direct product, p-group, elementary abelian, monomial
Aliases: C73, SmallGroup(343,5)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C73 |
| C1 — C73 |
| C1 — C73 |
Generators and relations for C73
G = < a,b,c | a7=b7=c7=1, ab=ba, ac=ca, bc=cb >
Subgroups: 116, all normal (2 characteristic)
C1, C7, C72, C73
Quotients: C1, C7, C72, C73
►Regular action on 343 points
Generators in S343
(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)
(1 11 258 209 160 111 62)(2 12 259 210 161 112 63)(3 13 253 204 155 106 57)(4 14 254 205 156 107 58)(5 8 255 206 157 108 59)(6 9 256 207 158 109 60)(7 10 257 208 159 110 61)(15 308 260 211 162 113 64)(16 302 261 212 163 114 65)(17 303 262 213 164 115 66)(18 304 263 214 165 116 67)(19 305 264 215 166 117 68)(20 306 265 216 167 118 69)(21 307 266 217 168 119 70)(22 315 267 218 169 120 71)(23 309 268 219 170 121 72)(24 310 269 220 171 122 73)(25 311 270 221 172 123 74)(26 312 271 222 173 124 75)(27 313 272 223 174 125 76)(28 314 273 224 175 126 77)(29 322 274 225 176 127 78)(30 316 275 226 177 128 79)(31 317 276 227 178 129 80)(32 318 277 228 179 130 81)(33 319 278 229 180 131 82)(34 320 279 230 181 132 83)(35 321 280 231 182 133 84)(36 329 281 232 183 134 85)(37 323 282 233 184 135 86)(38 324 283 234 185 136 87)(39 325 284 235 186 137 88)(40 326 285 236 187 138 89)(41 327 286 237 188 139 90)(42 328 287 238 189 140 91)(43 336 288 239 190 141 92)(44 330 289 240 191 142 93)(45 331 290 241 192 143 94)(46 332 291 242 193 144 95)(47 333 292 243 194 145 96)(48 334 293 244 195 146 97)(49 335 294 245 196 147 98)(50 343 295 246 197 148 99)(51 337 296 247 198 149 100)(52 338 297 248 199 150 101)(53 339 298 249 200 151 102)(54 340 299 250 201 152 103)(55 341 300 251 202 153 104)(56 342 301 252 203 154 105)
(1 342 48 41 34 27 20)(2 343 49 42 35 28 21)(3 337 43 36 29 22 15)(4 338 44 37 30 23 16)(5 339 45 38 31 24 17)(6 340 46 39 32 25 18)(7 341 47 40 33 26 19)(8 298 331 324 317 310 303)(9 299 332 325 318 311 304)(10 300 333 326 319 312 305)(11 301 334 327 320 313 306)(12 295 335 328 321 314 307)(13 296 336 329 322 315 308)(14 297 330 323 316 309 302)(50 98 91 84 77 70 63)(51 92 85 78 71 64 57)(52 93 86 79 72 65 58)(53 94 87 80 73 66 59)(54 95 88 81 74 67 60)(55 96 89 82 75 68 61)(56 97 90 83 76 69 62)(99 147 140 133 126 119 112)(100 141 134 127 120 113 106)(101 142 135 128 121 114 107)(102 143 136 129 122 115 108)(103 144 137 130 123 116 109)(104 145 138 131 124 117 110)(105 146 139 132 125 118 111)(148 196 189 182 175 168 161)(149 190 183 176 169 162 155)(150 191 184 177 170 163 156)(151 192 185 178 171 164 157)(152 193 186 179 172 165 158)(153 194 187 180 173 166 159)(154 195 188 181 174 167 160)(197 245 238 231 224 217 210)(198 239 232 225 218 211 204)(199 240 233 226 219 212 205)(200 241 234 227 220 213 206)(201 242 235 228 221 214 207)(202 243 236 229 222 215 208)(203 244 237 230 223 216 209)(246 294 287 280 273 266 259)(247 288 281 274 267 260 253)(248 289 282 275 268 261 254)(249 290 283 276 269 262 255)(250 291 284 277 270 263 256)(251 292 285 278 271 264 257)(252 293 286 279 272 265 258)
