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G = C9×C45order 405 = 34·5

Abelian group of type [9,45]

direct product, abelian, monomial, 3-elementary

Aliases: C9×C45, SmallGroup(405,2)

Series: Derived Chief Lower central Upper central

C1 — C9×C45
C1C3C32C3×C15C3×C45 — C9×C45
C1 — C9×C45
C1 — C9×C45

Generators and relations for C9×C45
 G = < a,b | a9=b45=1, ab=ba >


Smallest permutation representation of C9×C45
Regular action on 405 points
Generators in S405
(1 182 398 260 344 179 273 110 53)(2 183 399 261 345 180 274 111 54)(3 184 400 262 346 136 275 112 55)(4 185 401 263 347 137 276 113 56)(5 186 402 264 348 138 277 114 57)(6 187 403 265 349 139 278 115 58)(7 188 404 266 350 140 279 116 59)(8 189 405 267 351 141 280 117 60)(9 190 361 268 352 142 281 118 61)(10 191 362 269 353 143 282 119 62)(11 192 363 270 354 144 283 120 63)(12 193 364 226 355 145 284 121 64)(13 194 365 227 356 146 285 122 65)(14 195 366 228 357 147 286 123 66)(15 196 367 229 358 148 287 124 67)(16 197 368 230 359 149 288 125 68)(17 198 369 231 360 150 289 126 69)(18 199 370 232 316 151 290 127 70)(19 200 371 233 317 152 291 128 71)(20 201 372 234 318 153 292 129 72)(21 202 373 235 319 154 293 130 73)(22 203 374 236 320 155 294 131 74)(23 204 375 237 321 156 295 132 75)(24 205 376 238 322 157 296 133 76)(25 206 377 239 323 158 297 134 77)(26 207 378 240 324 159 298 135 78)(27 208 379 241 325 160 299 91 79)(28 209 380 242 326 161 300 92 80)(29 210 381 243 327 162 301 93 81)(30 211 382 244 328 163 302 94 82)(31 212 383 245 329 164 303 95 83)(32 213 384 246 330 165 304 96 84)(33 214 385 247 331 166 305 97 85)(34 215 386 248 332 167 306 98 86)(35 216 387 249 333 168 307 99 87)(36 217 388 250 334 169 308 100 88)(37 218 389 251 335 170 309 101 89)(38 219 390 252 336 171 310 102 90)(39 220 391 253 337 172 311 103 46)(40 221 392 254 338 173 312 104 47)(41 222 393 255 339 174 313 105 48)(42 223 394 256 340 175 314 106 49)(43 224 395 257 341 176 315 107 50)(44 225 396 258 342 177 271 108 51)(45 181 397 259 343 178 272 109 52)
(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)

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

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

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

405 conjugacy classes

class 1 3A···3H5A5B5C5D9A···9BT15A···15AF45A···45KB
order13···355559···915···1545···45
size11···111111···11···11···1

405 irreducible representations

dim111111
type+
imageC1C3C5C9C15C45
kernelC9×C45C3×C45C92C45C3×C9C9
# reps1847232288

Matrix representation of C9×C45 in GL2(𝔽181) generated by

10
080
,
1610
043
G:=sub<GL(2,GF(181))| [1,0,0,80],[161,0,0,43] >;

C9×C45 in GAP, Magma, Sage, TeX

C_9\times C_{45}
% in TeX

G:=Group("C9xC45");
// GroupNames label

G:=SmallGroup(405,2);
// by ID

G=gap.SmallGroup(405,2);
# by ID

G:=PCGroup([5,-3,-3,-5,-3,-3,225,456]);
// Polycyclic

G:=Group<a,b|a^9=b^45=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C9×C45 in TeX

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