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G = C14×C28order 392 = 23·72

Abelian group of type [14,28]

direct product, abelian, monomial

Aliases: C14×C28, SmallGroup(392,33)

Series: Derived Chief Lower central Upper central

C1 — C14×C28
C1C2C14C7×C14C7×C28 — C14×C28
C1 — C14×C28
C1 — C14×C28

Generators and relations for C14×C28
 G = < a,b | a14=b28=1, ab=ba >

Subgroups: 80, all normal (8 characteristic)
C1, C2, C2 [×2], C4 [×2], C22, C7 [×8], C2×C4, C14 [×24], C28 [×16], C2×C14 [×8], C72, C2×C28 [×8], C7×C14, C7×C14 [×2], C7×C28 [×2], C142, C14×C28
Quotients: C1, C2 [×3], C4 [×2], C22, C7 [×8], C2×C4, C14 [×24], C28 [×16], C2×C14 [×8], C72, C2×C28 [×8], C7×C14 [×3], C7×C28 [×2], C142, C14×C28

Smallest permutation representation of C14×C28
Regular action on 392 points
Generators in S392
(1 59 368 55 268 317 199 230 175 133 158 286 348 88)(2 60 369 56 269 318 200 231 176 134 159 287 349 89)(3 61 370 29 270 319 201 232 177 135 160 288 350 90)(4 62 371 30 271 320 202 233 178 136 161 289 351 91)(5 63 372 31 272 321 203 234 179 137 162 290 352 92)(6 64 373 32 273 322 204 235 180 138 163 291 353 93)(7 65 374 33 274 323 205 236 181 139 164 292 354 94)(8 66 375 34 275 324 206 237 182 140 165 293 355 95)(9 67 376 35 276 325 207 238 183 113 166 294 356 96)(10 68 377 36 277 326 208 239 184 114 167 295 357 97)(11 69 378 37 278 327 209 240 185 115 168 296 358 98)(12 70 379 38 279 328 210 241 186 116 141 297 359 99)(13 71 380 39 280 329 211 242 187 117 142 298 360 100)(14 72 381 40 253 330 212 243 188 118 143 299 361 101)(15 73 382 41 254 331 213 244 189 119 144 300 362 102)(16 74 383 42 255 332 214 245 190 120 145 301 363 103)(17 75 384 43 256 333 215 246 191 121 146 302 364 104)(18 76 385 44 257 334 216 247 192 122 147 303 337 105)(19 77 386 45 258 335 217 248 193 123 148 304 338 106)(20 78 387 46 259 336 218 249 194 124 149 305 339 107)(21 79 388 47 260 309 219 250 195 125 150 306 340 108)(22 80 389 48 261 310 220 251 196 126 151 307 341 109)(23 81 390 49 262 311 221 252 169 127 152 308 342 110)(24 82 391 50 263 312 222 225 170 128 153 281 343 111)(25 83 392 51 264 313 223 226 171 129 154 282 344 112)(26 84 365 52 265 314 224 227 172 130 155 283 345 85)(27 57 366 53 266 315 197 228 173 131 156 284 346 86)(28 58 367 54 267 316 198 229 174 132 157 285 347 87)
(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)

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

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

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

392 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7AV14A···14EN28A···28GJ
order122244447···714···1428···28
size111111111···11···11···1

392 irreducible representations

dim11111111
type+++
imageC1C2C2C4C7C14C14C28
kernelC14×C28C7×C28C142C7×C14C2×C28C28C2×C14C14
# reps1214489648192

Matrix representation of C14×C28 in GL2(𝔽29) generated by

50
01
,
190
019
G:=sub<GL(2,GF(29))| [5,0,0,1],[19,0,0,19] >;

C14×C28 in GAP, Magma, Sage, TeX

C_{14}\times C_{28}
% in TeX

G:=Group("C14xC28");
// GroupNames label

G:=SmallGroup(392,33);
// by ID

G=gap.SmallGroup(392,33);
# by ID

G:=PCGroup([5,-2,-2,-7,-7,-2,980]);
// Polycyclic

G:=Group<a,b|a^14=b^28=1,a*b=b*a>;
// generators/relations

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