direct product, abelian, monomial
Aliases: C6×C66, SmallGroup(396,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C66 |
| C1 — C6×C66 |
| C1 — C6×C66 |
Generators and relations for C6×C66
G = < a,b | a6=b66=1, ab=ba >
Subgroups: 60, all normal (8 characteristic)
C1, C2, C3, C22, C6, C32, C11, C2×C6, C3×C6, C22, C33, C62, C2×C22, C66, C3×C33, C2×C66, C3×C66, C6×C66
Quotients: C1, C2, C3, C22, C6, C32, C11, C2×C6, C3×C6, C22, C33, C62, C2×C22, C66, C3×C33, C2×C66, C3×C66, C6×C66
►Regular action on 396 points
Generators in S396
(1 222 194 292 114 363)(2 223 195 293 115 364)(3 224 196 294 116 365)(4 225 197 295 117 366)(5 226 198 296 118 367)(6 227 133 297 119 368)(7 228 134 298 120 369)(8 229 135 299 121 370)(9 230 136 300 122 371)(10 231 137 301 123 372)(11 232 138 302 124 373)(12 233 139 303 125 374)(13 234 140 304 126 375)(14 235 141 305 127 376)(15 236 142 306 128 377)(16 237 143 307 129 378)(17 238 144 308 130 379)(18 239 145 309 131 380)(19 240 146 310 132 381)(20 241 147 311 67 382)(21 242 148 312 68 383)(22 243 149 313 69 384)(23 244 150 314 70 385)(24 245 151 315 71 386)(25 246 152 316 72 387)(26 247 153 317 73 388)(27 248 154 318 74 389)(28 249 155 319 75 390)(29 250 156 320 76 391)(30 251 157 321 77 392)(31 252 158 322 78 393)(32 253 159 323 79 394)(33 254 160 324 80 395)(34 255 161 325 81 396)(35 256 162 326 82 331)(36 257 163 327 83 332)(37 258 164 328 84 333)(38 259 165 329 85 334)(39 260 166 330 86 335)(40 261 167 265 87 336)(41 262 168 266 88 337)(42 263 169 267 89 338)(43 264 170 268 90 339)(44 199 171 269 91 340)(45 200 172 270 92 341)(46 201 173 271 93 342)(47 202 174 272 94 343)(48 203 175 273 95 344)(49 204 176 274 96 345)(50 205 177 275 97 346)(51 206 178 276 98 347)(52 207 179 277 99 348)(53 208 180 278 100 349)(54 209 181 279 101 350)(55 210 182 280 102 351)(56 211 183 281 103 352)(57 212 184 282 104 353)(58 213 185 283 105 354)(59 214 186 284 106 355)(60 215 187 285 107 356)(61 216 188 286 108 357)(62 217 189 287 109 358)(63 218 190 288 110 359)(64 219 191 289 111 360)(65 220 192 290 112 361)(66 221 193 291 113 362)
(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)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)
G:=sub<Sym(396)| (1,222,194,292,114,363)(2,223,195,293,115,364)(3,224,196,294,116,365)(4,225,197,295,117,366)(5,226,198,296,118,367)(6,227,133,297,119,368)(7,228,134,298,120,369)(8,229,135,299,121,370)(9,230,136,300,122,371)(10,231,137,301,123,372)(11,232,138,302,124,373)(12,233,139,303,125,374)(13,234,140,304,126,375)(14,235,141,305,127,376)(15,236,142,306,128,377)(16,237,143,307,129,378)(17,238,144,308,130,379)(18,239,145,309,131,380)(19,240,146,310,132,381)(20,241,147,311,67,382)(21,242,148,312,68,383)(22,243,149,313,69,384)(23,244,150,314,70,385)(24,245,151,315,71,386)(25,246,152,316,72,387)(26,247,153,317,73,388)(27,248,154,318,74,389)(28,249,155,319,75,390)(29,250,156,320,76,391)(30,251,157,321,77,392)(31,252,158,322,78,393)(32,253,159,323,79,394)(33,254,160,324,80,395)(34,255,161,325,81,396)(35,256,162,326,82,331)(36,257,163,327,83,332)(37,258,164,328,84,333)(38,259,165,329,85,334)(39,260,166,330,86,335)(40,261,167,265,87,336)(41,262,168,266,88,337)(42,263,169,267,89,338)(43,264,170,268,90,339)(44,199,171,269,91,340)(45,200,172,270,92,341)(46,201,173,271,93,342)(47,202,174,272,94,343)(48,203,175,273,95,344)(49,204,176,274,96,345)(50,205,177,275,97,346)(51,206,178,276,98,347)(52,207,179,277,99,348)(53,208,180,278,100,349)(54,209,181,279,101,350)(55,210,182,280,102,351)(56,211,183,281,103,352)(57,212,184,282,104,353)(58,213,185,283,105,354)(59,214,186,284,106,355)(60,215,187,285,107,356)(61,216,188,286,108,357)(62,217,189,287,109,358)(63,218,190,288,110,359)(64,219,191,289,111,360)(65,220,192,290,112,361)(66,221,193,291,113,362), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)>;
