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G = C7×C56order 392 = 23·72

Abelian group of type [7,56]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C56, SmallGroup(392,16)

Series: Derived Chief Lower central Upper central

C1 — C7×C56
C1C2C4C28C7×C28 — C7×C56
C1 — C7×C56
C1 — C7×C56

Generators and relations for C7×C56
 G = < a,b | a7=b56=1, ab=ba >


Smallest permutation representation of C7×C56
Regular action on 392 points
Generators in S392
(1 386 177 158 237 289 62)(2 387 178 159 238 290 63)(3 388 179 160 239 291 64)(4 389 180 161 240 292 65)(5 390 181 162 241 293 66)(6 391 182 163 242 294 67)(7 392 183 164 243 295 68)(8 337 184 165 244 296 69)(9 338 185 166 245 297 70)(10 339 186 167 246 298 71)(11 340 187 168 247 299 72)(12 341 188 113 248 300 73)(13 342 189 114 249 301 74)(14 343 190 115 250 302 75)(15 344 191 116 251 303 76)(16 345 192 117 252 304 77)(17 346 193 118 253 305 78)(18 347 194 119 254 306 79)(19 348 195 120 255 307 80)(20 349 196 121 256 308 81)(21 350 197 122 257 309 82)(22 351 198 123 258 310 83)(23 352 199 124 259 311 84)(24 353 200 125 260 312 85)(25 354 201 126 261 313 86)(26 355 202 127 262 314 87)(27 356 203 128 263 315 88)(28 357 204 129 264 316 89)(29 358 205 130 265 317 90)(30 359 206 131 266 318 91)(31 360 207 132 267 319 92)(32 361 208 133 268 320 93)(33 362 209 134 269 321 94)(34 363 210 135 270 322 95)(35 364 211 136 271 323 96)(36 365 212 137 272 324 97)(37 366 213 138 273 325 98)(38 367 214 139 274 326 99)(39 368 215 140 275 327 100)(40 369 216 141 276 328 101)(41 370 217 142 277 329 102)(42 371 218 143 278 330 103)(43 372 219 144 279 331 104)(44 373 220 145 280 332 105)(45 374 221 146 225 333 106)(46 375 222 147 226 334 107)(47 376 223 148 227 335 108)(48 377 224 149 228 336 109)(49 378 169 150 229 281 110)(50 379 170 151 230 282 111)(51 380 171 152 231 283 112)(52 381 172 153 232 284 57)(53 382 173 154 233 285 58)(54 383 174 155 234 286 59)(55 384 175 156 235 287 60)(56 385 176 157 236 288 61)
(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)| (1,386,177,158,237,289,62)(2,387,178,159,238,290,63)(3,388,179,160,239,291,64)(4,389,180,161,240,292,65)(5,390,181,162,241,293,66)(6,391,182,163,242,294,67)(7,392,183,164,243,295,68)(8,337,184,165,244,296,69)(9,338,185,166,245,297,70)(10,339,186,167,246,298,71)(11,340,187,168,247,299,72)(12,341,188,113,248,300,73)(13,342,189,114,249,301,74)(14,343,190,115,250,302,75)(15,344,191,116,251,303,76)(16,345,192,117,252,304,77)(17,346,193,118,253,305,78)(18,347,194,119,254,306,79)(19,348,195,120,255,307,80)(20,349,196,121,256,308,81)(21,350,197,122,257,309,82)(22,351,198,123,258,310,83)(23,352,199,124,259,311,84)(24,353,200,125,260,312,85)(25,354,201,126,261,313,86)(26,355,202,127,262,314,87)(27,356,203,128,263,315,88)(28,357,204,129,264,316,89)(29,358,205,130,265,317,90)(30,359,206,131,266,318,91)(31,360,207,132,267,319,92)(32,361,208,133,268,320,93)(33,362,209,134,269,321,94)(34,363,210,135,270,322,95)(35,364,211,136,271,323,96)(36,365,212,137,272,324,97)(37,366,213,138,273,325,98)(38,367,214,139,274,326,99)(39,368,215,140,275,327,100)(40,369,216,141,276,328,101)(41,370,217,142,277,329,102)(42,371,218,143,278,330,103)(43,372,219,144,279,331,104)(44,373,220,145,280,332,105)(45,374,221,146,225,333,106)(46,375,222,147,226,334,107)(47,376,223,148,227,335,108)(48,377,224,149,228,336,109)(49,378,169,150,229,281,110)(50,379,170,151,230,282,111)(51,380,171,152,231,283,112)(52,381,172,153,232,284,57)(53,382,173,154,233,285,58)(54,383,174,155,234,286,59)(55,384,175,156,235,287,60)(56,385,176,157,236,288,61), (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( (1,386,177,158,237,289,62)(2,387,178,159,238,290,63)(3,388,179,160,239,291,64)(4,389,180,161,240,292,65)(5,390,181,162,241,293,66)(6,391,182,163,242,294,67)(7,392,183,164,243