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G = C3×C114order 342 = 2·32·19

Abelian group of type [3,114]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C114, SmallGroup(342,18)

Series: Derived Chief Lower central Upper central

C1 — C3×C114
C1C19C57C3×C57 — C3×C114
C1 — C3×C114
C1 — C3×C114

Generators and relations for C3×C114
 G = < a,b | a3=b114=1, ab=ba >


Smallest permutation representation of C3×C114
Regular action on 342 points
Generators in S342
(1 169 265)(2 170 266)(3 171 267)(4 172 268)(5 173 269)(6 174 270)(7 175 271)(8 176 272)(9 177 273)(10 178 274)(11 179 275)(12 180 276)(13 181 277)(14 182 278)(15 183 279)(16 184 280)(17 185 281)(18 186 282)(19 187 283)(20 188 284)(21 189 285)(22 190 286)(23 191 287)(24 192 288)(25 193 289)(26 194 290)(27 195 291)(28 196 292)(29 197 293)(30 198 294)(31 199 295)(32 200 296)(33 201 297)(34 202 298)(35 203 299)(36 204 300)(37 205 301)(38 206 302)(39 207 303)(40 208 304)(41 209 305)(42 210 306)(43 211 307)(44 212 308)(45 213 309)(46 214 310)(47 215 311)(48 216 312)(49 217 313)(50 218 314)(51 219 315)(52 220 316)(53 221 317)(54 222 318)(55 223 319)(56 224 320)(57 225 321)(58 226 322)(59 227 323)(60 228 324)(61 115 325)(62 116 326)(63 117 327)(64 118 328)(65 119 329)(66 120 330)(67 121 331)(68 122 332)(69 123 333)(70 124 334)(71 125 335)(72 126 336)(73 127 337)(74 128 338)(75 129 339)(76 130 340)(77 131 341)(78 132 342)(79 133 229)(80 134 230)(81 135 231)(82 136 232)(83 137 233)(84 138 234)(85 139 235)(86 140 236)(87 141 237)(88 142 238)(89 143 239)(90 144 240)(91 145 241)(92 146 242)(93 147 243)(94 148 244)(95 149 245)(96 150 246)(97 151 247)(98 152 248)(99 153 249)(100 154 250)(101 155 251)(102 156 252)(103 157 253)(104 158 254)(105 159 255)(106 160 256)(107 161 257)(108 162 258)(109 163 259)(110 164 260)(111 165 261)(112 166 262)(113 167 263)(114 168 264)
(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)

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

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

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

342 conjugacy classes

class 1  2 3A···3H6A···6H19A···19R38A···38R57A···57EN114A···114EN
order123···36···619···1938···3857···57114···114
size111···11···11···11···11···11···1

342 irreducible representations

dim11111111
type++
imageC1C2C3C6C19C38C57C114
kernelC3×C114C3×C57C114C57C3×C6C32C6C3
# reps11881818144144

Matrix representation of C3×C114 in GL2(𝔽229) generated by

1340
094
,
1460
015
G:=sub<GL(2,GF(229))| [134,0,0,94],[146,0,0,15] >;

C3×C114 in GAP, Magma, Sage, TeX

C_3\times C_{114}
% in TeX

G:=Group("C3xC114");
// GroupNames label

G:=SmallGroup(342,18);
// by ID

G=gap.SmallGroup(342,18);
# by ID

G:=PCGroup([4,-2,-3,-3,-19]);
// Polycyclic

G:=Group<a,b|a^3=b^114=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C114 in TeX

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