direct product, abelian, monomial, 2-elementary
Aliases: C2×C164, SmallGroup(328,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C164 |
| C1 — C2×C164 |
| C1 — C2×C164 |
Generators and relations for C2×C164
G = < a,b | a2=b164=1, ab=ba >
Smallest permutation representation of C2×C164
►Regular action on 328 points
Generators in S328
(1 288)(2 289)(3 290)(4 291)(5 292)(6 293)(7 294)(8 295)(9 296)(10 297)(11 298)(12 299)(13 300)(14 301)(15 302)(16 303)(17 304)(18 305)(19 306)(20 307)(21 308)(22 309)(23 310)(24 311)(25 312)(26 313)(27 314)(28 315)(29 316)(30 317)(31 318)(32 319)(33 320)(34 321)(35 322)(36 323)(37 324)(38 325)(39 326)(40 327)(41 328)(42 165)(43 166)(44 167)(45 168)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 180)(58 181)(59 182)(60 183)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 197)(75 198)(76 199)(77 200)(78 201)(79 202)(80 203)(81 204)(82 205)(83 206)(84 207)(85 208)(86 209)(87 210)(88 211)(89 212)(90 213)(91 214)(92 215)(93 216)(94 217)(95 218)(96 219)(97 220)(98 221)(99 222)(100 223)(101 224)(102 225)(103 226)(104 227)(105 228)(106 229)(107 230)(108 231)(109 232)(110 233)(111 234)(112 235)(113 236)(114 237)(115 238)(116 239)(117 240)(118 241)(119 242)(120 243)(121 244)(122 245)(123 246)(124 247)(125 248)(126 249)(127 250)(128 251)(129 252)(130 253)(131 254)(132 255)(133 256)(134 257)(135 258)(136 259)(137 260)(138 261)(139 262)(140 263)(141 264)(142 265)(143 266)(144 267)(145 268)(146 269)(147 270)(148 271)(149 272)(150 273)(151 274)(152 275)(153 276)(154 277)(155 278)(156 279)(157 280)(158 281)(159 282)(160 283)(161 284)(162 285)(163 286)(164 287)
(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)
G:=sub<Sym(328)| (1,288)(2,289)(3,290)(4,291)(5,292)(6,293)(7,294)(8,295)(9,296)(10,297)(11,298)(12,299)(13,300)(14,301)(15,302)(16,303)(17,304)(18,305)(19,306)(20,307)(21,308)(22,309)(23,310)(24,311)(25,312)(26,313)(27,314)(28,315)(29,316)(30,317)(31,318)(32,319)(33,320)(34,321)(35,322)(36,323)(37,324)(38,325)(39,326)(40,327)(41,328)(42,165)(43,166)(44,167)(45,168)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,181)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,221)(99,222)(100,223)(101,224)(102,225)(103,226)(104,227)(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)(120,243)(121,244)(122,245)(123,246)(124,247)(125,248)(126,249)(127,250)(128,251)(129,252)(130,253)(131,254)(132,255)(133,256)(134,257)(135,258)(136,259)(137,260)(138,261)(139,262)(140,263)(141,264)(142,265)(143,266)(144,267)(145,268)(146,269)(147,270)(148,271)(149,272)(150,273)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,280)(158,281)(159,282)(160,283)(161,284)(162,285)(163,286)(164,287), (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)>;
