direct product, abelian, monomial, 2-elementary
Aliases: C2×C168, SmallGroup(336,109)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C168 |
| C1 — C2×C168 |
| C1 — C2×C168 |
Generators and relations for C2×C168
G = < a,b | a2=b168=1, ab=ba >
Smallest permutation representation of C2×C168
►Regular action on 336 points
Generators in S336
(1 328)(2 329)(3 330)(4 331)(5 332)(6 333)(7 334)(8 335)(9 336)(10 169)(11 170)(12 171)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 184)(26 185)(27 186)(28 187)(29 188)(30 189)(31 190)(32 191)(33 192)(34 193)(35 194)(36 195)(37 196)(38 197)(39 198)(40 199)(41 200)(42 201)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 211)(53 212)(54 213)(55 214)(56 215)(57 216)(58 217)(59 218)(60 219)(61 220)(62 221)(63 222)(64 223)(65 224)(66 225)(67 226)(68 227)(69 228)(70 229)(71 230)(72 231)(73 232)(74 233)(75 234)(76 235)(77 236)(78 237)(79 238)(80 239)(81 240)(82 241)(83 242)(84 243)(85 244)(86 245)(87 246)(88 247)(89 248)(90 249)(91 250)(92 251)(93 252)(94 253)(95 254)(96 255)(97 256)(98 257)(99 258)(100 259)(101 260)(102 261)(103 262)(104 263)(105 264)(106 265)(107 266)(108 267)(109 268)(110 269)(111 270)(112 271)(113 272)(114 273)(115 274)(116 275)(117 276)(118 277)(119 278)(120 279)(121 280)(122 281)(123 282)(124 283)(125 284)(126 285)(127 286)(128 287)(129 288)(130 289)(131 290)(132 291)(133 292)(134 293)(135 294)(136 295)(137 296)(138 297)(139 298)(140 299)(141 300)(142 301)(143 302)(144 303)(145 304)(146 305)(147 306)(148 307)(149 308)(150 309)(151 310)(152 311)(153 312)(154 313)(155 314)(156 315)(157 316)(158 317)(159 318)(160 319)(161 320)(162 321)(163 322)(164 323)(165 324)(166 325)(167 326)(168 327)
(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)
G:=sub<Sym(336)| (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,187)(29,188)(30,189)(31,190)(32,191)(33,192)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,225)(67,226)(68,227)(69,228)(70,229)(71,230)(72,231)(73,232)(74,233)(75,234)(76,235)(77,236)(78,237)(79,238)(80,239)(81,240)(82,241)(83,242)(84,243)(85,244)(86,245)(87,246)(88,247)(89,248)(90,249)(91,250)(92,251)(93,252)(94,253)(95,254)(96,255)(97,256)(98,257)(99,258)(100,259)(101,260)(102,261)(103,262)(104,263)(105,264)(106,265)(107,266)(108,267)(109,268)(110,269)(111,270)(112,271)(113,272)(114,273)(115,274)(116,275)(117,276)(118,277)(119,278)(120,279)(121,280)(122,281)(123,282)(124,283)(125,284)(126,285)(127,286)(128,287)(129,288)(130,289)(131,290)(132,291)(133,292)(134,293)(135,294)(136,295)(137,296)(138,297)(139,298)(140,299)(141,300)(142,301)(143,302)(144,303)(145,304)(146,305)(147,306)(148,307)(149,308)(150,309)(151,310)(152,311)(153,312)(154,313)(155,314)(156,315)(157,316)(158,317)(159,318)(160,319)(161,320)(162,321)(163,322)(164,323)(165,324)(166,325)(167,326)(168,327), (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)>;
G:=Group( (1,328)(2,329)(3,330)(4,331)(5,332)(6,333)(7,334)(8,335)(9,336)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,187)(29,188)(30,189)(31,190)(32,191)(33,192)(34,193)(35,194)(36,195)(37,196)(38,197)(39,198)(40,199)(41,200)(42,201)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,211)(53,212)(54,213)(55,214)(56,215)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,225)(67,226)(68,227)(69,228)(70,229)(71,230)(72,231)(73,232)(74,233)(75,234)(76,235)(77,236)(78,237)(79,238)(80,239)(81,240)(82,241)(83,242)(84,243)(85,244)(86,245)(87,246)(88,247)(89,248)(90,249)(91,250)(92,251)(93,252)(94,253)(95,254)(96,255)(97,256)(98,257)(99,258)(100,259)(101,260)(102,261)(103,262)(104,263)(105,264)(106,265)(107,266)(108,267)(109,268)(110,269