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G = C6×C72order 432 = 24·33

Abelian group of type [6,72]

direct product, abelian, monomial

Aliases: C6×C72, SmallGroup(432,209)

Series: Derived Chief Lower central Upper central

C1 — C6×C72
C1C2C6C12C3×C12C3×C36C3×C72 — C6×C72
C1 — C6×C72
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 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×11], C8 [×2], C2×C4, C9 [×3], C32, C12 [×2], C12 [×6], C2×C6, C2×C6 [×3], C2×C8, C18 [×9], C3×C6, C3×C6 [×2], C24 [×8], C2×C12, C2×C12 [×3], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C2×C24, C2×C24 [×3], C3×C18, C3×C18 [×2], C72 [×6], C2×C36 [×3], C3×C24 [×2], C6×C12, C3×C36 [×2], C6×C18, C2×C72 [×3], C6×C24, C3×C72 [×2], C6×C36, C6×C72
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C9 [×3], C32, C12 [×8], C2×C6 [×4], C2×C8, C18 [×9], C3×C6 [×3], C24 [×8], C2×C12 [×4], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C2×C24 [×4], C3×C18 [×3], C72 [×6], C2×C36 [×3], C3×C24 [×2], C6×C12, C3×C36 [×2], C6×C18, C2×C72 [×3], C6×C24, C3×C72 [×2], C6×C36, C6×C72

