direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary
Aliases: C6×C7⋊C9, C42⋊C9, C21⋊4C18, C42.4C32, C14⋊2(C3×C9), C7⋊4(C3×C18), (C3×C42).2C3, (C3×C21).7C6, C21.10(C3×C6), C3.2(C6×C7⋊C3), C6.4(C3×C7⋊C3), (C3×C6).2(C7⋊C3), C32.3(C2×C7⋊C3), SmallGroup(378,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C6×C7⋊C9 |
Generators and relations for C6×C7⋊C9
G = < a,b,c | a6=b7=c9=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C6×C7⋊C9
►Regular action on 378 points
Generators in S378
(1 254 298 201 124 92)(2 255 299 202 125 93)(3 256 300 203 126 94)(4 257 301 204 118 95)(5 258 302 205 119 96)(6 259 303 206 120 97)(7 260 304 207 121 98)(8 261 305 199 122 99)(9 253 306 200 123 91)(10 50 235 323 364 135)(11 51 236 324 365 127)(12 52 237 316 366 128)(13 53 238 317 367 129)(14 54 239 318 368 130)(15 46 240 319 369 131)(16 47 241 320 361 132)(17 48 242 321 362 133)(18 49 243 322 363 134)(19 354 339 292 151 185)(20 355 340 293 152 186)(21 356 341 294 153 187)(22 357 342 295 145 188)(23 358 334 296 146 189)(24 359 335 297 147 181)(25 360 336 289 148 182)(26 352 337 290 149 183)(27 353 338 291 150 184)(28 89 57 251 225 196)(29 90 58 252 217 197)(30 82 59 244 218 198)(31 83 60 245 219 190)(32 84 61 246 220 191)(33 85 62 247 221 192)(34 86 63 248 222 193)(35 87 55 249 223 194)(36 88 56 250 224 195)(37 231 103 272 110 376)(38 232 104 273 111 377)(39 233 105 274 112 378)(40 234 106 275 113 370)(41 226 107 276 114 371)(42 227 108 277 115 372)(43 228 100 278 116 373)(44 229 101 279 117 374)(45 230 102 271 109 375)(64 171 76 270 314 208)(65 163 77 262 315 209)(66 164 78 263 307 210)(67 165 79 264 308 211)(68 166 80 265 309 212)(69 167 81 266 310 213)(70 168 73 267 311 214)(71 169 74 268 312 215)(72 170 75 269 313 216)(136 176 158 343 333 283)(137 177 159 344 325 284)(138 178 160 345 326 285)(139 179 161 346 327 286)(140 180 162 347 328 287)(141 172 154 348 329 288)(142 173 155 349 330 280)(143 174 156 350 331 281)(144 175 157 351 332 282)
(1 114 314 127 329 58 336)(2 330 115 59 315 337 128)(3 307 331 338 116 129 60)(4 117 308 130 332 61 339)(5 333 109 62 309 340 131)(6 310 325 341 110 132 63)(7 111 311 133 326 55 342)(8 327 112 56 312 334 134)(9 313 328 335 113 135 57)(10 251 253 216 287 297 370)(11 288 252 289 254 371 208)(12 255 280 372 244 209 290)(13 245 256 210 281 291 373)(14 282 246 292 257 374 211)(15 258 283 375 247 212 293)(16 248 259 213 284 294 376)(17 285 249 295 260 377 214)(18 261 286 378 250 215 296)(19 118 101 79 318 157 32)(20 319 119 158 102 33 80)(21 103 320 34 120 81 159)(22 121 104 73 321 160 35)(23 322 122 161 105 36 74)(24 106 323 28 123 75 162)(25 124 107 76 324 154 29)(26 316 125 155 108 30 77)(27 100 317 31 126 78 156)(37 47 222 303 69 137 153)(38 70 48 138 223 145 304)(39 224 71 146 49 305 139)(40 50 225 306 72 140 147)(41 64 51 141 217 148 298)(42 218 65 149 52 299 142)(43 53 219 300 66 143 150)(44 67 54 144 220 151 301)(45 221 68 152 46 302 136)(82 262 352 366 93 349 277)(83 94 263 350 353 278 367)(84 354 95 279 264 368 351)(85 265 355 369 96 343 271)(86 97 266 344 356 272 361)(87 357 98 273 267 362 345)(88 268 358 363 99 346 274)(89 91 269 347 359 275 364)(90 360 92 276 270 365 348)(163 183 237 202 173 227 198)(164 174 184 228 238 190 203)(165 239 175 191 185 204 229)(166 186 240 205 176 230 192)(167 177 187 231 241 193 206)(168 242 178 194 188 207 232)(169 189 243 199 179 233 195)(170 180 181 234 235 196 200)(171 236 172 197 182 201 226)
(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)
