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G = C15×C30order 450 = 2·32·52

Abelian group of type [15,30]

direct product, abelian, monomial

Aliases: C15×C30, SmallGroup(450,34)

Series: Derived Chief Lower central Upper central

C1 — C15×C30
C1C5C52C5×C15C152 — C15×C30
C1 — C15×C30
C1 — C15×C30

Generators and relations for C15×C30
 G = < a,b | a15=b30=1, ab=ba >

Subgroups: 96, all normal (8 characteristic)
C1, C2, C3 [×4], C5 [×6], C6 [×4], C32, C10 [×6], C15 [×24], C3×C6, C52, C30 [×24], C3×C15 [×6], C5×C10, C5×C15 [×4], C3×C30 [×6], C5×C30 [×4], C152, C15×C30
Quotients: C1, C2, C3 [×4], C5 [×6], C6 [×4], C32, C10 [×6], C15 [×24], C3×C6, C52, C30 [×24], C3×C15 [×6], C5×C10, C5×C15 [×4], C3×C30 [×6], C5×C30 [×4], C152, C15×C30

Smallest permutation representation of C15×C30
Regular action on 450 points
Generators in S450
(1 362 132 351 193 160 240 426 281 304 65 404 263 44 95)(2 363 133 352 194 161 211 427 282 305 66 405 264 45 96)(3 364 134 353 195 162 212 428 283 306 67 406 265 46 97)(4 365 135 354 196 163 213 429 284 307 68 407 266 47 98)(5 366 136 355 197 164 214 430 285 308 69 408 267 48 99)(6 367 137 356 198 165 215 431 286 309 70 409 268 49 100)(7 368 138 357 199 166 216 432 287 310 71 410 269 50 101)(8 369 139 358 200 167 217 433 288 311 72 411 270 51 102)(9 370 140 359 201 168 218 434 289 312 73 412 241 52 103)(10 371 141 360 202 169 219 435 290 313 74 413 242 53 104)(11 372 142 331 203 170 220 436 291 314 75 414 243 54 105)(12 373 143 332 204 171 221 437 292 315 76 415 244 55 106)(13 374 144 333 205 172 222 438 293 316 77 416 245 56 107)(14 375 145 334 206 173 223 439 294 317 78 417 246 57 108)(15 376 146 335 207 174 224 440 295 318 79 418 247 58 109)(16 377 147 336 208 175 225 441 296 319 80 419 248 59 110)(17 378 148 337 209 176 226 442 297 320 81 420 249 60 111)(18 379 149 338 210 177 227 443 298 321 82 391 250 31 112)(19 380 150 339 181 178 228 444 299 322 83 392 251 32 113)(20 381 121 340 182 179 229 445 300 323 84 393 252 33 114)(21 382 122 341 183 180 230 446 271 324 85 394 253 34 115)(22 383 123 342 184 151 231 447 272 325 86 395 254 35 116)(23 384 124 343 185 152 232 448 273 326 87 396 255 36 117)(24 385 125 344 186 153 233 449 274 327 88 397 256 37 118)(25 386 126 345 187 154 234 450 275 328 89 398 257 38 119)(26 387 127 346 188 155 235 421 276 329 90 399 258 39 120)(27 388 128 347 189 156 236 422 277 330 61 400 259 40 91)(28 389 129 348 190 157 237 423 278 301 62 401 260 41 92)(29 390 130 349 191 158 238 424 279 302 63 402 261 42 93)(30 361 131 350 192 159 239 425 280 303 64 403 262 43 94)
(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)(421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450)