Smallest permutation representation of C6×C72
Regular action on 432 points
Generators in S432
(1 224 305 419 125 149)(2 225 306 420 126 150)(3 226 307 421 127 151)(4 227 308 422 128 152)(5 228 309 423 129 153)(6 229 310 424 130 154)(7 230 311 425 131 155)(8 231 312 426 132 156)(9 232 313 427 133 157)(10 233 314 428 134 158)(11 234 315 429 135 159)(12 235 316 430 136 160)(13 236 317 431 137 161)(14 237 318 432 138 162)(15 238 319 361 139 163)(16 239 320 362 140 164)(17 240 321 363 141 165)(18 241 322 364 142 166)(19 242 323 365 143 167)(20 243 324 366 144 168)(21 244 325 367 73 169)(22 245 326 368 74 170)(23 246 327 369 75 171)(24 247 328 370 76 172)(25 248 329 371 77 173)(26 249 330 372 78 174)(27 250 331 373 79 175)(28 251 332 374 80 176)(29 252 333 375 81 177)(30 253 334 376 82 178)(31 254 335 377 83 179)(32 255 336 378 84 180)(33 256 337 379 85 181)(34 257 338 380 86 182)(35 258 339 381 87 183)(36 259 340 382 88 184)(37 260 341 383 89 185)(38 261 342 384 90 186)(39 262 343 385 91 187)(40 263 344 386 92 188)(41 264 345 387 93 189)(42 265 346 388 94 190)(43 266 347 389 95 191)(44 267 348 390 96 192)(45 268 349 391 97 193)(46 269 350 392 98 194)(47 270 351 393 99 195)(48 271 352 394 100 196)(49 272 353 395 101 197)(50 273 354 396 102 198)(51 274 355 397 103 199)(52 275 356 398 104 200)(53 276 357 399 105 201)(54 277 358 400 106 202)(55 278 359 401 107 203)(56 279 360 402 108 204)(57 280 289 403 109 205)(58 281 290 404 110 206)(59 282 291 405 111 207)(60 283 292 406 112 208)(61 284 293 407 113 209)(62 285 294 408 114 210)(63 286 295 409 115 211)(64 287 296 410 116 212)(65 288 297 411 117 213)(66 217 298 412 118 214)(67 218 299 413 119 215)(68 219 300 414 120 216)(69 220 301 415 121 145)(70 221 302 416 122 146)(71 222 303 417 123 147)(72 223 304 418 124 148)
(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)| (1,224,305,419,125,149)(2,225,306,420,126,150)(3,226,307,421,127,151)(4,227,308,422,128,152)(5,228,309,423,129,153)(6,229,310,424,130,154)(7,230,311,425,131,155)(8,231,312,426,132,156)(9,232,313,427,133,157)(10,233,314,428,134,158)(11,234,315,429,135,159)(12,235,316,430,136,160)(13,236,317,431,137,161)(14,237,318,432,138,162)(15,238,319,361,139,163)(16,239,320,362,140,164)(17,240,321,363,141,165)(18,241,322,364,142,166)(19,242,323,365,143,167)(20,243,324,366,144,168)(21,244,325,367,73,169)(22,245,326,368,74,170)(23,246,327,369,75,171)(24,247,328,370,76,172)(25,248,329,371,77,173)(26,249,330,372,78,174)(27,250,331,373,79,175)(28,251,332,374,80,176)(29,252,333,375,81,177)(30,253,334,376,82,178)(31,254,335,377,83,179)(32,255,336,378,84,180)(33,256,337,379,85,181)(34,257,338,380,86,182)(35,258,339,381,87,183)(36,259,340,382,88,184)(37,260,341,383,89,185)(38,261,342,384,90,186)(39,262,343,385,91,187)(40,263,344,386,92,188)(41,264,345,387,93,189)(42,265,346,388,94,190)(43,266,347,389,95,191)(44,267,348,390,96,192)(45,268,349,391,97,193)(46,269,350,392,98,194)(47,270,351,393,99,195)(48,271,352,394,100,196)(49,272,353,395,101,197)(50,273,354,396,102,198)(51,274,355,397,103,199)(52,275,356,398,104,200)(53,276,357,399,105,201)(54,277,358,400,106,202)(55,278,359,401,107,203)(56,279,360,402,108,204)(57,280,289,403,109,205)(58,281,290,404,110,206)(59,282,291,405,111,207)(60,283,292,406,112,208)(61,284,293,407,113,209)(62,285,294,408,114,210)(63,286,295,409,115,211)(64,287,296,410,116,212)(65,288,297,411,117,213)(66,217,298,412,118,214)(67,218,299,413,119,215)(68,219,300,414,120,216)(69,220,301,415,121,145)(70,221,302,416,122,146)(71,222,303,417,123,147)(72,223,304,418,124,148), (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( (1,224,305,419,125,149)(2,225,306,420,126,150)(3,226,307,421,127,151)(4,227,308,422,128,152)(5,228,309,423,129,153)(6,229,310,424,130,154)(7,230,311,425,131,155)(8,231,312,426,132,156)(9,232,313,427,133,157)(10,233,314,428,134,158)(11,234,315,429,135,159)(12,235,316,430,136,160)(13,236,317,431,137,161)(14,237,318