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## G = C3×C114order 342 = 2·32·19

### Abelian group of type [3,114]

Aliases: C3×C114, SmallGroup(342,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C114
 Chief series C1 — C19 — C57 — C3×C57 — C3×C114
 Lower central C1 — C3×C114
 Upper central 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
(1 243 139)(2 244 140)(3 245 141)(4 246 142)(5 247 143)(6 248 144)(7 249 145)(8 250 146)(9 251 147)(10 252 148)(11 253 149)(12 254 150)(13 255 151)(14 256 152)(15 257 153)(16 258 154)(17 259 155)(18 260 156)(19 261 157)(20 262 158)(21 263 159)(22 264 160)(23 265 161)(24 266 162)(25 267 163)(26 268 164)(27 269 165)(28 270 166)(29 271 167)(30 272 168)(31 273 169)(32 274 170)(33 275 171)(34 276 172)(35 277 173)(36 278 174)(37 279 175)(38 280 176)(39 281 177)(40 282 178)(41 283 179)(42 284 180)(43 285 181)(44 286 182)(45 287 183)(46 288 184)(47 289 185)(48 290 186)(49 291 187)(50 292 188)(51 293 189)(52 294 190)(53 295 191)(54 296 192)(55 297 193)(56 298 194)(57 299 195)(58 300 196)(59 301 197)(60 302 198)(61 303 199)(62 304 200)(63 305 201)(64 306 202)(65 307 203)(66 308 204)(67 309 205)(68 310 206)(69 311 207)(70 312 208)(71 313 209)(72 314 210)(73 315 211)(74 316 212)(75 317 213)(76 318 214)(77 319 215)(78 320 216)(79 321 217)(80 322 218)(81 323 219)(82 324 220)(83 325 221)(84 326 222)(85 327 223)(86 328 224)(87 329 225)(88 330 226)(89 331 227)(90 332 228)(91 333 115)(92 334 116)(93 335 117)(94 336 118)(95 337 119)(96 338 120)(97 339 121)(98 340 122)(99 341 123)(100 342 124)(101 229 125)(102 230 126)(103 231 127)(104 232 128)(105 233 129)(106 234 130)(107 235 131)(108 236 132)(109 237 133)(110 238 134)(111 239 135)(112 240 136)(113 241 137)(114 242 138)
(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)| (1,243,139)(2,244,140)(3,245,141)(4,246,142)(5,247,143)(6,248,144)(7,249,145)(8,250,146)(9,251,147)(10,252,148)(11,253,149)(12,254,150)(13,255,151)(14,256,152)(15,257,153)(16,258,154)(17,259,155)(18,260,156)(19,261,157)(20,262,158)(21,263,159)(22,264,160)(23,265,161)(24,266,162)(25,267,163)(26,268,164)(27,269,165)(28,270,166)(29,271,167)(30,272,168)(31,273,169)(32,274,170)(33,275,171)(34,276,172)(35,277,173)(36,278,174)(37,279,175)(38,280,176)(39,281,177)(40,282,178)(41,283,179)(42,284,180)(43,285,181)(44,286,182)(45,287,183)(46,288,184)(47,289,185)(48,290,186)(49,291,187)(50,292,188)(51,293,189)(52,294,190)(53,295,191)(54,296,192)(55,297,193)(56,298,194)(57,299,195)(58,300,196)(59,301,197)(60,302,198)(61,303,199)(62,304,200)(63,305,201)(64,306,202)(65,307,203)(66,308,204)(67,309,205)(68,310,206)(69,311,207)(70,312,208)(71,313,209)(72,314,210)(73,315,211)(74,316,212)(75,317,213)(76,318,214)(77,319,215)(78,320,216)(79,321,217)(80,322,218)(81,323,219)(82,324,220)(83,325,221)(84,326,222)(85,327,223)(86,328,224)(87,329,225)(88,330,226)(89,331,227)(90,332,228)(91,333,115)(92,334,116)(93,335,117)(94,336,118)(95,337,119)(96,338,120)(97,339,121)(98,340,122)(99,341,123)(100,342,124)(101,229,125)(102,230,126)(103,231,127)(104,232,128)(105,233,129)(106,234,130)(107,235,131)(108,236,132)(109,237,133)(110,238,134)(111,239,135)(112,240,136)(113,241,137)(114,242,138), (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( (1,243,139)(2,244,140)(3,245,141)(4,246,142)(5,247,143)(6,248,144)(7,249,145)(8,250,146)(9,251,147)(10,252,148)(11,253,149)(12,254,150)(13,255,151)