G:=sub<Sym(343)| (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), (1,11,258,209,160,111,62)(2,12,259,210,161,112,63)(3,13,253,204,155,106,57)(4,14,254,205,156,107,58)(5,8,255,206,157,108,59)(6,9,256,207,158,109,60)(7,10,257,208,159,110,61)(15,308,260,211,162,113,64)(16,302,261,212,163,114,65)(17,303,262,213,164,115,66)(18,304,263,214,165,116,67)(19,305,264,215,166,117,68)(20,306,265,216,167,118,69)(21,307,266,217,168,119,70)(22,315,267,218,169,120,71)(23,309,268,219,170,121,72)(24,310,269,220,171,122,73)(25,311,270,221,172,123,74)(26,312,271,222,173,124,75)(27,313,272,223,174,125,76)(28,314,273,224,175,126,77)(29,322,274,225,176,127,78)(30,316,275,226,177,128,79)(31,317,276,227,178,129,80)(32,318,277,228,179,130,81)(33,319,278,229,180,131,82)(34,320,279,230,181,132,83)(35,321,280,231,182,133,84)(36,329,281,232,183,134,85)(37,323,282,233,184,135,86)(38,324,283,234,185,136,87)(39,325,284,235,186,137,88)(40,326,285,236,187,138,89)(41,327,286,237,188,139,90)(42,328,287,238,189,140,91)(43,336,288,239,190,141,92)(44,330,289,240,191,142,93)(45,331,290,241,192,143,94)(46,332,291,242,193,144,95)(47,333,292,243,194,145,96)(48,334,293,244,195,146,97)(49,335,294,245,196,147,98)(50,343,295,246,197,148,99)(51,337,296,247,198,149,100)(52,338,297,248,199,150,101)(53,339,298,249,200,151,102)(54,340,299,250,201,152,103)(55,341,300,251,202,153,104)(56,342,301,252,203,154,105), (1,342,48,41,34,27,20)(2,343,49,42,35,28,21)(3,337,43,36,29,22,15)(4,338,44,37,30,23,16)(5,339,45,38,31,24,17)(6,340,46,39,32,25,18)(7,341,47,40,33,26,19)(8,298,331,324,317,310,303)(9,299,332,325,318,311,304)(10,300,333,326,319,312,305)(11,301,334,327,320,313,306)(12,295,335,328,321,314,307)(13,296,336,329,322,315,308)(14,297,330,323,316,309,302)(50,98,91,84,77,70,63)(51,92,85,78,71,64,57)(52,93,86,79,72,65,58)(53,94,87,80,73,66,59)(54,95,88,81,74,67,60)(55,96,89,82,75,68,61)(56,97,90,83,76,69,62)(99,147,140,133,126,119,112)(100,141,134,127,120,113,106)(101,142,135,128,121,114,107)(102,143,136,129,122,115,108)(103,144,137,130,123,116,109)(104,145,138,131,124,117,110)(105,146,139,132,125,118,111)(148,196,189,182,175,168,161)(149,190,183,176,169,162,155)(150,191,184,177,170,163,156)(151,192,185,178,171,164,157)(152,193,186,179,172,165,158)(153,194,187,180,173,166,159)(154,195,188,181,174,167,160)(197,245,238,231,224,217,210)(198,239,232,225,218,211,204)(199,240,233,226,219,212,205)(200,241,234,227,220,213,206)(201,242,235,228,221,214,207)(202,243,236,229,222,215,208)(203,244,237,230,223,216,209)(246,294,287,280,273,266,259)(247,288,281,274,267,260,253)(248,289,282,275,268,261,254)(249,290,283,276,269,262,255)(250,291,284,277,270,263,256)(251,292,285,278,271,264,257)(252,293,286,279,272,265,258)>;