,295,68)(8,337,184,165,244,296,69)(9,338,185,166,245,297,70)(10,339,186,167,246,298,71)(11,340,187,168,247,299,72)(12,341,188,113,248,300,73)(13,342,189,114,249,301,74)(14,343,190,115,250,302,75)(15,344,191,116,251,303,76)(16,345,192,117,252,304,77)(17,346,193,118,253,305,78)(18,347,194,119,254,306,79)(19,348,195,120,255,307,80)(20,349,196,121,256,308,81)(21,350,197,122,257,309,82)(22,351,198,123,258,310,83)(23,352,199,124,259,311,84)(24,353,200,125,260,312,85)(25,354,201,126,261,313,86)(26,355,202,127,262,314,87)(27,356,203,128,263,315,88)(28,357,204,129,264,316,89)(29,358,205,130,265,317,90)(30,359,206,131,266,318,91)(31,360,207,132,267,319,92)(32,361,208,133,268,320,93)(33,362,209,134,269,321,94)(34,363,210,135,270,322,95)(35,364,211,136,271,323,96)(36,365,212,137,272,324,97)(37,366,213,138,273,325,98)(38,367,214,139,274,326,99)(39,368,215,140,275,327,100)(40,369,216,141,276,328,101)(41,370,217,142,277,329,102)(42,371,218,143,278,330,103)(43,372,219,144,279,331,104)(44,373,220,145,280,332,105)(45,374,221,146,225,333,106)(46,375,222,147,226,334,107)(47,376,223,148,227,335,108)(48,377,224,149,228,336,109)(49,378,169,150,229,281,110)(50,379,170,151,230,282,111)(51,380,171,152,231,283,112)(52,381,172,153,232,284,57)(53,382,173,154,233,285,58)(54,383,174,155,234,286,59)(55,384,175,156,235,287,60)(56,385,176,157,236,288,61), (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([(1,386,177,158,237,289,62),(2,387,178,159,238,290,63),(3,388,179,160,239,291,64),(4,389,180,161,240,292,65),(5,390,181,162,241,293,66),(6,391,182,163,242,294,67),(7,392,183,164,243,295,68),(8,337,184,165,244,296,69),(9,338,185,166,245,297,70),(10,339,186,167,246,298,71),(11,340,187,168,247,299,72),(12,341,188,113,248,300,73),(13,342,189,114,249,301,74),(14,343,190,115,250,302,75),(15,344,191,116,251,303,76),(16,345,192,117,252,304,77),(17,346,193,118,253,305,78),(18,347,194,119,254,306,79),(19,348,195,120,255,307,80),(20,349,196,121,256,308,81),(21,350,197,122,257,309,82),(22,351,198,123,258,310,83),(23,352,199,124,259,311,84),(24,353,200,125,260,312,85),(25,354,201,126,261,313,86),(26,355,202,127,262,314,87),(27,356,203,128,263,315,88),(28,357,204,129,264,316,89),(29,358,205,130,265,317,90),(30,359,206,131,266,318,91),(31,360,207,132,267,319,92),(32,361,208,133,268,320,93),(33,362,209,134,269,321,94),(34,363,210,135,270,322,95),(35,364,211,136,271,323,96),(36,365,212,137,272,324,97),(37,366,213,138,273,325,98),(38,367,214,139,274,326,99),(39,368,215,140,275,327,100),(40,369,216,141,276,328,101),(41,370,217,142,277,329,102),(42,371,218,143,278,330,103),(43,372,219,144,279,331,104),(44,373,220,145,280,332,105),(45,374,221,146,225,333,106),(46,375,222,147,226,334,107),(47,376,223,148,227,335,108),(48,377,224,149,228,336,109),(49,378,169,150,229,281,110),(50,379,170,151,230,282,111),(51,380,171,152,231,283,112),(52,381,172,153,232,284,57),(53,382,173,154,233,285,58),(54,383,174,155,234,286,59),(55,384,175,156,235,287,60),(56,385,176,157,236,288,61)], [(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  2 4A4B7A···7AV8A8B8C8D14A···14AV28A···28CR56A···56GJ
order12447···7888814···1428···2856···56
size11111···111111···11···11···1

392 irreducible representations

dim11111111
type++
imageC1C2C4C7C8C14C28C56
kernelC7×C56C7×C28C7×C14C56C72C28C14C7
# reps1124844896192

Matrix representation of C7×C56 in GL2(𝔽113) generated by

1060
049
,
440
014
G:=sub<GL(2,GF(113))| [106,0,0,49],[44,0,0,14] >;

C7×C56 in GAP, Magma, Sage, TeX

C_7\times C_{56}
% in TeX

G:=Group("C7xC56");
// GroupNames label

G:=SmallGroup(392,16);
// by ID

G=gap.SmallGroup(392,16);
# by ID

G:=PCGroup([5,-2,-7,-7,-2,-2,490,58]);
// Polycyclic

G:=Group<a,b|a^7=b^56=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C56 in TeX

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