direct product, abelian, monomial, 2-elementary
Aliases: C2×C174, SmallGroup(348,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C174 |
| C1 — C2×C174 |
| C1 — C2×C174 |
Generators and relations for C2×C174
G = < a,b | a2=b174=1, ab=ba >
Smallest permutation representation of C2×C174
►Regular action on 348 points
Generators in S348
(1 207)(2 208)(3 209)(4 210)(5 211)(6 212)(7 213)(8 214)(9 215)(10 216)(11 217)(12 218)(13 219)(14 220)(15 221)(16 222)(17 223)(18 224)(19 225)(20 226)(21 227)(22 228)(23 229)(24 230)(25 231)(26 232)(27 233)(28 234)(29 235)(30 236)(31 237)(32 238)(33 239)(34 240)(35 241)(36 242)(37 243)(38 244)(39 245)(40 246)(41 247)(42 248)(43 249)(44 250)(45 251)(46 252)(47 253)(48 254)(49 255)(50 256)(51 257)(52 258)(53 259)(54 260)(55 261)(56 262)(57 263)(58 264)(59 265)(60 266)(61 267)(62 268)(63 269)(64 270)(65 271)(66 272)(67 273)(68 274)(69 275)(70 276)(71 277)(72 278)(73 279)(74 280)(75 281)(76 282)(77 283)(78 284)(79 285)(80 286)(81 287)(82 288)(83 289)(84 290)(85 291)(86 292)(87 293)(88 294)(89 295)(90 296)(91 297)(92 298)(93 299)(94 300)(95 301)(96 302)(97 303)(98 304)(99 305)(100 306)(101 307)(102 308)(103 309)(104 310)(105 311)(106 312)(107 313)(108 314)(109 315)(110 316)(111 317)(112 318)(113 319)(114 320)(115 321)(116 322)(117 323)(118 324)(119 325)(120 326)(121 327)(122 328)(123 329)(124 330)(125 331)(126 332)(127 333)(128 334)(129 335)(130 336)(131 337)(132 338)(133 339)(134 340)(135 341)(136 342)(137 343)(138 344)(139 345)(140 346)(141 347)(142 348)(143 175)(144 176)(145 177)(146 178)(147 179)(148 180)(149 181)(150 182)(151 183)(152 184)(153 185)(154 186)(155 187)(156 188)(157 189)(158 190)(159 191)(160 192)(161 193)(162 194)(163 195)(164 196)(165 197)(166 198)(167 199)(168 200)(169 201)(170 202)(171 203)(172 204)(173 205)(174 206)
(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)
G:=sub<Sym(348)| (1,207)(2,208)(3,209)(4,210)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,221)(16,222)(17,223)(18,224)(19,225)(20,226)(21,227)(22,228)(23,229)(24,230)(25,231)(26,232)(27,233)(28,234)(29,235)(30,236)(31,237)(32,238)(33,239)(34,240)(35,241)(36,242)(37,243)(38,244)(39,245)(40,246)(41,247)(42,248)(43,249)(44,250)(45,251)(46,252)(47,253)(48,254)(49,255)(50,256)(51,257)(52,258)(53,259)(54,260)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(61,267)(62,268)(63,269)(64,270)(65,271)(66,272)(67,273)(68,274)(69,275)(70,276)(71,277)(72,278)(73,279)(74,280)(75,281)(76,282)(77,283)(78,284)(79,285)(80,286)(81,287)(82,288)(83,289)(84,290)(85,291)(86,292)(87,293)(88,294)(89,295)(90,296)(91,297)(92,298)(93,299)(94,300)(95,301)(96,302)(97,303)(98,304)(99,305)(100,306)(101,307)(102,308)(103,309)(104,310)(105,311)(106,312)(107,313)(108,314)(109,315)(110,316)(111,317)(112,318)(113,319)(114,320)(115,321)(116,322)(117,323)(118,324)(119,325)(120,326)(121,327)(122,328)(123,329)(124,330)(125,331)(126,332)(127,333)(128,334)(129,335)(130,336)(131,337)(132,338)(133,339)(134,340)(135,341)(136,342)(137,343)(138,344)(139,345)(140,346)(141,347)(142,348)(143,175)(144,176)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206), (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)>;