(14,256,152)(15,257,153)(16,258,154)(17,259,155)(18,260,156)(19,261,157)(20,262,158)(21,263,159)(22,264,160)(23,265,161)(24,266,162)(25,267,163)(26,268,164)(27,269,165)(28,270,166)(29,271,167)(30,272,168)(31,273,169)(32,274,170)(33,275,171)(34,276,172)(35,277,173)(36,278,174)(37,279,175)(38,280,176)(39,281,177)(40,282,178)(41,283,179)(42,284,180)(43,285,181)(44,286,182)(45,287,183)(46,288,184)(47,289,185)(48,290,186)(49,291,187)(50,292,188)(51,293,189)(52,294,190)(53,295,191)(54,296,192)(55,297,193)(56,298,194)(57,299,195)(58,300,196)(59,301,197)(60,302,198)(61,303,199)(62,304,200)(63,305,201)(64,306,202)(65,307,203)(66,308,204)(67,309,205)(68,310,206)(69,311,207)(70,312,208)(71,313,209)(72,314,210)(73,315,211)(74,316,212)(75,317,213)(76,318,214)(77,319,215)(78,320,216)(79,321,217)(80,322,218)(81,323,219)(82,324,220)(83,325,221)(84,326,222)(85,327,223)(86,328,224)(87,329,225)(88,330,226)(89,331,227)(90,332,228)(91,333,115)(92,334,116)(93,335,117)(94,336,118)(95,337,119)(96,338,120)(97,339,121)(98,340,122)(99,341,123)(100,342,124)(101,229,125)(102,230,126)(103,231,127)(104,232,128)(105,233,129)(106,234,130)(107,235,131)(108,236,132)(109,237,133)(110,238,134)(111,239,135)(112,240,136)(113,241,137)(114,242,138), (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([[(1,243,139),(2,244,140),(3,245,141),(4,246,142),(5,247,143),(6,248,144),(7,249,145),(8,250,146),(9,251,147),(10,252,148),(11,253,149),(12,254,150),(13,255,151),(14,256,152),(15,257,153),(16,258,154),(17,259,155),(18,260,156),(19,261,157),(20,262,158),(21,263,159),(22,264,160),(23,265,161),(24,266,162),(25,267,163),(26,268,164),(27,269,165),(28,270,166),(29,271,167),(30,272,168),(31,273,169),(32,274,170),(33,275,171),(34,276,172),(35,277,173),(36,278,174),(37,279,175),(38,280,176),(39,281,177),(40,282,178),(41,283,179),(42,284,180),(43,285,181),(44,286,182),(45,287,183),(46,288,184),(47,289,185),(48,290,186),(49,291,187),(50,292,188),(51,293,189),(52,294,190),(53,295,191),(54,296,192),(55,297,193),(56,298,194),(57,299,195),(58,300,196),(59,301,197),(60,302,198),(61,303,199),(62,304,200),(63,305,201),(64,306,202),(65,307,203),(66,308,204),(67,309,205),(68,310,206),(69,311,207),(70,312,208),(71,313,209),(72,314,210),(73,315,211),(74,316,212),(75,317,213),(76,318,214),(77,319,215),(78,320,216),(79,321,217),(80,322,218),(81,323,219),(82,324,220),(83,325,221),(84,326,222),(85,327,223),(86,328,224),(87,329,225),(88,330,226),(89,331,227),(90,332,228),(91,333,115),(92,334,116),(93,335,117),(94,336,118),(95,337,119),(96,338,120),(97,339,121),(98,340,122),(99,341,123),(100,342,124),(101,229,125),(102,230,126),(103,231,127),(104,232,128),(105,233,129),(106,234,130),(107,235,131),(108,236,132),(109,237,133),(110,238,134),(111,239,135),(112,240,136),(113,241,137),(114,242,138)], [(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 ··· 3H 6A ··· 6H 19A ··· 19R 38A ··· 38R 57A ··· 57EN 114A ··· 114EN order 1 2 3 ··· 3 6 ··· 6 19 ··· 19 38 ··· 38 57 ··· 57 114 ··· 114 size 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

342 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C19 C38 C57 C114 kernel C3×C114 C3×C57 C114 C57 C3×C6 C32 C6 C3 # reps 1 1 8 8 18 18 144 144

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

 134 0 0 94
,
 146 0 0 15
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

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