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G = C73order 343 = 73

Elementary abelian group of type [7,7,7]

direct product, p-group, elementary abelian, monomial

Aliases: C73, SmallGroup(343,5)

Series: Derived Chief Lower central Upper central Jennings

C1 — C73
C1C7C72 — C73
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 [×57], C72 [×57], C73
Quotients: C1, C7 [×57], C72 [×57], C73

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

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

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

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

343 conjugacy classes

class 1 7A···7MD
order17···7
size11···1

343 irreducible representations

dim11
type+
imageC1C7
kernelC73C72
# reps1342

Matrix representation of C73 in GL3(𝔽29) generated by

2000
010
001
,
2400
0250
0020
,
100
010
0024
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

׿
×
𝔽