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G = C3×C123order 369 = 32·41

Abelian group of type [3,123]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C123, SmallGroup(369,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C123
C1C41C123 — C3×C123
C1 — C3×C123
C1 — C3×C123

Generators and relations for C3×C123
 G = < a,b | a3=b123=1, ab=ba >


Smallest permutation representation of C3×C123
Regular action on 369 points
Generators in S369
(1 326 195)(2 327 196)(3 328 197)(4 329 198)(5 330 199)(6 331 200)(7 332 201)(8 333 202)(9 334 203)(10 335 204)(11 336 205)(12 337 206)(13 338 207)(14 339 208)(15 340 209)(16 341 210)(17 342 211)(18 343 212)(19 344 213)(20 345 214)(21 346 215)(22 347 216)(23 348 217)(24 349 218)(25 350 219)(26 351 220)(27 352 221)(28 353 222)(29 354 223)(30 355 224)(31 356 225)(32 357 226)(33 358 227)(34 359 228)(35 360 229)(36 361 230)(37 362 231)(38 363 232)(39 364 233)(40 365 234)(41 366 235)(42 367 236)(43 368 237)(44 369 238)(45 247 239)(46 248 240)(47 249 241)(48 250 242)(49 251 243)(50 252 244)(51 253 245)(52 254 246)(53 255 124)(54 256 125)(55 257 126)(56 258 127)(57 259 128)(58 260 129)(59 261 130)(60 262 131)(61 263 132)(62 264 133)(63 265 134)(64 266 135)(65 267 136)(66 268 137)(67 269 138)(68 270 139)(69 271 140)(70 272 141)(71 273 142)(72 274 143)(73 275 144)(74 276 145)(75 277 146)(76 278 147)(77 279 148)(78 280 149)(79 281 150)(80 282 151)(81 283 152)(82 284 153)(83 285 154)(84 286 155)(85 287 156)(86 288 157)(87 289 158)(88 290 159)(89 291 160)(90 292 161)(91 293 162)(92 294 163)(93 295 164)(94 296 165)(95 297 166)(96 298 167)(97 299 168)(98 300 169)(99 301 170)(100 302 171)(101 303 172)(102 304 173)(103 305 174)(104 306 175)(105 307 176)(106 308 177)(107 309 178)(108 310 179)(109 311 180)(110 312 181)(111 313 182)(112 314 183)(113 315 184)(114 316 185)(115 317 186)(116 318 187)(117 319 188)(118 320 189)(119 321 190)(120 322 191)(121 323 192)(122 324 193)(123 325 194)
(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)

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

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

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

369 conjugacy classes

class 1 3A···3H41A···41AN123A···123LH
order13···341···41123···123
size11···11···11···1

369 irreducible representations

dim1111
type+
imageC1C3C41C123
kernelC3×C123C123C32C3
# reps1840320

Matrix representation of C3×C123 in GL2(𝔽739) generated by

10
0320
,
4330
0433
G:=sub<GL(2,GF(739))| [1,0,0,320],[433,0,0,433] >;

C3×C123 in GAP, Magma, Sage, TeX

C_3\times C_{123}
% in TeX

G:=Group("C3xC123");
// GroupNames label

G:=SmallGroup(369,2);
// by ID

G=gap.SmallGroup(369,2);
# by ID

G:=PCGroup([3,-3,-3,-41]);
// Polycyclic

G:=Group<a,b|a^3=b^123=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C123 in TeX

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