G:=Group( (1,288)(2,289)(3,290)(4,291)(5,292)(6,293)(7,294)(8,295)(9,296)(10,297)(11,298)(12,299)(13,300)(14,301)(15,302)(16,303)(17,304)(18,305)(19,306)(20,307)(21,308)(22,309)(23,310)(24,311)(25,312)(26,313)(27,314)(28,315)(29,316)(30,317)(31,318)(32,319)(33,320)(34,321)(35,322)(36,323)(37,324)(38,325)(39,326)(40,327)(41,328)(42,165)(43,166)(44,167)(45,168)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,180)(58,181)(59,182)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,201)(79,202)(80,203)(81,204)(82,205)(83,206)(84,207)(85,208)(86,209)(87,210)(88,211)(89,212)(90,213)(91,214)(92,215)(93,216)(94,217)(95,218)(96,219)(97,220)(98,221)(99,222)(100,223)(101,224)(102,225)(103,226)(104,227)(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)(120,243)(121,244)(122,245)(123,246)(124,247)(125,248)(126,249)(127,250)(128,251)(129,252)(130,253)(131,254)(132,255)(133,256)(134,257)(135,258)(136,259)(137,260)(138,261)(139,262)(140,263)(141,264)(142,265)(143,266)(144,267)(145,268)(146,269)(147,270)(148,271)(149,272)(150,273)(151,274)(152,275)(153,276)(154,277)(155,278)(156,279)(157,280)(158,281)(159,282)(160,283)(161,284)(162,285)(163,286)(164,287), (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) );
G=PermutationGroup([[(1,288),(2,289),(3,290),(4,291),(5,292),(6,293),(7,294),(8,295),(9,296),(10,297),(11,298),(12,299),(13,300),(14,301),(15,302),(16,303),(17,304),(18,305),(19,306),(20,307),(21,308),(22,309),(23,310),(24,311),(25,312),(26,313),(27,314),(28,315),(29,316),(30,317),(31,318),(32,319),(33,320),(34,321),(35,322),(36,323),(37,324),(38,325),(39,326),(40,327),(41,328),(42,165),(43,166),(44,167),(45,168),(46,169),(47,170),(48,171),(49,172),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,180),(58,181),(59,182),(60,183),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,197),(75,198),(76,199),(77,200),(78,201),(79,202),(80,203),(81,204),(82,205),(83,206),(84,207),(85,208),(86,209),(87,210),(88,211),(89,212),(90,213),(91,214),(92,215),(93,216),(94,217),(95,218),(96,219),(97,220),(98,221),(99,222),(100,223),(101,224),(102,225),(103,226),(104,227),(105,228),(106,229),(107,230),(108,231),(109,232),(110,233),(111,234),(112,235),(113,236),(114,237),(115,238),(116,239),(117,240),(118,241),(119,242),(120,243),(121,244),(122,245),(123,246),(124,247),(125,248),(126,249),(127,250),(128,251),(129,252),(130,253),(131,254),(132,255),(133,256),(134,257),(135,258),(136,259),(137,260),(138,261),(139,262),(140,263),(141,264),(142,265),(143,266),(144,267),(145,268),(146,269),(147,270),(148,271),(149,272),(150,273),(151,274),(152,275),(153,276),(154,277),(155,278),(156,279),(157,280),(158,281),(159,282),(160,283),(161,284),(162,285),(163,286),(164,287)], [(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)]])
328 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 41A | ··· | 41AN | 82A | ··· | 82DP | 164A | ··· | 164FD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 41 | ··· | 41 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
328 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C41 | C82 | C82 | C164 |
| kernel | C2×C164 | C164 | C2×C82 | C82 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 40 | 80 | 40 | 160 |
Matrix representation of C2×C164 ►in GL2(𝔽821) generated by
| 820 | 0 |
| 0 | 1 |
| 386 | 0 |
| 0 | 659 |
G:=sub<GL(2,GF(821))| [820,0,0,1],[386,0,0,659] >;
C2×C164 in GAP, Magma, Sage, TeX
C_2\times C_{164} % in TeX
G:=Group("C2xC164"); // GroupNames label
G:=SmallGroup(328,9);
// by ID
G=gap.SmallGroup(328,9);
# by ID
G:=PCGroup([4,-2,-2,-41,-2,656]);
// Polycyclic
G:=Group<a,b|a^2=b^164=1,a*b=b*a>;
// generators/relations
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