G:=sub<Sym(378)| (1,254,298,201,124,92)(2,255,299,202,125,93)(3,256,300,203,126,94)(4,257,301,204,118,95)(5,258,302,205,119,96)(6,259,303,206,120,97)(7,260,304,207,121,98)(8,261,305,199,122,99)(9,253,306,200,123,91)(10,50,235,323,364,135)(11,51,236,324,365,127)(12,52,237,316,366,128)(13,53,238,317,367,129)(14,54,239,318,368,130)(15,46,240,319,369,131)(16,47,241,320,361,132)(17,48,242,321,362,133)(18,49,243,322,363,134)(19,354,339,292,151,185)(20,355,340,293,152,186)(21,356,341,294,153,187)(22,357,342,295,145,188)(23,358,334,296,146,189)(24,359,335,297,147,181)(25,360,336,289,148,182)(26,352,337,290,149,183)(27,353,338,291,150,184)(28,89,57,251,225,196)(29,90,58,252,217,197)(30,82,59,244,218,198)(31,83,60,245,219,190)(32,84,61,246,220,191)(33,85,62,247,221,192)(34,86,63,248,222,193)(35,87,55,249,223,194)(36,88,56,250,224,195)(37,231,103,272,110,376)(38,232,104,273,111,377)(39,233,105,274,112,378)(40,234,106,275,113,370)(41,226,107,276,114,371)(42,227,108,277,115,372)(43,228,100,278,116,373)(44,229,101,279,117,374)(45,230,102,271,109,375)(64,171,76,270,314,208)(65,163,77,262,315,209)(66,164,78,263,307,210)(67,165,79,264,308,211)(68,166,80,265,309,212)(69,167,81,266,310,213)(70,168,73,267,311,214)(71,169,74,268,312,215)(72,170,75,269,313,216)(136,176,158,343,333,283)(137,177,159,344,325,284)(138,178,160,345,326,285)(139,179,161,346,327,286)(140,180,162,347,328,287)(141,172,154,348,329,288)(142,173,155,349,330,280)(143,174,156,350,331,281)(144,175,157,351,332,282), (1,114,314,127,329,58,336)(2,330,115,59,315,337,128)(3,307,331,338,116,129,60)(4,117,308,130,332,61,339)(5,333,109,62,309,340,131)(6,310,325,341,110,132,63)(7,111,311,133,326,55,342)(8,327,112,56,312,334,134)(9,313,328,335,113,135,57)(10,251,253,216,287,297,370)(11,288,252,289,254,371,208)(12,255,280,372,244,209,290)(13,245,256,210,281,291,373)(14,282,246,292,257,374,211)(15,258,283,375,247,212,293)(16,248,259,213,284,294,376)(17,285,249,295,260,377,214)(18,261,286,378,250,215,296)(19,118,101,79,318,157,32)(20,319,119,158,102,33,80)(21,103,320,34,120,81,159)(22,121,104,73,321,160,35)(23,322,122,161,105,36,74)(24,106,323,28,123,75,162)(25,124,107,76,324,154,29)(26,316,125,155,108,30,77)(27,100,317,31,126,78,156)(37,47,222,303,69,137,153)(38,70,48,138,223,145,304)(39,224,71,146,49,305,139)(40,50,225,306,72,140,147)(41,64,51,141,217,148,298)(42,218,65,149,52,299,142)(43,53,219,300,66,143,150)(44,67,54,144,220,151,301)(45,221,68,152,46,302,136)(82,262,352,366,93,349,277)(83,94,263,350,353,278,367)(84,354,95,279,264,368,351)(85,265,355,369,96,343,271)(86,97,266,344,356,272,361)(87,357,98,273,267,362,345)(88,268,358,363,99,346,274)(89,91,269,347,359,275,364)(90,360,92,276,270,365,348)(163,183,237,202,173,227,198)(164,174,184,228,238,190,203)(165,239,175,191,185,204,229)(166,186,240,205,176,230,192)(167,177,187,231,241,193,206)(168,242,178,194,188,207,232)(169,189,243,199,179,233,195)(170,180,181,234,235,196,200)(171,236,172,197,182,201,226), (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)>;
G:=Group( (1,254,298,201,124,92)(2,255,299,202,125,93)(3,256,300,203,126,94)(4,257,301,204,118,95)(5,258,302,205,119,96)(6,259,303,206,120,97)(7,260,304,207,121,98)(8,261,305,199,122,99)(9,253,306,200,123,91)(10,50,235,323,364,135)(11,51,236,324,365,127)(12,52,237,316,366,128)(13,53,238,317,367,129)(14,54,239,318,368,130)(15,46,240,319,369,131)(16,47,241,320,361,132)(17,48,242,321,362,133)(18,49,243,322,363,134)(19,354,339,292,151,185)(20,355,340,293,152,186)(21,356,341,294,153,187)(22,357,342,295,145,188)(23,358,334,296,146,189)(24,359,335,297,147,181)(25