Copied to
clipboard

G = C6×C66order 396 = 22·32·11

Abelian group of type [6,66]

direct product, abelian, monomial

Aliases: C6×C66, SmallGroup(396,30)

Series: Derived Chief Lower central Upper central

C1 — C6×C66
C1C11C33C3×C33C3×C66 — 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 [×3], C3 [×4], C22, C6 [×12], C32, C11, C2×C6 [×4], C3×C6 [×3], C22 [×3], C33 [×4], C62, C2×C22, C66 [×12], C3×C33, C2×C66 [×4], C3×C66 [×3], C6×C66
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C11, C2×C6 [×4], C3×C6 [×3], C22 [×3], C33 [×4], C62, C2×C22, C66 [×12], C3×C33, C2×C66 [×4], C3×C66 [×3], C6×C66

Smallest permutation representation of C6×C66
Regular action on 396 points
Generators in S396
(1 178 72 286 240 344)(2 179 73 287 241 345)(3 180 74 288 242 346)(4 181 75 289 243 347)(5 182 76 290 244 348)(6 183 77 291 245 349)(7 184 78 292 246 350)(8 185 79 293 247 351)(9 186 80 294 248 352)(10 187 81 295 249 353)(11 188 82 296 250 354)(12 189 83 297 251 355)(13 190 84 298 252 356)(14 191 85 299 253 357)(15 192 86 300 254 358)(16 193 87 301 255 359)(17 194 88 302 256 360)(18 195 89 303 257 361)(19 196 90 304 258 362)(20 197 91 305 259 363)(21 198 92 306 260 364)(22 133 93 307 261 365)(23 134 94 308 262 366)(24 135 95 309 263 367)(25 136 96 310 264 368)(26 137 97 311 199 369)(27 138 98 312 200 370)(28 139 99 313 201 371)(29 140 100 314 202 372)(30 141 101 315 203 373)(31 142 102 316 204 374)(32 143 103 317 205 375)(33 144 104 318 206 376)(34 145 105 319 207 377)(35 146 106 320 208 378)(36 147 107 321 209 379)(37 148 108 322 210 380)(38 149 109 323 211 381)(39 150 110 324 212 382)(40 151 111 325 213 383)(41 152 112 326 214 384)(42 153 113 327 215 385)(43 154 114 328 216 386)(44 155 115 329 217 387)(45 156 116 330 218 388)(46 157 117 265 219 389)(47 158 118 266 220 390)(48 159 119 267 221 391)(49 160 120 268 222 392)(50 161 121 269 223 393)(51 162 122 270 224 394)(52 163 123 271 225 395)(53 164 124 272 226 396)(54 165 125 273 227 331)(55 166 126 274 228 332)(56 167 127 275 229 333)(57 168 128 276 230 334)(58 169 129 277 231 335)(59 170 130 278 232 336)(60 171 131 279 233 337)(61 172 132 280 234 338)(62 173 67 281 235 339)(63 174 68 282 236 340)(64 175 69 283 237 341)(65 176 70 284 238 342)(66 177 71 285 239 343)
(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)(331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)

G:=sub<Sym(396)| (1,178,72,286,240,344)(2,179,73,287,241,345)(3,180,74,288,242,346)(4,181,75,289,243,347)(5,182,76,290,244,348)(6,183,77,291,245,349)(7,184,78,292,246,350)(8,185,79,293,247,351)(9,186,80,294,248,352)(10,187,81,295,249,353)(11,188,82,296,250,354)(12,189,83,297,251,355)(13,190,84,298,252,356)(14,191,85,299,253,357)(15,192,86,300,254,358)(16,193,87,301,255,359)(17,194,88,302,256,360)(18,195,89,303,257,361)(19,196,90,304,258,362)(20,197,91,305,259,363)(21,198,92,306,260,364)(22,133,93,307,261,365)(23,134,94,308,262,366)(24,135,95,309,263,367)(25,136,96,310,264,368)(26,137,97,311,199,369)(27,138,98,312,200,370)(28,139,99,313,201,371)(29,140,100,314,202,372)(30,141,101,315,203,373)(31,142,102,316,204,374)(32,143,103,317,205,375)(33,144,104,318,206,376)(34,145,105,319,207,377)(35,146,106,320,208,378)(36,147,107,321,209,379)(37,148,108,322,210,380)(38,149,109,323,211,381)(39,150,110,324,212,382)(40,151,111,325,213,383)(41,152,112,326,214,384)(42,153,113,327,215,385)(43,154,114,328,216,386)(44,155,115,329,217,387)(45,156,116,330,218,388)(46,157,117,265,219,389)(47,158,118,266,220,390)(48,159,119,267,221,391)(49,160,120,268,222,392)(50,161,121,269,223,393)(51,162,122,270,224,394)(52,163,123,271,225,395)(53,164,124,272,226,396)(54,165,125,273,227,331)(55,166,126,274,228,332)(56,167,127,275,229,333)(57,168,128,276,230,334)(58,169,129,277,231,335)(59,170,130,278,232,336)(60,171,131,279,233,337)(61,172,132,280,234,338)(62,173,67,281,235,339)(63,174,68,282,236,340)(64,175,69,283,237,341)(65,176,70,284,238,342)(66,177,71,285,239,343), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)>;