G:=Group( (1,222,194,292,114,363)(2,223,195,293,115,364)(3,224,196,294,116,365)(4,225,197,295,117,366)(5,226,198,296,118,367)(6,227,133,297,119,368)(7,228,134,298,120,369)(8,229,135,299,121,370)(9,230,136,300,122,371)(10,231,137,301,123,372)(11,232,138,302,124,373)(12,233,139,303,125,374)(13,234,140,304,126,375)(14,235,141,305,127,376)(15,236,142,306,128,377)(16,237,143,307,129,378)(17,238,144,308,130,379)(18,239,145,309,131,380)(19,240,146,310,132,381)(20,241,147,311,67,382)(21,242,148,312,68,383)(22,243,149,313,69,384)(23,244,150,314,70,385)(24,245,151,315,71,386)(25,246,152,316,72,387)(26,247,153,317,73,388)(27,248,154,318,74,389)(28,249,155,319,75,390)(29,250,156,320,76,391)(30,251,157,321,77,392)(31,252,158,322,78,393)(32,253,159,323,79,394)(33,254,160,324,80,395)(34,255,161,325,81,396)(35,256,162,326,82,331)(36,257,163,327,83,332)(37,258,164,328,84,333)(38,259,165,329,85,334)(39,260,166,330,86,335)(40,261,167,265,87,336)(41,262,168,266,88,337)(42,263,169,267,89,338)(43,264,170,268,90,339)(44,199,171,269,91,340)(45,200,172,270,92,341)(46,201,173,271,93,342)(47,202,174,272,94,343)(48,203,175,273,95,344)(49,204,176,274,96,345)(50,205,177,275,97,346)(51,206,178,276,98,347)(52,207,179,277,99,348)(53,208,180,278,100,349)(54,209,181,279,101,350)(55,210,182,280,102,351)(56,211,183,281,103,352)(57,212,184,282,104,353)(58,213,185,283,105,354)(59,214,186,284,106,355)(60,215,187,285,107,356)(61,216,188,286,108,357)(62,217,189,287,109,358)(63,218,190,288,110,359)(64,219,191,289,111,360)(65,220,192,290,112,361)(66,221,193,291,113,362), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396) );
G=PermutationGroup([[(1,222,194,292,114,363),(2,223,195,293,115,364),(3,224,196,294,116,365),(4,225,197,295,117,366),(5,226,198,296,118,367),(6,227,133,297,119,368),(7,228,134,298,120,369),(8,229,135,299,121,370),(9,230,136,300,122,371),(10,231,137,301,123,372),(11,232,138,302,124,373),(12,233,139,303,125,374),(13,234,140,304,126,375),(14,235,141,305,127,376),(15,236,142,306,128,377),(16,237,143,307,129,378),(17,238,144,308,130,379),(18,239,145,309,131,380),(19,240,146,310,132,381),(20,241,147,311,67,382),(21,242,148,312,68,383),(22,243,149,313,69,384),(23,244,150,314,70,385),(24,245,151,315,71,386),(25,246,152,316,72,387),(26,247,153,317,73,388),(27,248,154,318,74,389),(28,249,155,319,75,390),(29,250,156,320,76,391),(30,251,157,321,77,392),(31,252,158,322,78,393),(32,253,159,323,79,394),(33,254,160,324,80,395),(34,255,161,325,81,396),(35,256,162,326,82,331),(36,257,163,327,83,332),(37,258,164,328,84,333),(38,259,165,329,85,334),(39,260,166,330,86,335),(40,261,167,265,87,336),(41,262,168,266,88,337),(42,263,169,267,89,338),(43,264,170,268,90,339),(44,199,171,269,91,340),(45,200,172,270,92,341),(46,201,173,271,93,342),(47,202,174,272,94,343),(48,203,175,273,95,344),(49,204,176,274,96,345),(50,205,177,275,97,346),(51,206,178,276,98,347),(52,207,179,277,99,348),(53,208,180,278,100,349),(54,209,181,279,101,350),(55,210,182,280,102,351),(56,211,183,281,103,352),(57,212,184,282,104,353),(58,213,185,283,105,354),(59,214,186,284,106,355),(60,215,187,285,107,356),(61,216,188,286,108,357),(62,217,189,287,109,358),(63,218,190,288,110,359),(64,219,191,289,111,360),(65,220,192,290,112,361),(66,221,193,291,113,362)], [(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),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)]])
396 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X | 11A | ··· | 11J | 22A | ··· | 22AD | 33A | ··· | 33CB | 66A | ··· | 66IF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
396 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 |
| kernel | C6×C66 | C3×C66 | C2×C66 | C66 | C62 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 8 | 24 | 10 | 30 | 80 | 240 |
Matrix representation of C6×C66 ►in GL3(𝔽67) generated by
| 1 | 0 | 0 |
| 0 | 29 | 0 |
| 0 | 0 | 66 |
| 34 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 45 |
G:=sub<GL(3,GF(67))| [1,0,0,0,29,0,0,0,66],[34,0,0,0,7,0,0,0,45] >;
C6×C66 in GAP, Magma, Sage, TeX
C_6\times C_{66} % in TeX
G:=Group("C6xC66"); // GroupNames label
G:=SmallGroup(396,30);
// by ID
G=gap.SmallGroup(396,30);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-11]);
// Polycyclic
G:=Group<a,b|a^6=b^66=1,a*b=b*a>;
// generators/relations