G:=Group( (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), (1,11,258,209,160,111,62)(2,12,259,210,161,112,63)(3,13,253,204,155,106,57)(4,14,254,205,156,107,58)(5,8,255,206,157,108,59)(6,9,256,207,158,109,60)(7,10,257,208,159,110,61)(15,308,260,211,162,113,64)(16,302,261,212,163,114,65)(17,303,262,213,164,115,66)(18,304,263,214,165,116,67)(19,305,264,215,166,117,68)(20,306,265,216,167,118,69)(21,307,266,217,168,119,70)(22,315,267,218,169,120,71)(23,309,268,219,170,121,72)(24,310,269,220,171,122,73)(25,311,270,221,172,123,74)(26,312,271,222,173,124,75)(27,313,272,223,174,125,76)(28,314,273,224,175,126,77)(29,322,274,225,176,127,78)(30,316,275,226,177,128,79)(31,317,276,227,178,129,80)(32,318,277,228,179,130,81)(33,319,278,229,180,131,82)(34,320,279,230,181,132,83)(35,321,280,231,182,133,84)(36,329,281,232,183,134,85)(37,323,282,233,184,135,86)(38,324,283,234,185,136,87)(39,325,284,235,186,137,88)(40,326,285,236,187,138,89)(41,327,286,237,188,139,90)(42,328,287,238,189,140,91)(43,336,288,239,190,141,92)(44,330,289,240,191,142,93)(45,331,290,241,192,143,94)(46,332,291,242,193,144,95)(47,333,292,243,194,145,96)(48,334,293,244,195,146,97)(49,335,294,245,196,147,98)(50,343,295,246,197,148,99)(51,337,296,247,198,149,100)(52,338,297,248,199,150,101)(53,339,298,249,200,151,102)(54,340,299,250,201,152,103)(55,341,300,251,202,153,104)(56,342,301,252,203,154,105), (1,342,48,41,34,27,20)(2,343,49,42,35,28,21)(3,337,43,36,29,22,15)(4,338,44,37,30,23,16)(5,339,45,38,31,24,17)(6,340,46,39,32,25,18)(7,341,47,40,33,26,19)(8,298,331,324,317,310,303)(9,299,332,325,318,311,304)(10,300,333,326,319,312,305)(11,301,334,327,320,313,306)(12,295,335,328,321,314,307)(13,296,336,329,322,315,308)(14,297,330,323,316,309,302)(50,98,91,84,77,70,63)(51,92,85,78,71,64,57)(52,93,86,79,72,65,58)(53,94,87,80,73,66,59)(54,95,88,81,74,67,60)(55,96,89,82,75,68,61)(56,97,90,83,76,69,62)(99,147,140,133,126,119,112)(100,141,134,127,120,113,106)(101,142,135,128,121,114,107)(102,143,136,129,122,115,108)(103,144,137,130,123,116,109)(104,145,138,131,124,117,110)(105,146,139,132,125,118,111)(148,196,189,182,175,168,161)(149,190,183,176,169,162,155)(150,191,184,177,170,163,156)(151,192,185,178,171,164,157)(152,193,186,179,172,165,158)(153,194,187,180,173,166,159)(154,195,188,181,174,167,160)(197,245,238,231,224,217,210)(198,239,232,225,218,211,204)(199,240,233,226,219,212,205)(200,241,234,227,220,213,206)(201,242,235,228,221,214,207)(202,243,236,229,222,215,208)(203,244,237,230,223,216,209)(246,294,287,280,273,266,259)(247,288,281,274,267,260,253)(248,289,282,275,268,261,254)(249,290,283,276,269,262,255)(250,291,284,277,270,263,256)(251,292,285,278,271,264,257)(252,293,286,279,272,265,258) );
G=PermutationGroup([[(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)], [(1,11,258,209,160,111,62),(2,12,259,210,161,112,63),(3,13,253,204,155,106,57),(4,14,254,205,156,107,58),(5,8,255,206,157,108,59),(6,9,256,207,158,109,60),(7,10,257,208,159,110,61),(15,308,260,211,162,113,64),(16,302,261,212,163,114,65),(17,303,262,213,164,115,66),(18,304,263,214,165,116,67),(19,305,264,215,166,117,68),(20,306,265,216,167,118,69),(21,307,266,217,168,119,70),(22,315,267,218,169,120,71),(23,309,268,219,170,121,72),(24,310,269,220,171,122,73),(25,311,270,221,172,123,74),(26,312,271,222,173,124,