Copied to
clipboard

G = C3×C117order 351 = 33·13

Abelian group of type [3,117]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C117, SmallGroup(351,9)

Series: Derived Chief Lower central Upper central

C1 — C3×C117
C1C3C39C117 — C3×C117
C1 — C3×C117
C1 — C3×C117

Generators and relations for C3×C117
 G = < a,b | a3=b117=1, ab=ba >


Smallest permutation representation of C3×C117
Regular action on 351 points
Generators in S351
(1 126 272)(2 127 273)(3 128 274)(4 129 275)(5 130 276)(6 131 277)(7 132 278)(8 133 279)(9 134 280)(10 135 281)(11 136 282)(12 137 283)(13 138 284)(14 139 285)(15 140 286)(16 141 287)(17 142 288)(18 143 289)(19 144 290)(20 145 291)(21 146 292)(22 147 293)(23 148 294)(24 149 295)(25 150 296)(26 151 297)(27 152 298)(28 153 299)(29 154 300)(30 155 301)(31 156 302)(32 157 303)(33 158 304)(34 159 305)(35 160 306)(36 161 307)(37 162 308)(38 163 309)(39 164 310)(40 165 311)(41 166 312)(42 167 313)(43 168 314)(44 169 315)(45 170 316)(46 171 317)(47 172 318)(48 173 319)(49 174 320)(50 175 321)(51 176 322)(52 177 323)(53 178 324)(54 179 325)(55 180 326)(56 181 327)(57 182 328)(58 183 329)(59 184 330)(60 185 331)(61 186 332)(62 187 333)(63 188 334)(64 189 335)(65 190 336)(66 191 337)(67 192 338)(68 193 339)(69 194 340)(70 195 341)(71 196 342)(72 197 343)(73 198 344)(74 199 345)(75 200 346)(76 201 347)(77 202 348)(78 203 349)(79 204 350)(80 205 351)(81 206 235)(82 207 236)(83 208 237)(84 209 238)(85 210 239)(86 211 240)(87 212 241)(88 213 242)(89 214 243)(90 215 244)(91 216 245)(92 217 246)(93 218 247)(94 219 248)(95 220 249)(96 221 250)(97 222 251)(98 223 252)(99 224 253)(100 225 254)(101 226 255)(102 227 256)(103 228 257)(104 229 258)(105 230 259)(106 231 260)(107 232 261)(108 233 262)(109 234 263)(110 118 264)(111 119 265)(112 120 266)(113 121 267)(114 122 268)(115 123 269)(116 124 270)(117 125 271)
(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)

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

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

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

351 conjugacy classes

class 1 3A···3H9A···9R13A···13L39A···39CR117A···117HH
order13···39···913···1339···39117···117
size11···11···11···11···11···1

351 irreducible representations

dim11111111
type+
imageC1C3C3C9C13C39C39C117
kernelC3×C117C117C3×C39C39C3×C9C9C32C3
# reps16218127224216

Matrix representation of C3×C117 in GL2(𝔽937) generated by

6140
0614
,
5050
0676
G:=sub<GL(2,GF(937))| [614,0,0,614],[505,0,0,676] >;

C3×C117 in GAP, Magma, Sage, TeX

C_3\times C_{117}
% in TeX

G:=Group("C3xC117");
// GroupNames label

G:=SmallGroup(351,9);
// by ID

G=gap.SmallGroup(351,9);
# by ID

G:=PCGroup([4,-3,-3,-13,-3,468]);
// Polycyclic

G:=Group<a,b|a^3=b^117=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C117 in TeX

׿
×
𝔽