,432,138,162)(15,238,319,361,139,163)(16,239,320,362,140,164)(17,240,321,363,141,165)(18,241,322,364,142,166)(19,242,323,365,143,167)(20,243,324,366,144,168)(21,244,325,367,73,169)(22,245,326,368,74,170)(23,246,327,369,75,171)(24,247,328,370,76,172)(25,248,329,371,77,173)(26,249,330,372,78,174)(27,250,331,373,79,175)(28,251,332,374,80,176)(29,252,333,375,81,177)(30,253,334,376,82,178)(31,254,335,377,83,179)(32,255,336,378,84,180)(33,256,337,379,85,181)(34,257,338,380,86,182)(35,258,339,381,87,183)(36,259,340,382,88,184)(37,260,341,383,89,185)(38,261,342,384,90,186)(39,262,343,385,91,187)(40,263,344,386,92,188)(41,264,345,387,93,189)(42,265,346,388,94,190)(43,266,347,389,95,191)(44,267,348,390,96,192)(45,268,349,391,97,193)(46,269,350,392,98,194)(47,270,351,393,99,195)(48,271,352,394,100,196)(49,272,353,395,101,197)(50,273,354,396,102,198)(51,274,355,397,103,199)(52,275,356,398,104,200)(53,276,357,399,105,201)(54,277,358,400,106,202)(55,278,359,401,107,203)(56,279,360,402,108,204)(57,280,289,403,109,205)(58,281,290,404,110,206)(59,282,291,405,111,207)(60,283,292,406,112,208)(61,284,293,407,113,209)(62,285,294,408,114,210)(63,286,295,409,115,211)(64,287,296,410,116,212)(65,288,297,411,117,213)(66,217,298,412,118,214)(67,218,299,413,119,215)(68,219,300,414,120,216)(69,220,301,415,121,145)(70,221,302,416,122,146)(71,222,303,417,123,147)(72,223,304,418,124,148), (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([(1,224,305,419,125,149),(2,225,306,420,126,150),(3,226,307,421,127,151),(4,227,308,422,128,152),(5,228,309,423,129,153),(6,229,310,424,130,154),(7,230,311,425,131,155),(8,231,312,426,132,156),(9,232,313,427,133,157),(10,233,314,428,134,158),(11,234,315,429,135,159),(12,235,316,430,136,160),(13,236,317,431,137,161),(14,237,318,432,138,162),(15,238,319,361,139,163),(16,239,320,362,140,164),(17,240,321,363,141,165),(18,241,322,364,142,166),(19,242,323,365,143,167),(20,243,324,366,144,168),(21,244,325,367,73,169),(22,245,326,368,74,170),(23,246,327,369,75,171),(24,247,328,370,76,172),(25,248,329,371,77,173),(26,249,330,372,78,174),(27,250,331,373,79,175),(28,251,332,374,80,176),(29,252,333,375,81,177),(30,253,334,376,82,178),(31,254,335,377,83,179),(32,255,336,378,84,180),(33,256,337,379,85,181),(34,257,338,380,86,182),(35,258,339,381,87,183),(36,259,340,382,88,184),(37,260,341,383,89,185),(38,261,342,384,90,186),(39,262,343,385,91,187),(40,263,344,386,92,188),(41,264,345,387,93,189),(42,265,346,388,94,190),(43,266,347,389,95,191),(44,267,348,390,96,192),(45,268,349,391,97,193),(46,269,350,392,98,194),(47,270,351,393,99,195),(48,271,352,394,100,196),(49,272,353,395,101,197),(50,273,354,396,102,198),(51,274,355,397,103,199),(52,275,356,398,104,200),(53,276,357,399,105,201),(54,277,358,400,106,202),(55,278,359,401,107,203),(56,279,360,402,108,204),(57,280,289,403,109,205),(58,281,290,404,110,206),(59,282,291,405,111,207),(60,283,292,406,112,208),(61,284,293,407,113,209),(62,285,294,408,114,210),(63,286,295,409,115,211),(64,287,296,410,116,212),(65,288,297,411,117,213),(66,217,298,412,118,214),(67,218,299,413,119,215),(68,219,300,414,120,216),(69,220,301,415,121,145),(70,221,302,416,122,146),(71,222,303,417,123,147),(72,223,304,418,124,148)], [(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 2A2B2C3A···3H4A4B4C4D6A···6X8A···8H9A···9R12A···12AF18A···18BB24A···24BL36A···36BT72A···72EN
order12223···344446···68···89···912···1218···1824···2436···3672···72
size11111···111111···11···11···11···11···11···11···11···1

432 irreducible representations

dim111111111111111111111111
type+++
imageC1C2C2C3C3C4C4C6C6C6C6C8C9C12C12C12C12C18C18C24C24C36C36C72
kernelC6×C72C3×C72C6×C36C2×C72C6×C24C3×C36C6×C18C72C2×C36C3×C24C6×C12C3×C18C2×C24C36C2×C18C3×C12C62C24C2×C12C18C3×C6C12C2×C6C6
# reps121622212642818121244361848163636144

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

720
08
,
690
059
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