75),(27,313,272,223,174,125,76),(28,314,273,224,175,126,77),(29,322,274,225,176,127,78),(30,316,275,226,177,128,79),(31,317,276,227,178,129,80),(32,318,277,228,179,130,81),(33,319,278,229,180,131,82),(34,320,279,230,181,132,83),(35,321,280,231,182,133,84),(36,329,281,232,183,134,85),(37,323,282,233,184,135,86),(38,324,283,234,185,136,87),(39,325,284,235,186,137,88),(40,326,285,236,187,138,89),(41,327,286,237,188,139,90),(42,328,287,238,189,140,91),(43,336,288,239,190,141,92),(44,330,289,240,191,142,93),(45,331,290,241,192,143,94),(46,332,291,242,193,144,95),(47,333,292,243,194,145,96),(48,334,293,244,195,146,97),(49,335,294,245,196,147,98),(50,343,295,246,197,148,99),(51,337,296,247,198,149,100),(52,338,297,248,199,150,101),(53,339,298,249,200,151,102),(54,340,299,250,201,152,103),(55,341,300,251,202,153,104),(56,342,301,252,203,154,105)], [(1,342,48,41,34,27,20),(2,343,49,42,35,28,21),(3,337,43,36,29,22,15),(4,338,44,37,30,23,16),(5,339,45,38,31,24,17),(6,340,46,39,32,25,18),(7,341,47,40,33,26,19),(8,298,331,324,317,310,303),(9,299,332,325,318,311,304),(10,300,333,326,319,312,305),(11,301,334,327,320,313,306),(12,295,335,328,321,314,307),(13,296,336,329,322,315,308),(14,297,330,323,316,309,302),(50,98,91,84,77,70,63),(51,92,85,78,71,64,57),(52,93,86,79,72,65,58),(53,94,87,80,73,66,59),(54,95,88,81,74,67,60),(55,96,89,82,75,68,61),(56,97,90,83,76,69,62),(99,147,140,133,126,119,112),(100,141,134,127,120,113,106),(101,142,135,128,121,114,107),(102,143,136,129,122,115,108),(103,144,137,130,123,116,109),(104,145,138,131,124,117,110),(105,146,139,132,125,118,111),(148,196,189,182,175,168,161),(149,190,183,176,169,162,155),(150,191,184,177,170,163,156),(151,192,185,178,171,164,157),(152,193,186,179,172,165,158),(153,194,187,180,173,166,159),(154,195,188,181,174,167,160),(197,245,238,231,224,217,210),(198,239,232,225,218,211,204),(199,240,233,226,219,212,205),(200,241,234,227,220,213,206),(201,242,235,228,221,214,207),(202,243,236,229,222,215,208),(203,244,237,230,223,216,209),(246,294,287,280,273,266,259),(247,288,281,274,267,260,253),(248,289,282,275,268,261,254),(249,290,283,276,269,262,255),(250,291,284,277,270,263,256),(251,292,285,278,271,264,257),(252,293,286,279,272,265,258)]])
343 conjugacy classes
| class | 1 | 7A | ··· | 7MD |
| order | 1 | 7 | ··· | 7 |
| size | 1 | 1 | ··· | 1 |
343 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C7 |
| kernel | C73 | C72 |
| # reps | 1 | 342 |
Matrix representation of C73 ►in GL3(𝔽29) generated by
| 20 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 24 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 20 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 24 |
G:=sub<GL(3,GF(29))| [20,0,0,0,1,0,0,0,1],[24,0,0,0,25,0,0,0,20],[1,0,0,0,1,0,0,0,24] >;
C73 in GAP, Magma, Sage, TeX
C_7^3
% in TeX
G:=Group("C7^3"); // GroupNames label
G:=SmallGroup(343,5);
// by ID
G=gap.SmallGroup(343,5);
# by ID
G:=PCGroup([3,-7,7,7]);
// Polycyclic
G:=Group<a,b,c|a^7=b^7=c^7=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations