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## G = C4×C108order 432 = 24·33

### Abelian group of type [4,108]

Aliases: C4×C108, SmallGroup(432,20)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C108
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×C54 — C2×C108 — C4×C108
 Lower central C1 — C4×C108
 Upper central C1 — C4×C108

Generators and relations for C4×C108
G = < a,b | a4=b108=1, ab=ba >

Subgroups: 60, all normal (12 characteristic)
C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C2×C4 [×3], C9, C12 [×6], C2×C6, C42, C18 [×3], C2×C12 [×3], C27, C36 [×6], C2×C18, C4×C12, C54 [×3], C2×C36 [×3], C108 [×6], C2×C54, C4×C36, C2×C108 [×3], C4×C108
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C6 [×3], C2×C4 [×3], C9, C12 [×6], C2×C6, C42, C18 [×3], C2×C12 [×3], C27, C36 [×6], C2×C18, C4×C12, C54 [×3], C2×C36 [×3], C108 [×6], C2×C54, C4×C36, C2×C108 [×3], C4×C108

Smallest permutation representation of C4×C108
Regular action on 432 points
Generators in S432
(1 209 337 297)(2 210 338 298)(3 211 339 299)(4 212 340 300)(5 213 341 301)(6 214 342 302)(7 215 343 303)(8 216 344 304)(9 109 345 305)(10 110 346 306)(11 111 347 307)(12 112 348 308)(13 113 349 309)(14 114 350 310)(15 115 351 311)(16 116 352 312)(17 117 353 313)(18 118 354 314)(19 119 355 315)(20 120 356 316)(21 121 357 317)(22 122 358 318)(23 123 359 319)(24 124 360 320)(25 125 361 321)(26 126 362 322)(27 127 363 323)(28 128 364 324)(29 129 365 217)(30 130 366 218)(31 131 367 219)(32 132 368 220)(33 133 369 221)(34 134 370 222)(35 135 371 223)(36 136 372 224)(37 137 373 225)(38 138 374 226)(39 139 375 227)(40 140 376 228)(41 141 377 229)(42 142 378 230)(43 143 379 231)(44 144 380 232)(45 145 381 233)(46 146 382 234)(47 147 383 235)(48 148 384 236)(49 149 385 237)(50 150 386 238)(51 151 387 239)(52 152 388 240)(53 153 389 241)(54 154 390 242)(55 155 391 243)(56 156 392 244)(57 157 393 245)(58 158 394 246)(59 159 395 247)(60 160 396 248)(61 161 397 249)(62 162 398 250)(63 163 399 251)(64 164 400 252)(65 165 401 253)(66 166 402 254)(67 167 403 255)(68 168 404 256)(69 169 405 257)(70 170 406 258)(71 171 407 259)(72 172 408 260)(73 173 409 261)(74 174 410 262)(75 175 411 263)(76 176 412 264)(77 177 413 265)(78 178 414 266)(79 179 415 267)(80 180 416 268)(81 181 417 269)(82 182 418 270)(83 183 419 271)(84 184 420 272)(85 185 421 273)(86 186 422 274)(87 187 423 275)(88 188 424 276)(89 189 425 277)(90 190 426 278)(91 191 427 279)(92 192 428 280)(93 193 429 281)(94 194 430 282)(95 195 431 283)(96 196 432 284)(97 197 325 285)(98 198 326 286)(99 199 327 287)(100 200 328 288)(101 201 329 289)(102 202 330 290)(103 203 331 291)(104 204 332 292)(105 205 333 293)(106 206 334 294)(107 207 335 295)(108 208 336 296)
(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)| (1,209,337,297)(2,210,338,298)(3,211,339,299)(4,212,340,300)(5,213,341,301)(6,214,342,302)(7,215,343,303)(8,216,344,304)(9,109,345,305)(10,110,346,306)(11,111,347,307)(12,112,348,308)(13,113,349,309)(14,114,350,310)(15,115,351,311)(16,116,352,312)(17,117,353,313)(18,118,354,314)(19,119,355,315)(20,120,356,316)(21,121,357,317)(22,122,358,318)(23,123,359,319)(24,124,360,320)(25,125,361,321)(26,126,362,322)(27,127,363,323)(28,128,364,324)(29,129,365,217)(30,130,366,218)(31,131,367,219)(32,132,368,220)(33,133,369,221)(34,134,370,222)(35,135,371,223)(36,136,372,224)(37,137,373,225)(38,138,374,226)(39,139,375,227)(40,140,376,228)(41,141,377,229)(42,142,378,230)(43,143,379,231)(44,144,380,232)(45,145,381,233)(46,146,382,234)(47,147,383,235)(48,148,384,236)(49,149,385,237)(50,150,386,238)(51,151,387,239)(52,152,388,240)(53,153,389,241)(54,154,390,242)(55,155,391,243)(56,156,392,244)(57,157,393,245)(58,158,394,246)(59,159,395,247)(60,160,396,248)(61,161,397,249)(62,162,398,250)(63,163,399,251)(64,164,400,252)(65,165,401,253)(66,166,402,254)(67,167,403,255)(68,168,404,256)(69,169,405,257)(70,170,406,258)(71,171,407,259)(72,172,408,260)(73,173,409,261)(74,174,410,262)(75,175,411,263)(76,176,412,264)(77,177,413,265)(78,178,414,266)(79,179,415,267)(80,180,416,268)(81,181,417,269)(82,182,418,270)(83,183,419,271)(84,184,420,272)(85,185,421,273)(86,186,422,274)(87,187,423,275)(88,188,424,276)(89,189,425,277)(90,190,426,278)(91,191,427,279)(92,192,428,280)(93,193,429,281)(94,194,430,282)(95,195,431,283)(96,196,432,284)(97,197,325,285)(98,198,326,286)(99,199,327,287)(100,200,328,288)(101,201,329,289)(102,202,330,290)(103,203,331,291)(104,204,332,292)(105,205,333,293)(106,206,334,294)(107,207,335,295)(108,208,336,296), (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( (1,209,337,297)(2,210,338,298)(3,211,339,299)(4,212,340,300)(5,213,341,301)(6,214,342,302)(7,215,343,303)(8,216,344,304)(9,109,345,305)(10,110,346,306)(11,111,347,307)(12,112,348,308)(13,113,349,309)(14,114,350,310)(15,115,351,311)(16,116,352,312)(17,117,353,313)(18,118,354,314)(19,119,355,315)(20,120,356,316)(21,121,357,317)(22,122,358,318)(23,123,359,319)(24,124,360,320)(25,125,361,321)(26,126,362,322)(27,127,363,323)(28,128,364,324)(29,129,365,217)(30,130,366,218)(31,131,367,219)(32,132,368,220)(33,133,369,221)(34,134,370,222)(35,135,371,223)(36,136,372,224)(37,137,373,225)(38,138,374,226)(39,139,375,227)(40,140,376,228)(41,141,377,229)(42,142,378,230)(43,143,379,231)(44,144,380,232)(45,145,381,233)(46,146,382,234)(47,147,383,235)(48,148,384,236)(49,149,385,237)(50,150,386,238)(51,151,387,239)(52,152,388,240)(53,153,389,241)(54,154,390,242)(55,155,391,243)(56,156,392,244)(57,157,393,245)(58,158,394,246)(59,159,395,247)(60,160,396,248)(61,161,397,249)(62,162,398,250)(63,163,399,251)(64,164,400,252)(65,165,401,253)(66,166,402,254)(67,167,403,255)(68,168,404,256)(69,169,405,257)(70,170,406,258)(71,171,407,259)(72,172,408,260)(73,173,409,261)(74,174,410,262)(75,175,411,263)(76,176,412,264)(77,177,413,265)(78,178,414,266)(79,179,415,267)(80,180,416,268)(81,181,417,269)(82,182,418,270)(83,183,419,271)(84,184,420,272)(85,185,421,273)(86,186,422,274)(87,187,423,275)(88,188,424,276)(89,189,425,277)(90,190,426,278)(91,191,427,279)(92,192,428,280)(93,193,429,281)(94,194,430,282)(95,195,431,283)(96,196,432,284