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## G = C4×C112order 448 = 26·7

### Abelian group of type [4,112]

Aliases: C4×C112, SmallGroup(448,149)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C112
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C56 — C2×C112 — C4×C112
 Lower central C1 — C4×C112
 Upper central C1 — C4×C112

Generators and relations for C4×C112
G = < a,b | a4=b112=1, ab=ba >

Smallest permutation representation of C4×C112
Regular action on 448 points
Generators in S448
(1 142 316 360)(2 143 317 361)(3 144 318 362)(4 145 319 363)(5 146 320 364)(6 147 321 365)(7 148 322 366)(8 149 323 367)(9 150 324 368)(10 151 325 369)(11 152 326 370)(12 153 327 371)(13 154 328 372)(14 155 329 373)(15 156 330 374)(16 157 331 375)(17 158 332 376)(18 159 333 377)(19 160 334 378)(20 161 335 379)(21 162 336 380)(22 163 225 381)(23 164 226 382)(24 165 227 383)(25 166 228 384)(26 167 229 385)(27 168 230 386)(28 169 231 387)(29 170 232 388)(30 171 233 389)(31 172 234 390)(32 173 235 391)(33 174 236 392)(34 175 237 393)(35 176 238 394)(36 177 239 395)(37 178 240 396)(38 179 241 397)(39 180 242 398)(40 181 243 399)(41 182 244 400)(42 183 245 401)(43 184 246 402)(44 185 247 403)(45 186 248 404)(46 187 249 405)(47 188 250 406)(48 189 251 407)(49 190 252 408)(50 191 253 409)(51 192 254 410)(52 193 255 411)(53 194 256 412)(54 195 257 413)(55 196 258 414)(56 197 259 415)(57 198 260 416)(58 199 261 417)(59 200 262 418)(60 201 263 419)(61 202 264 420)(62 203 265 421)(63 204 266 422)(64 205 267 423)(65 206 268 424)(66 207 269 425)(67 208 270 426)(68 209 271 427)(69 210 272 428)(70 211 273 429)(71 212 274 430)(72 213 275 431)(73 214 276 432)(74 215 277 433)(75 216 278 434)(76 217 279 435)(77 218 280 436)(78 219 281 437)(79 220 282 438)(80 221 283 439)(81 222 284 440)(82 223 285 441)(83 224 286 442)(84 113 287 443)(85 114 288 444)(86 115 289 445)(87 116 290 446)(88 117 291 447)(89 118 292 448)(90 119 293 337)(91 120 294 338)(92 121 295 339)(93 122 296 340)(94 123 297 341)(95 124 298 342)(96 125 299 343)(97 126 300 344)(98 127 301 345)(99 128 302 346)(100 129 303 347)(101 130 304 348)(102 131 305 349)(103 132 306 350)(104 133 307 351)(105 134 308 352)(106 135 309 353)(107 136 310 354)(108 137 311 355)(109 138 312 356)(110 139 313 357)(111 140 314 358)(112 141 315 359)
(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)

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

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

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

448 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 7A ··· 7F 8A ··· 8P 14A ··· 14R 16A ··· 16AF 28A ··· 28BT 56A ··· 56CR 112A ··· 112GJ order 1 2 2 2 4 ··· 4 7 ··· 7 8 ··· 8 14 ··· 14 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

448 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C4 C7 C8 C8 C14 C14 C16 C28 C28 C28 C56 C56 C112 kernel C4×C112 C4×C56 C2×C112 C112 C4×C28 C2×C56 C4×C16 C56 C2×C28 C4×C8 C2×C16 C28 C16 C42 C2×C8 C8 C2×C4 C4 # reps 1 1 2 8 2 2 6 8 8 6 12 32 48 12 12 48 48 192

Matrix representation of C4×C112 in GL3(𝔽113) generated by

 98 0 0 0 1 0 0 0 98
,
 1 0 0 0 13 0 0 0 35
G:=sub<GL(3,GF(113))| [98,0,0,0,1,0,0,0,98],[1,0,0,0,13,0,0,0,35] >;

C4×C112 in GAP, Magma, Sage, TeX

C_4\times C_{112}
% in TeX

G:=Group("C4xC112");
// GroupNames label

G:=SmallGroup(448,149);
// by ID

G=gap.SmallGroup(448,149);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,400,136,124]);
// Polycyclic

G:=Group<a,b|a^4=b^112=1,a*b=b*a>;
// generators/relations

Export

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