direct product, abelian, monomial, 2-elementary
Aliases: C2×C166, SmallGroup(332,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C166 |
| C1 — C2×C166 |
| C1 — C2×C166 |
Generators and relations for C2×C166
G = < a,b | a2=b166=1, ab=ba >
Smallest permutation representation of C2×C166
►Regular action on 332 points
Generators in S332
(1 300)(2 301)(3 302)(4 303)(5 304)(6 305)(7 306)(8 307)(9 308)(10 309)(11 310)(12 311)(13 312)(14 313)(15 314)(16 315)(17 316)(18 317)(19 318)(20 319)(21 320)(22 321)(23 322)(24 323)(25 324)(26 325)(27 326)(28 327)(29 328)(30 329)(31 330)(32 331)(33 332)(34 167)(35 168)(36 169)(37 170)(38 171)(39 172)(40 173)(41 174)(42 175)(43 176)(44 177)(45 178)(46 179)(47 180)(48 181)(49 182)(50 183)(51 184)(52 185)(53 186)(54 187)(55 188)(56 189)(57 190)(58 191)(59 192)(60 193)(61 194)(62 195)(63 196)(64 197)(65 198)(66 199)(67 200)(68 201)(69 202)(70 203)(71 204)(72 205)(73 206)(74 207)(75 208)(76 209)(77 210)(78 211)(79 212)(80 213)(81 214)(82 215)(83 216)(84 217)(85 218)(86 219)(87 220)(88 221)(89 222)(90 223)(91 224)(92 225)(93 226)(94 227)(95 228)(96 229)(97 230)(98 231)(99 232)(100 233)(101 234)(102 235)(103 236)(104 237)(105 238)(106 239)(107 240)(108 241)(109 242)(110 243)(111 244)(112 245)(113 246)(114 247)(115 248)(116 249)(117 250)(118 251)(119 252)(120 253)(121 254)(122 255)(123 256)(124 257)(125 258)(126 259)(127 260)(128 261)(129 262)(130 263)(131 264)(132 265)(133 266)(134 267)(135 268)(136 269)(137 270)(138 271)(139 272)(140 273)(141 274)(142 275)(143 276)(144 277)(145 278)(146 279)(147 280)(148 281)(149 282)(150 283)(151 284)(152 285)(153 286)(154 287)(155 288)(156 289)(157 290)(158 291)(159 292)(160 293)(161 294)(162 295)(163 296)(164 297)(165 298)(166 299)
(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)
G:=sub<Sym(332)| (1,300)(2,301)(3,302)(4,303)(5,304)(6,305)(7,306)(8,307)(9,308)(10,309)(11,310)(12,311)(13,312)(14,313)(15,314)(16,315)(17,316)(18,317)(19,318)(20,319)(21,320)(22,321)(23,322)(24,323)(25,324)(26,325)(27,326)(28,327)(29,328)(30,329)(31,330)(32,331)(33,332)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,174)(42,175)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,183)(51,184)(52,185)(53,186)(54,187)(55,188)(56,189)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,211)(79,212)(80,213)(81,214)(82,215)(83,216)(84,217)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,225)(93,226)(94,227)(95,228)(96,229)(97,230)(98,231)(99,232)(100,233)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,241)(109,242)(110,243)(111,244)(112,245)(113,246)(114,247)(115,248)(116,249)(117,250)(118,251)(119,252)(120,253)(121,254)(122,255)(123,256)(124,257)(125,258)(126,259)(127,260)(128,261)(129,262)(130,263)(131,264)(132,265)(133,266)(134,267)(135,268)(136,269)(137,270)(138,271)(139,272)(140,273)(141,274)(142,275)(143,276)(144,277)(145,278)(146,279)(147,280)(148,281)(149,282)(150,283)(151,284)(152,285)(153,286)(154,287)(155,288)(156,289)(157,290)(158,291)(159,292)(160,293)(161,294)(162,295)(163,296)(164,297)(165,298)(166,299), (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)>;