G:=Group( (1,178,72,286,240,344)(2,179,73,287,241,345)(3,180,74,288,242,346)(4,181,75,289,243,347)(5,182,76,290,244,348)(6,183,77,291,245,349)(7,184,78,292,246,350)(8,185,79,293,247,351)(9,186,80,294,248,352)(10,187,81,295,249,353)(11,188,82,296,250,354)(12,189,83,297,251,355)(13,190,84,298,252,356)(14,191,85,299,253,357)(15,192,86,300,254,358)(16,193,87,301,255,359)(17,194,88,302,256,360)(18,195,89,303,257,361)(19,196,90,304,258,362)(20,197,91,305,259,363)(21,198,92,306,260,364)(22,133,93,307,261,365)(23,134,94,308,262,366)(24,135,95,309,263,367)(25,136,96,310,264,368)(26,137,97,311,199,369)(27,138,98,312,200,370)(28,139,99,313,201,371)(29,140,100,314,202,372)(30,141,101,315,203,373)(31,142,102,316,204,374)(32,143,103,317,205,375)(33,144,104,318,206,376)(34,145,105,319,207,377)(35,146,106,320,208,378)(36,147,107,321,209,379)(37,148,108,322,210,380)(38,149,109,323,211,381)(39,150,110,324,212,382)(40,151,111,325,213,383)(41,152,112,326,214,384)(42,153,113,327,215,385)(43,154,114,328,216,386)(44,155,115,329,217,387)(45,156,116,330,218,388)(46,157,117,265,219,389)(47,158,118,266,220,390)(48,159,119,267,221,391)(49,160,120,268,222,392)(50,161,121,269,223,393)(51,162,122,270,224,394)(52,163,123,271,225,395)(53,164,124,272,226,396)(54,165,125,273,227,331)(55,166,126,274,228,332)(56,167,127,275,229,333)(57,168,128,276,230,334)(58,169,129,277,231,335)(59,170,130,278,232,336)(60,171,131,279,233,337)(61,172,132,280,234,338)(62,173,67,281,235,339)(63,174,68,282,236,340)(64,175,69,283,237,341)(65,176,70,284,238,342)(66,177,71,285,239,343), (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)(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396) );

G=PermutationGroup([(1,178,72,286,240,344),(2,179,73,287,241,345),(3,180,74,288,242,346),(4,181,75,289,243,347),(5,182,76,290,244,348),(6,183,77,291,245,349),(7,184,78,292,246,350),(8,185,79,293,247,351),(9,186,80,294,248,352),(10,187,81,295,249,353),(11,188,82,296,250,354),(12,189,83,297,251,355),(13,190,84,298,252,356),(14,191,85,299,253,357),(15,192,86,300,254,358),(16,193,87,301,255,359),(17,194,88,302,256,360),(18,195,89,303,257,361),(19,196,90,304,258,362),(20,197,91,305,259,363),(21,198,92,306,260,364),(22,133,93,307,261,365),(23,134,94,308,262,366),(24,135,95,309,263,367),(25,136,96,310,264,368),(26,137,97,311,199,369),(27,138,98,312,200,370),(28,139,99,313,201,371),(29,140,100,314,202,372),(30,141,101,315,203,373),(31,142,102,316,204,374),(32,143,103,317,205,375),(33,144,104,318,206,376),(34,145,105,319,207,377),(35,146,106,320,208,378),(36,147,107,321,209,379),(37,148,108,322,210,380),(38,149,109,323,211,381),(39,150,110,324,212,382),(40,151,111,325,213,383),(41,152,112,326,214,384),(42,153,113,327,215,385),(43,154,114,328,216,386),(44,155,115,329,217,387),(45,156,116,330,218,388),(46,157,117,265,219,389),(47,158,118,266,220,390),(48,159,119,267,221,391),(49,160,120,268,222,392),(50,161,121,269,223,393),(51,162,122,270,224,394),(52,163,123,271,225,395),(53,164,124,272,226,396),(54,165,125,273,227,331),(55,166,126,274,228,332),(56,167,127,275,229,333),(57,168,128,276,230,334),(58,169,129,277,231,335),(59,170,130,278,232,336),(60,171,131,279,233,337),(61,172,132,280,234,338),(62,173,67,281,235,339),(63,174,68,282,236,340),(64,175,69,283,237,341),(65,176,70,284,238,342),(66,177,71,285,239,343)], [(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),(331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)])

396 conjugacy classes

class 1 2A2B2C3A···3H6A···6X11A···11J22A···22AD33A···33CB66A···66IF
order12223···36···611···1122···2233···3366···66
size11111···11···11···11···11···11···1

396 irreducible representations

dim11111111
type++
imageC1C2C3C6C11C22C33C66
kernelC6×C66C3×C66C2×C66C66C62C3×C6C2×C6C6
# reps13824103080240

Matrix representation of C6×C66 in GL3(𝔽67) generated by

100
0290
0066
,
3400
070
0045
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

׿
×
𝔽