G:=sub<Sym(450)| (1,362,132,351,193,160,240,426,281,304,65,404,263,44,95)(2,363,133,352,194,161,211,427,282,305,66,405,264,45,96)(3,364,134,353,195,162,212,428,283,306,67,406,265,46,97)(4,365,135,354,196,163,213,429,284,307,68,407,266,47,98)(5,366,136,355,197,164,214,430,285,308,69,408,267,48,99)(6,367,137,356,198,165,215,431,286,309,70,409,268,49,100)(7,368,138,357,199,166,216,432,287,310,71,410,269,50,101)(8,369,139,358,200,167,217,433,288,311,72,411,270,51,102)(9,370,140,359,201,168,218,434,289,312,73,412,241,52,103)(10,371,141,360,202,169,219,435,290,313,74,413,242,53,104)(11,372,142,331,203,170,220,436,291,314,75,414,243,54,105)(12,373,143,332,204,171,221,437,292,315,76,415,244,55,106)(13,374,144,333,205,172,222,438,293,316,77,416,245,56,107)(14,375,145,334,206,173,223,439,294,317,78,417,246,57,108)(15,376,146,335,207,174,224,440,295,318,79,418,247,58,109)(16,377,147,336,208,175,225,441,296,319,80,419,248,59,110)(17,378,148,337,209,176,226,442,297,320,81,420,249,60,111)(18,379,149,338,210,177,227,443,298,321,82,391,250,31,112)(19,380,150,339,181,178,228,444,299,322,83,392,251,32,113)(20,381,121,340,182,179,229,445,300,323,84,393,252,33,114)(21,382,122,341,183,180,230,446,271,324,85,394,253,34,115)(22,383,123,342,184,151,231,447,272,325,86,395,254,35,116)(23,384,124,343,185,152,232,448,273,326,87,396,255,36,117)(24,385,125,344,186,153,233,449,274,327,88,397,256,37,118)(25,386,126,345,187,154,234,450,275,328,89,398,257,38,119)(26,387,127,346,188,155,235,421,276,329,90,399,258,39,120)(27,388,128,347,189,156,236,422,277,330,61,400,259,40,91)(28,389,129,348,190,157,237,423,278,301,62,401,260,41,92)(29,390,130,349,191,158,238,424,279,302,63,402,261,42,93)(30,361,131,350,192,159,239,425,280,303,64,403,262,43,94), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)>;

G:=Group( (1,362,132,351,193,160,240,426,281,304,65,404,263,44,95)(2,363,133,352,194,161,211,427,282,305,66,405,264,45,96)(3,364,134,353,195,162,212,428,283,306,67,406,265,46,97)(4,365,135,354,196,163,213,429,284,307,68,407,266,47,98)(5,366,136,355,197,164,214,430,285,308,69,408,267,48,99)(6,367,137,356,198,165,215,431,286,309,70,409,268,49,100)(7,368,138,357,199,166,216,432,287,310,71,410,269,50,101)(8,369,139,358,200,167,217,433,288,311,72,411,270,51,102)(9,370,140,359,201,168,218,434,289,312,73,412,241,52,103)(10,371,141,360,202,169,219,435,290,313,74,413,242,53,104)(11,372,142,331,203,170,220,436,291,314,75,414,243,54,105)(12,373,143,332,204,171,221,437,292,315,76,415,244,55,106)(13,374,144,333,205,172,222,438,293,316,77,416,245,56,107)(14,375,145,334,206,173,223,439,294,317,78,417,246,57,108)(15,376,146,335,207,174,224,440,295,318,79,418,247,58,109)(16,377,147,336,208,175,225,441,296,319,80,419,248,59,110)(17,378,148,337,209,176,226,442,297,320,81,420,249,60,111)(18,379,149,338,210,177,227,443,298,321,82,391,250,31,112)(19,380,150,339,181,178,228,444,299,322,83,392,251,32,113)(20,381,121,340,182,179,229,445,300,323,84,393,252,33,114)(21,382,122,341,183,180,230,446,271,324,85,394,253,34,115)(22,383,123,342,184,151,231,447,272,325,86,395,254,35,116)(23,384,124,343,185,152,232,448,273,326,87,396,255,36,117)(24,385,125,344,186,153,233,449,274,327,88,397,256,37,118)(25,386,126,345,187,154,234,450,275,328,89,398,257,38,119)(26,387,127,346,188,155,235,421,276,329,90,399,258,39,120)(27,388,128,347,189,156,236,422,277,330,61,400,259,40,91)(28,389,129,348,190,157,237,423,278,301,62,401,260,41,92)(29,390,130,349,191,158,238,424,279,302,63,402,261,42,93)(30,361,131,350,192,159,239,425,280,303,64,403,262,43,94), (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)(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450) );