G:=Group( (1,207)(2,208)(3,209)(4,210)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,221)(16,222)(17,223)(18,224)(19,225)(20,226)(21,227)(22,228)(23,229)(24,230)(25,231)(26,232)(27,233)(28,234)(29,235)(30,236)(31,237)(32,238)(33,239)(34,240)(35,241)(36,242)(37,243)(38,244)(39,245)(40,246)(41,247)(42,248)(43,249)(44,250)(45,251)(46,252)(47,253)(48,254)(49,255)(50,256)(51,257)(52,258)(53,259)(54,260)(55,261)(56,262)(57,263)(58,264)(59,265)(60,266)(61,267)(62,268)(63,269)(64,270)(65,271)(66,272)(67,273)(68,274)(69,275)(70,276)(71,277)(72,278)(73,279)(74,280)(75,281)(76,282)(77,283)(78,284)(79,285)(80,286)(81,287)(82,288)(83,289)(84,290)(85,291)(86,292)(87,293)(88,294)(89,295)(90,296)(91,297)(92,298)(93,299)(94,300)(95,301)(96,302)(97,303)(98,304)(99,305)(100,306)(101,307)(102,308)(103,309)(104,310)(105,311)(106,312)(107,313)(108,314)(109,315)(110,316)(111,317)(112,318)(113,319)(114,320)(115,321)(116,322)(117,323)(118,324)(119,325)(120,326)(121,327)(122,328)(123,329)(124,330)(125,331)(126,332)(127,333)(128,334)(129,335)(130,336)(131,337)(132,338)(133,339)(134,340)(135,341)(136,342)(137,343)(138,344)(139,345)(140,346)(141,347)(142,348)(143,175)(144,176)(145,177)(146,178)(147,179)(148,180)(149,181)(150,182)(151,183)(152,184)(153,185)(154,186)(155,187)(156,188)(157,189)(158,190)(159,191)(160,192)(161,193)(162,194)(163,195)(164,196)(165,197)(166,198)(167,199)(168,200)(169,201)(170,202)(171,203)(172,204)(173,205)(174,206), (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) );
G=PermutationGroup([[(1,207),(2,208),(3,209),(4,210),(5,211),(6,212),(7,213),(8,214),(9,215),(10,216),(11,217),(12,218),(13,219),(14,220),(15,221),(16,222),(17,223),(18,224),(19,225),(20,226),(21,227),(22,228),(23,229),(24,230),(25,231),(26,232),(27,233),(28,234),(29,235),(30,236),(31,237),(32,238),(33,239),(34,240),(35,241),(36,242),(37,243),(38,244),(39,245),(40,246),(41,247),(42,248),(43,249),(44,250),(45,251),(46,252),(47,253),(48,254),(49,255),(50,256),(51,257),(52,258),(53,259),(54,260),(55,261),(56,262),(57,263),(58,264),(59,265),(60,266),(61,267),(62,268),(63,269),(64,270),(65,271),(66,272),(67,273),(68,274),(69,275),(70,276),(71,277),(72,278),(73,279),(74,280),(75,281),(76,282),(77,283),(78,284),(79,285),(80,286),(81,287),(82,288),(83,289),(84,290),(85,291),(86,292),(87,293),(88,294),(89,295),(90,296),(91,297),(92,298),(93,299),(94,300),(95,301),(96,302),(97,303),(98,304),(99,305),(100,306),(101,307),(102,308),(103,309),(104,310),(105,311),(106,312),(107,313),(108,314),(109,315),(110,316),(111,317),(112,318),(113,319),(114,320),(115,321),(116,322),(117,323),(118,324),(119,325),(120,326),(121,327),(122,328),(123,329),(124,330),(125,331),(126,332),(127,333),(128,334),(129,335),(130,336),(131,337),(132,338),(133,339),(134,340),(135,341),(136,342),(137,343),(138,344),(139,345),(140,346),(141,347),(142,348),(143,175),(144,176),(145,177),(146,178),(147,179),(148,180),(149,181),(150,182),(151,183),(152,184),(153,185),(154,186),(155,187),(156,188),(157,189),(158,190),(159,191),(160,192),(161,193),(162,194),(163,195),(164,196),(165,197),(166,198),(167,199),(168,200),(169,201),(170,202),(171,203),(172,204),(173,205),(174,206)], [(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)]])
348 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 29A | ··· | 29AB | 58A | ··· | 58CF | 87A | ··· | 87BD | 174A | ··· | 174FL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 | 174 | ··· | 174 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
348 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C29 | C58 | C87 | C174 |
| kernel | C2×C174 | C174 | C2×C58 | C58 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 28 | 84 | 56 | 168 |
Matrix representation of C2×C174 ►in GL2(𝔽349) generated by
| 348 | 0 |
| 0 | 348 |
| 83 | 0 |
| 0 | 322 |
G:=sub<GL(2,GF(349))| [348,0,0,348],[83,0,0,322] >;
C2×C174 in GAP, Magma, Sage, TeX
C_2\times C_{174} % in TeX
G:=Group("C2xC174"); // GroupNames label
G:=SmallGroup(348,12);
// by ID
G=gap.SmallGroup(348,12);
# by ID
G:=PCGroup([4,-2,-2,-3,-29]);
// Polycyclic
G:=Group<a,b|a^2=b^174=1,a*b=b*a>;
// generators/relations
Export