)(111,270)(112,271)(113,272)(114,273)(115,274)(116,275)(117,276)(118,277)(119,278)(120,279)(121,280)(122,281)(123,282)(124,283)(125,284)(126,285)(127,286)(128,287)(129,288)(130,289)(131,290)(132,291)(133,292)(134,293)(135,294)(136,295)(137,296)(138,297)(139,298)(140,299)(141,300)(142,301)(143,302)(144,303)(145,304)(146,305)(147,306)(148,307)(149,308)(150,309)(151,310)(152,311)(153,312)(154,313)(155,314)(156,315)(157,316)(158,317)(159,318)(160,319)(161,320)(162,321)(163,322)(164,323)(165,324)(166,325)(167,326)(168,327), (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) );
G=PermutationGroup([[(1,328),(2,329),(3,330),(4,331),(5,332),(6,333),(7,334),(8,335),(9,336),(10,169),(11,170),(12,171),(13,172),(14,173),(15,174),(16,175),(17,176),(18,177),(19,178),(20,179),(21,180),(22,181),(23,182),(24,183),(25,184),(26,185),(27,186),(28,187),(29,188),(30,189),(31,190),(32,191),(33,192),(34,193),(35,194),(36,195),(37,196),(38,197),(39,198),(40,199),(41,200),(42,201),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,208),(50,209),(51,210),(52,211),(53,212),(54,213),(55,214),(56,215),(57,216),(58,217),(59,218),(60,219),(61,220),(62,221),(63,222),(64,223),(65,224),(66,225),(67,226),(68,227),(69,228),(70,229),(71,230),(72,231),(73,232),(74,233),(75,234),(76,235),(77,236),(78,237),(79,238),(80,239),(81,240),(82,241),(83,242),(84,243),(85,244),(86,245),(87,246),(88,247),(89,248),(90,249),(91,250),(92,251),(93,252),(94,253),(95,254),(96,255),(97,256),(98,257),(99,258),(100,259),(101,260),(102,261),(103,262),(104,263),(105,264),(106,265),(107,266),(108,267),(109,268),(110,269),(111,270),(112,271),(113,272),(114,273),(115,274),(116,275),(117,276),(118,277),(119,278),(120,279),(121,280),(122,281),(123,282),(124,283),(125,284),(126,285),(127,286),(128,287),(129,288),(130,289),(131,290),(132,291),(133,292),(134,293),(135,294),(136,295),(137,296),(138,297),(139,298),(140,299),(141,300),(142,301),(143,302),(144,303),(145,304),(146,305),(147,306),(148,307),(149,308),(150,309),(151,310),(152,311),(153,312),(154,313),(155,314),(156,315),(157,316),(158,317),(159,318),(160,319),(161,320),(162,321),(163,322),(164,323),(165,324),(166,325),(167,326),(168,327)], [(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)]])
336 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 7A | ··· | 7F | 8A | ··· | 8H | 12A | ··· | 12H | 14A | ··· | 14R | 21A | ··· | 21L | 24A | ··· | 24P | 28A | ··· | 28X | 42A | ··· | 42AJ | 56A | ··· | 56AV | 84A | ··· | 84AV | 168A | ··· | 168CR |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 7 | ··· | 7 | 8 | ··· | 8 | 12 | ··· | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | ··· | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
336 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C7 | C8 | C12 | C12 | C14 | C14 | C21 | C24 | C28 | C28 | C42 | C42 | C56 | C84 | C84 | C168 |
| kernel | C2×C168 | C168 | C2×C84 | C2×C56 | C84 | C2×C42 | C56 | C2×C28 | C2×C24 | C42 | C28 | C2×C14 | C24 | C2×C12 | C2×C8 | C14 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 6 | 8 | 4 | 4 | 12 | 6 | 12 | 16 | 12 | 12 | 24 | 12 | 48 | 24 | 24 | 96 |
Matrix representation of C2×C168 ►in GL2(𝔽337) generated by
| 1 | 0 |
| 0 | 336 |
| 137 | 0 |
| 0 | 169 |
G:=sub<GL(2,GF(337))| [1,0,0,336],[137,0,0,169] >;
C2×C168 in GAP, Magma, Sage, TeX
C_2\times C_{168} % in TeX
G:=Group("C2xC168"); // GroupNames label
G:=SmallGroup(336,109);
// by ID
G=gap.SmallGroup(336,109);
# by ID
G:=PCGroup([6,-2,-2,-3,-7,-2,-2,504,88]);
// Polycyclic
G:=Group<a,b|a^2=b^168=1,a*b=b*a>;
// generators/relations
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