G=PermutationGroup([(1,362,132,351,193,160,240,426,281,304,65,404,263,44,95),(2,363,133,352,194,161,211,427,282,305,66,405,264,45,96),(3,364,134,353,195,162,212,428,283,306,67,406,265,46,97),(4,365,135,354,196,163,213,429,284,307,68,407,266,47,98),(5,366,136,355,197,164,214,430,285,308,69,408,267,48,99),(6,367,137,356,198,165,215,431,286,309,70,409,268,49,100),(7,368,138,357,199,166,216,432,287,310,71,410,269,50,101),(8,369,139,358,200,167,217,433,288,311,72,411,270,51,102),(9,370,140,359,201,168,218,434,289,312,73,412,241,52,103),(10,371,141,360,202,169,219,435,290,313,74,413,242,53,104),(11,372,142,331,203,170,220,436,291,314,75,414,243,54,105),(12,373,143,332,204,171,221,437,292,315,76,415,244,55,106),(13,374,144,333,205,172,222,438,293,316,77,416,245,56,107),(14,375,145,334,206,173,223,439,294,317,78,417,246,57,108),(15,376,146,335,207,174,224,440,295,318,79,418,247,58,109),(16,377,147,336,208,175,225,441,296,319,80,419,248,59,110),(17,378,148,337,209,176,226,442,297,320,81,420,249,60,111),(18,379,149,338,210,177,227,443,298,321,82,391,250,31,112),(19,380,150,339,181,178,228,444,299,322,83,392,251,32,113),(20,381,121,340,182,179,229,445,300,323,84,393,252,33,114),(21,382,122,341,183,180,230,446,271,324,85,394,253,34,115),(22,383,123,342,184,151,231,447,272,325,86,395,254,35,116),(23,384,124,343,185,152,232,448,273,326,87,396,255,36,117),(24,385,125,344,186,153,233,449,274,327,88,397,256,37,118),(25,386,126,345,187,154,234,450,275,328,89,398,257,38,119),(26,387,127,346,188,155,235,421,276,329,90,399,258,39,120),(27,388,128,347,189,156,236,422,277,330,61,400,259,40,91),(28,389,129,348,190,157,237,423,278,301,62,401,260,41,92),(29,390,130,349,191,158,238,424,279,302,63,402,261,42,93),(30,361,131,350,192,159,239,425,280,303,64,403,262,43,94)], [(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),(421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450)])

450 conjugacy classes

class 1  2 3A···3H5A···5X6A···6H10A···10X15A···15GJ30A···30GJ
order123···35···56···610···1015···1530···30
size111···11···11···11···11···11···1

450 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC15×C30C152C5×C30C3×C30C5×C15C3×C15C30C15
# reps11824824192192

Matrix representation of C15×C30 in GL2(𝔽31) generated by

140
09
,
230
05
G:=sub<GL(2,GF(31))| [14,0,0,9],[23,0,0,5] >;

C15×C30 in GAP, Magma, Sage, TeX

C_{15}\times C_{30}
% in TeX

G:=Group("C15xC30");
// GroupNames label

G:=SmallGroup(450,34);
// by ID

G=gap.SmallGroup(450,34);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-5]);
// Polycyclic

G:=Group<a,b|a^15=b^30=1,a*b=b*a>;
// generators/relations

׿
×
𝔽