representation of C6×C72 in GL2(𝔽73) generated by

 72 0 0 8
,
 69 0 0 59
G:=sub<GL(2,GF(73))| [72,0,0,8],[69,0,0,59] >;

C6×C72 in GAP, Magma, Sage, TeX

C_6\times C_{72}
% in TeX

G:=Group("C6xC72");
// GroupNames label

G:=SmallGroup(432,209);
// by ID

G=gap.SmallGroup(432,209);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-3,-2,252,394,242]);
// Polycyclic

G:=Group<a,b|a^6=b^72=1,a*b=b*a>;
// generators/relations

׿
×
𝔽