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G = C7×C49order 343 = 73

Abelian group of type [7,49]

direct product, p-group, abelian, monomial

Aliases: C7×C49, SmallGroup(343,2)

Series: Derived Chief Lower central Upper central Jennings

C1 — C7×C49
C1C7C72 — C7×C49
C1 — C7×C49
C1 — C7×C49
C1C7C7C7C7C7C7 — C7×C49

Generators and relations for C7×C49
 G = < a,b | a7=b49=1, ab=ba >


Smallest permutation representation of C7×C49
Regular action on 343 points
Generators in S343
(1 226 342 275 188 142 63)(2 227 343 276 189 143 64)(3 228 295 277 190 144 65)(4 229 296 278 191 145 66)(5 230 297 279 192 146 67)(6 231 298 280 193 147 68)(7 232 299 281 194 99 69)(8 233 300 282 195 100 70)(9 234 301 283 196 101 71)(10 235 302 284 148 102 72)(11 236 303 285 149 103 73)(12 237 304 286 150 104 74)(13 238 305 287 151 105 75)(14 239 306 288 152 106 76)(15 240 307 289 153 107 77)(16 241 308 290 154 108 78)(17 242 309 291 155 109 79)(18 243 310 292 156 110 80)(19 244 311 293 157 111 81)(20 245 312 294 158 112 82)(21 197 313 246 159 113 83)(22 198 314 247 160 114 84)(23 199 315 248 161 115 85)(24 200 316 249 162 116 86)(25 201 317 250 163 117 87)(26 202 318 251 164 118 88)(27 203 319 252 165 119 89)(28 204 320 253 166 120 90)(29 205 321 254 167 121 91)(30 206 322 255 168 122 92)(31 207 323 256 169 123 93)(32 208 324 257 170 124 94)(33 209 325 258 171 125 95)(34 210 326 259 172 126 96)(35 211 327 260 173 127 97)(36 212 328 261 174 128 98)(37 213 329 262 175 129 50)(38 214 330 263 176 130 51)(39 215 331 264 177 131 52)(40 216 332 265 178 132 53)(41 217 333 266 179 133 54)(42 218 334 267 180 134 55)(43 219 335 268 181 135 56)(44 220 336 269 182 136 57)(45 221 337 270 183 137 58)(46 222 338 271 184 138 59)(47 223 339 272 185 139 60)(48 224 340 273 186 140 61)(49 225 341 274 187 141 62)
(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)

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

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

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

343 conjugacy classes

class 1 7A···7AV49A···49KH
order17···749···49
size11···11···1

343 irreducible representations

dim1111
type+
imageC1C7C7C49
kernelC7×C49C49C72C7
# reps1426294

Matrix representation of C7×C49 in GL2(𝔽197) generated by

10
0178
,
590
0154
G:=sub<GL(2,GF(197))| [1,0,0,178],[59,0,0,154] >;

C7×C49 in GAP, Magma, Sage, TeX

C_7\times C_{49}
% in TeX

G:=Group("C7xC49");
// GroupNames label

G:=SmallGroup(343,2);
// by ID

G=gap.SmallGroup(343,2);
# by ID

G:=PCGroup([3,-7,7,-7,147]);
// Polycyclic

G:=Group<a,b|a^7=b^49=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C49 in TeX

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