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G = C11×C33order 363 = 3·112

Abelian group of type [11,33]

direct product, abelian, monomial, 11-elementary

Aliases: C11×C33, SmallGroup(363,3)

Series: Derived Chief Lower central Upper central

C1 — C11×C33
C1C11C112 — C11×C33
C1 — C11×C33
C1 — C11×C33

Generators and relations for C11×C33
 G = < a,b | a11=b33=1, ab=ba >


Smallest permutation representation of C11×C33
Regular action on 363 points
Generators in S363
(1 144 242 323 99 102 285 62 342 215 193)(2 145 243 324 67 103 286 63 343 216 194)(3 146 244 325 68 104 287 64 344 217 195)(4 147 245 326 69 105 288 65 345 218 196)(5 148 246 327 70 106 289 66 346 219 197)(6 149 247 328 71 107 290 34 347 220 198)(7 150 248 329 72 108 291 35 348 221 166)(8 151 249 330 73 109 292 36 349 222 167)(9 152 250 298 74 110 293 37 350 223 168)(10 153 251 299 75 111 294 38 351 224 169)(11 154 252 300 76 112 295 39 352 225 170)(12 155 253 301 77 113 296 40 353 226 171)(13 156 254 302 78 114 297 41 354 227 172)(14 157 255 303 79 115 265 42 355 228 173)(15 158 256 304 80 116 266 43 356 229 174)(16 159 257 305 81 117 267 44 357 230 175)(17 160 258 306 82 118 268 45 358 231 176)(18 161 259 307 83 119 269 46 359 199 177)(19 162 260 308 84 120 270 47 360 200 178)(20 163 261 309 85 121 271 48 361 201 179)(21 164 262 310 86 122 272 49 362 202 180)(22 165 263 311 87 123 273 50 363 203 181)(23 133 264 312 88 124 274 51 331 204 182)(24 134 232 313 89 125 275 52 332 205 183)(25 135 233 314 90 126 276 53 333 206 184)(26 136 234 315 91 127 277 54 334 207 185)(27 137 235 316 92 128 278 55 335 208 186)(28 138 236 317 93 129 279 56 336 209 187)(29 139 237 318 94 130 280 57 337 210 188)(30 140 238 319 95 131 281 58 338 211 189)(31 141 239 320 96 132 282 59 339 212 190)(32 142 240 321 97 100 283 60 340 213 191)(33 143 241 322 98 101 284 61 341 214 192)
(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)

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

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

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

363 conjugacy classes

class 1 3A3B11A···11DP33A···33IF
order13311···1133···33
size1111···11···1

363 irreducible representations

dim1111
type+
imageC1C3C11C33
kernelC11×C33C112C33C11
# reps12120240

Matrix representation of C11×C33 in GL2(𝔽67) generated by

90
022
,
220
035
G:=sub<GL(2,GF(67))| [9,0,0,22],[22,0,0,35] >;

C11×C33 in GAP, Magma, Sage, TeX

C_{11}\times C_{33}
% in TeX

G:=Group("C11xC33");
// GroupNames label

G:=SmallGroup(363,3);
// by ID

G=gap.SmallGroup(363,3);
# by ID

G:=PCGroup([3,-3,-11,-11]);
// Polycyclic

G:=Group<a,b|a^11=b^33=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C11×C33 in TeX

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