)(97,197,325,285)(98,198,326,286)(99,199,327,287)(100,200,328,288)(101,201,329,289)(102,202,330,290)(103,203,331,291)(104,204,332,292)(105,205,333,293)(106,206,334,294)(107,207,335,295)(108,208,336,296), (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([(1,209,337,297),(2,210,338,298),(3,211,339,299),(4,212,340,300),(5,213,341,301),(6,214,342,302),(7,215,343,303),(8,216,344,304),(9,109,345,305),(10,110,346,306),(11,111,347,307),(12,112,348,308),(13,113,349,309),(14,114,350,310),(15,115,351,311),(16,116,352,312),(17,117,353,313),(18,118,354,314),(19,119,355,315),(20,120,356,316),(21,121,357,317),(22,122,358,318),(23,123,359,319),(24,124,360,320),(25,125,361,321),(26,126,362,322),(27,127,363,323),(28,128,364,324),(29,129,365,217),(30,130,366,218),(31,131,367,219),(32,132,368,220),(33,133,369,221),(34,134,370,222),(35,135,371,223),(36,136,372,224),(37,137,373,225),(38,138,374,226),(39,139,375,227),(40,140,376,228),(41,141,377,229),(42,142,378,230),(43,143,379,231),(44,144,380,232),(45,145,381,233),(46,146,382,234),(47,147,383,235),(48,148,384,236),(49,149,385,237),(50,150,386,238),(51,151,387,239),(52,152,388,240),(53,153,389,241),(54,154,390,242),(55,155,391,243),(56,156,392,244),(57,157,393,245),(58,158,394,246),(59,159,395,247),(60,160,396,248),(61,161,397,249),(62,162,398,250),(63,163,399,251),(64,164,400,252),(65,165,401,253),(66,166,402,254),(67,167,403,255),(68,168,404,256),(69,169,405,257),(70,170,406,258),(71,171,407,259),(72,172,408,260),(73,173,409,261),(74,174,410,262),(75,175,411,263),(76,176,412,264),(77,177,413,265),(78,178,414,266),(79,179,415,267),(80,180,416,268),(81,181,417,269),(82,182,418,270),(83,183,419,271),(84,184,420,272),(85,185,421,273),(86,186,422,274),(87,187,423,275),(88,188,424,276),(89,189,425,277),(90,190,426,278),(91,191,427,279),(92,192,428,280),(93,193,429,281),(94,194,430,282),(95,195,431,283),(96,196,432,284),(97,197,325,285),(98,198,326,286),(99,199,327,287),(100,200,328,288),(101,201,329,289),(102,202,330,290),(103,203,331,291),(104,204,332,292),(105,205,333,293),(106,206,334,294),(107,207,335,295),(108,208,336,296)], [(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 3B 4A ··· 4L 6A ··· 6F 9A ··· 9F 12A ··· 12X 18A ··· 18R 27A ··· 27R 36A ··· 36BT 54A ··· 54BB 108A ··· 108HH order 1 2 2 2 3 3 4 ··· 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 108 ··· 108 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

432 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C9 C12 C18 C27 C36 C54 C108 kernel C4×C108 C2×C108 C4×C36 C108 C2×C36 C4×C12 C36 C2×C12 C42 C12 C2×C4 C4 # reps 1 3 2 12 6 6 24 18 18 72 54 216

Matrix representation of C4×C108 in GL2(𝔽109) generated by

 1 0 0 76
,
 77 0 0 29
G:=sub<GL(2,GF(109))| [1,0,0,76],[77,0,0,29] >;

C4×C108 in GAP, Magma, Sage, TeX

C_4\times C_{108}
% in TeX

G:=Group("C4xC108");
// GroupNames label

G:=SmallGroup(432,20);
// by ID

G=gap.SmallGroup(432,20);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,268,166]);
// Polycyclic

G:=Group<a,b|a^4=b^108=1,a*b=b*a>;
// generators/relations

׿
×
𝔽