,360,336,289,148,182)(26,352,337,290,149,183)(27,353,338,291,150,184)(28,89,57,251,225,196)(29,90,58,252,217,197)(30,82,59,244,218,198)(31,83,60,245,219,190)(32,84,61,246,220,191)(33,85,62,247,221,192)(34,86,63,248,222,193)(35,87,55,249,223,194)(36,88,56,250,224,195)(37,231,103,272,110,376)(38,232,104,273,111,377)(39,233,105,274,112,378)(40,234,106,275,113,370)(41,226,107,276,114,371)(42,227,108,277,115,372)(43,228,100,278,116,373)(44,229,101,279,117,374)(45,230,102,271,109,375)(64,171,76,270,314,208)(65,163,77,262,315,209)(66,164,78,263,307,210)(67,165,79,264,308,211)(68,166,80,265,309,212)(69,167,81,266,310,213)(70,168,73,267,311,214)(71,169,74,268,312,215)(72,170,75,269,313,216)(136,176,158,343,333,283)(137,177,159,344,325,284)(138,178,160,345,326,285)(139,179,161,346,327,286)(140,180,162,347,328,287)(141,172,154,348,329,288)(142,173,155,349,330,280)(143,174,156,350,331,281)(144,175,157,351,332,282), (1,114,314,127,329,58,336)(2,330,115,59,315,337,128)(3,307,331,338,116,129,60)(4,117,308,130,332,61,339)(5,333,109,62,309,340,131)(6,310,325,341,110,132,63)(7,111,311,133,326,55,342)(8,327,112,56,312,334,134)(9,313,328,335,113,135,57)(10,251,253,216,287,297,370)(11,288,252,289,254,371,208)(12,255,280,372,244,209,290)(13,245,256,210,281,291,373)(14,282,246,292,257,374,211)(15,258,283,375,247,212,293)(16,248,259,213,284,294,376)(17,285,249,295,260,377,214)(18,261,286,378,250,215,296)(19,118,101,79,318,157,32)(20,319,119,158,102,33,80)(21,103,320,34,120,81,159)(22,121,104,73,321,160,35)(23,322,122,161,105,36,74)(24,106,323,28,123,75,162)(25,124,107,76,324,154,29)(26,316,125,155,108,30,77)(27,100,317,31,126,78,156)(37,47,222,303,69,137,153)(38,70,48,138,223,145,304)(39,224,71,146,49,305,139)(40,50,225,306,72,140,147)(41,64,51,141,217,148,298)(42,218,65,149,52,299,142)(43,53,219,300,66,143,150)(44,67,54,144,220,151,301)(45,221,68,152,46,302,136)(82,262,352,366,93,349,277)(83,94,263,350,353,278,367)(84,354,95,279,264,368,351)(85,265,355,369,96,343,271)(86,97,266,344,356,272,361)(87,357,98,273,267,362,345)(88,268,358,363,99,346,274)(89,91,269,347,359,275,364)(90,360,92,276,270,365,348)(163,183,237,202,173,227,198)(164,174,184,228,238,190,203)(165,239,175,191,185,204,229)(166,186,240,205,176,230,192)(167,177,187,231,241,193,206)(168,242,178,194,188,207,232)(169,189,243,199,179,233,195)(170,180,181,234,235,196,200)(171,236,172,197,182,201,226), (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) );
G=PermutationGroup([[(1,254,298,201,124,92),(2,255,299,202,125,93),(3,256,300,203,126,94),(4,257,301,204,118,95),(5,258,302,205,119,96),(6,259,303,206,120,97),(7,260,304,207,121,98),(8,261,305,199,122,99),(9,253,306,200,123,91),(10,50,235,323,364,135),(11,51,236,324,365,127),(12,52,237,316,366,128),(13,53,238,317,367,129),(14,54,239,318,368,130),(15,46,240,319,369,131),(16,47,241,320,361,132),(17,48,242,321,362,133),(18,49,243,322,363,134),(19,354,339,292,151,185),(20,355,340,293,152,186),(21,356,341,294,153,187),(22,357,342,295,145,188),(23,358,334,296,146,189),(24,359,335,297,147,181),(25,360,336,289,148,182),(26,352,337,290,149,183),(27,353,338,291,150,184),(28,89,57,251,225,196),(29,90,58,252,217,197),(30,82,59,244,218,198),(31,83,60,245,219,190),(32,84,61,246,220,191),(33,85,62,247,221,192),(34,86,63,248,222,193),(35,87,55,249,223,194),(36,88,56,250,224,195),(37,231,103,272,110,376),(38,232,104,273,111,377),(39,233,105,274,112,378),(40,234,106,275,113,370),(41,226,107,276,114,371),(42,227,108,277,115,372),(43,228,100,278,116,373),(44,229,101,279,117,374),(45,230,102,271,109,375),(64,171,76,270,314,208),(65,163,77,262,315,209),(66,164,78,263,307,210),(67,165,79,264,308,211),(68,166,80,265,309,212),(69,167,81,266,310,213),(70,168,73,267,311,214),(71,169,74,268,312,215),(72,170,75,269,313,216),(136,176,158,343,333,283),(137,177,159,344,325,284),(138,178,160,345,326,285),(139,179,161,346,327,286),(140,180,162,347,328,287),(141,172,154,348,329,288),(142,173,155,349,330,280),(143,174,156,350,331,281),(144,175,157,351,332,282)], [(1,114,314,127,329,58,336),(2,330,115,59,315,337,128),(3,307,331,338,116,129,60),(4,117,308,130,332,61,339),(5,333,109,62,309,340,131),(6,310,325,341,110,132,63),(7,111,311,133,326,55,342),(8,327,112,56,312,334,134),(9,313,328,335,113,135,57),(10,251,253,216,287,297,370),(11,288,252,289,254,371,208),(12,255,280,372,244,209,290),(13,245,256,210,281,291,373),(14,282,246,292,257,374,211),(15,258,283,375,247,212,293),(16,248,259,213,284,294,376),(17,285,249,295,260,377,214),(18,261,286,378,250,215,296),(19,118,101,79,318,157,32),(20,319,119,158,102,33,80),(21,103,320,34,120,81,159),(22,121,104,73,321,160,35),(23,322,122,161,105,36,74),(24,106,323,28,123,75,162),(25,124,107,76,324,154,29),(26,316,125,155,108,30,77),(27,100,317,31,126,78,156),(37,47,222,303,69,137,153),(38,70,48,138,223,145,304),(39,224,71,146,49,305,139),(40,50,225,306,72,140,147),(41,64,51,141,217,148,298),(42,218,65,149,52,299,142),(43,53,219,300,66,143,150),(44,67,54,144,220,151,301),(45,221,68,152,46,302,136),(82,262,352,366,93,349,277),(83,94,263,350,353,278,367),(84,354,95,279,264,368,351),(85,265,355,369,96,343,271),(86,97,266,344,356,272,361),(87,357,98,273,267,362,345),(88,268,358,363,99,346,274),(89,91,269,347,359,275,364),(90,360,92,276,270,365,348),(163,183,237,202,173,227,198),(164,174,184,228,238,190,203),(165,239,175,191,185,204,229),(166,186,240,205,176,230,192),(167,177,187,231,241,193,206),(168,242,178,194,188,207,232),(169,189,243,199,179,233,195),(170,180,181,234,235,196,200),(171,236,172,197,182,201,226)], [(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)]])
90 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 7A | 7B | 9A | ··· | 9R | 14A | 14B | 18A | ··· | 18R | 21A | ··· | 21P | 42A | ··· | 42P |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 7 | 7 | 9 | ··· | 9 | 14 | 14 | 18 | ··· | 18 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | 3 | 7 | ··· | 7 | 3 | 3 | 7 | ··· | 7 | 3 | ··· | 3 | 3 | ··· | 3 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 | C7⋊C3 | C2×C7⋊C3 | C7⋊C9 | C3×C7⋊C3 | C2×C7⋊C9 | C6×C7⋊C3 |
| kernel | C6×C7⋊C9 | C3×C7⋊C9 | C2×C7⋊C9 | C3×C42 | C7⋊C9 | C3×C21 | C42 | C21 | C3×C6 | C32 | C6 | C6 | C3 | C3 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 18 | 18 | 2 | 2 | 12 | 4 | 12 | 4 |
Matrix representation of C6×C7⋊C9 ►in GL4(𝔽127) generated by
| 108 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 108 | 0 |
| 0 | 0 | 0 | 108 |
| 1 | 0 | 0 | 0 |
| 0 | 22 | 23 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 107 | 0 | 0 | 0 |
| 0 | 91 | 8 | 116 |
| 0 | 70 | 29 | 74 |
| 0 | 116 | 79 | 7 |
G:=sub<GL(4,GF(127))| [108,0,0,0,0,108,0,0,0,0,108,0,0,0,0,108],[1,0,0,0,0,22,1,0,0,23,0,1,0,1,0,0],[107,0,0,0,0,91,70,116,0,8,29,79,0,116,74,7] >;
C6×C7⋊C9 in GAP, Magma, Sage, TeX
C_6\times C_7\rtimes C_9
% in TeX
G:=Group("C6xC7:C9"); // GroupNames label
G:=SmallGroup(378,26);
// by ID
G=gap.SmallGroup(378,26);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,96,1359]);
// Polycyclic
G:=Group<a,b,c|a^6=b^7=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export