G:=Group( (1,300)(2,301)(3,302)(4,303)(5,304)(6,305)(7,306)(8,307)(9,308)(10,309)(11,310)(12,311)(13,312)(14,313)(15,314)(16,315)(17,316)(18,317)(19,318)(20,319)(21,320)(22,321)(23,322)(24,323)(25,324)(26,325)(27,326)(28,327)(29,328)(30,329)(31,330)(32,331)(33,332)(34,167)(35,168)(36,169)(37,170)(38,171)(39,172)(40,173)(41,174)(42,175)(43,176)(44,177)(45,178)(46,179)(47,180)(48,181)(49,182)(50,183)(51,184)(52,185)(53,186)(54,187)(55,188)(56,189)(57,190)(58,191)(59,192)(60,193)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,201)(69,202)(70,203)(71,204)(72,205)(73,206)(74,207)(75,208)(76,209)(77,210)(78,211)(79,212)(80,213)(81,214)(82,215)(83,216)(84,217)(85,218)(86,219)(87,220)(88,221)(89,222)(90,223)(91,224)(92,225)(93,226)(94,227)(95,228)(96,229)(97,230)(98,231)(99,232)(100,233)(101,234)(102,235)(103,236)(104,237)(105,238)(106,239)(107,240)(108,241)(109,242)(110,243)(111,244)(112,245)(113,246)(114,247)(115,248)(116,249)(117,250)(118,251)(119,252)(120,253)(121,254)(122,255)(123,256)(124,257)(125,258)(126,259)(127,260)(128,261)(129,262)(130,263)(131,264)(132,265)(133,266)(134,267)(135,268)(136,269)(137,270)(138,271)(139,272)(140,273)(141,274)(142,275)(143,276)(144,277)(145,278)(146,279)(147,280)(148,281)(149,282)(150,283)(151,284)(152,285)(153,286)(154,287)(155,288)(156,289)(157,290)(158,291)(159,292)(160,293)(161,294)(162,295)(163,296)(164,297)(165,298)(166,299), (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) );
G=PermutationGroup([[(1,300),(2,301),(3,302),(4,303),(5,304),(6,305),(7,306),(8,307),(9,308),(10,309),(11,310),(12,311),(13,312),(14,313),(15,314),(16,315),(17,316),(18,317),(19,318),(20,319),(21,320),(22,321),(23,322),(24,323),(25,324),(26,325),(27,326),(28,327),(29,328),(30,329),(31,330),(32,331),(33,332),(34,167),(35,168),(36,169),(37,170),(38,171),(39,172),(40,173),(41,174),(42,175),(43,176),(44,177),(45,178),(46,179),(47,180),(48,181),(49,182),(50,183),(51,184),(52,185),(53,186),(54,187),(55,188),(56,189),(57,190),(58,191),(59,192),(60,193),(61,194),(62,195),(63,196),(64,197),(65,198),(66,199),(67,200),(68,201),(69,202),(70,203),(71,204),(72,205),(73,206),(74,207),(75,208),(76,209),(77,210),(78,211),(79,212),(80,213),(81,214),(82,215),(83,216),(84,217),(85,218),(86,219),(87,220),(88,221),(89,222),(90,223),(91,224),(92,225),(93,226),(94,227),(95,228),(96,229),(97,230),(98,231),(99,232),(100,233),(101,234),(102,235),(103,236),(104,237),(105,238),(106,239),(107,240),(108,241),(109,242),(110,243),(111,244),(112,245),(113,246),(114,247),(115,248),(116,249),(117,250),(118,251),(119,252),(120,253),(121,254),(122,255),(123,256),(124,257),(125,258),(126,259),(127,260),(128,261),(129,262),(130,263),(131,264),(132,265),(133,266),(134,267),(135,268),(136,269),(137,270),(138,271),(139,272),(140,273),(141,274),(142,275),(143,276),(144,277),(145,278),(146,279),(147,280),(148,281),(149,282),(150,283),(151,284),(152,285),(153,286),(154,287),(155,288),(156,289),(157,290),(158,291),(159,292),(160,293),(161,294),(162,295),(163,296),(164,297),(165,298),(166,299)], [(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)]])
332 conjugacy classes
| class | 1 | 2A | 2B | 2C | 83A | ··· | 83CD | 166A | ··· | 166IL |
| order | 1 | 2 | 2 | 2 | 83 | ··· | 83 | 166 | ··· | 166 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
332 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C83 | C166 |
| kernel | C2×C166 | C166 | C22 | C2 |
| # reps | 1 | 3 | 82 | 246 |
Matrix representation of C2×C166 ►in GL2(𝔽167) generated by
| 166 | 0 |
| 0 | 166 |
| 2 | 0 |
| 0 | 138 |
G:=sub<GL(2,GF(167))| [166,0,0,166],[2,0,0,138] >;
C2×C166 in GAP, Magma, Sage, TeX
C_2\times C_{166} % in TeX
G:=Group("C2xC166"); // GroupNames label
G:=SmallGroup(332,4);
// by ID
G=gap.SmallGroup(332,4);
# by ID
G:=PCGroup([3,-2,-2,-83]);
// Polycyclic
G:=Group<a,b|a^2=b^166=1,a*b=b*a>;
// generators/relations
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