direct product, abelian, monomial, 2-elementary
Aliases: C2×C158, SmallGroup(316,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C158 |
| C1 — C2×C158 |
| C1 — C2×C158 |
Generators and relations for C2×C158
G = < a,b | a2=b158=1, ab=ba >
Smallest permutation representation of C2×C158
►Regular action on 316 points
Generators in S316
(1 247)(2 248)(3 249)(4 250)(5 251)(6 252)(7 253)(8 254)(9 255)(10 256)(11 257)(12 258)(13 259)(14 260)(15 261)(16 262)(17 263)(18 264)(19 265)(20 266)(21 267)(22 268)(23 269)(24 270)(25 271)(26 272)(27 273)(28 274)(29 275)(30 276)(31 277)(32 278)(33 279)(34 280)(35 281)(36 282)(37 283)(38 284)(39 285)(40 286)(41 287)(42 288)(43 289)(44 290)(45 291)(46 292)(47 293)(48 294)(49 295)(50 296)(51 297)(52 298)(53 299)(54 300)(55 301)(56 302)(57 303)(58 304)(59 305)(60 306)(61 307)(62 308)(63 309)(64 310)(65 311)(66 312)(67 313)(68 314)(69 315)(70 316)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)(89 177)(90 178)(91 179)(92 180)(93 181)(94 182)(95 183)(96 184)(97 185)(98 186)(99 187)(100 188)(101 189)(102 190)(103 191)(104 192)(105 193)(106 194)(107 195)(108 196)(109 197)(110 198)(111 199)(112 200)(113 201)(114 202)(115 203)(116 204)(117 205)(118 206)(119 207)(120 208)(121 209)(122 210)(123 211)(124 212)(125 213)(126 214)(127 215)(128 216)(129 217)(130 218)(131 219)(132 220)(133 221)(134 222)(135 223)(136 224)(137 225)(138 226)(139 227)(140 228)(141 229)(142 230)(143 231)(144 232)(145 233)(146 234)(147 235)(148 236)(149 237)(150 238)(151 239)(152 240)(153 241)(154 242)(155 243)(156 244)(157 245)(158 246)
(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)
G:=sub<Sym(316)| (1,247)(2,248)(3,249)(4,250)(5,251)(6,252)(7,253)(8,254)(9,255)(10,256)(11,257)(12,258)(13,259)(14,260)(15,261)(16,262)(17,263)(18,264)(19,265)(20,266)(21,267)(22,268)(23,269)(24,270)(25,271)(26,272)(27,273)(28,274)(29,275)(30,276)(31,277)(32,278)(33,279)(34,280)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,303)(58,304)(59,305)(60,306)(61,307)(62,308)(63,309)(64,310)(65,311)(66,312)(67,313)(68,314)(69,315)(70,316)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,193)(106,194)(107,195)(108,196)(109,197)(110,198)(111,199)(112,200)(113,201)(114,202)(115,203)(116,204)(117,205)(118,206)(119,207)(120,208)(121,209)(122,210)(123,211)(124,212)(125,213)(126,214)(127,215)(128,216)(129,217)(130,218)(131,219)(132,220)(133,221)(134,222)(135,223)(136,224)(137,225)(138,226)(139,227)(140,228)(141,229)(142,230)(143,231)(144,232)(145,233)(146,234)(147,235)(148,236)(149,237)(150,238)(151,239)(152,240)(153,241)(154,242)(155,243)(156,244)(157,245)(158,246), (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)>;
G:=Group( (1,247)(2,248)(3,249)(4,250)(5,251)(6,252)(7,253)(8,254)(9,255)(10,256)(11,257)(12,258)(13,259)(14,260)(15,261)(16,262)(17,263)(18,264)(19,265)(20,266)(21,267)(22,268)(23,269)(24,270)(25,271)(26,272)(27,273)(28,274)(29,275)(30,276)(31,277)(32,278)(33,279)(34,280)(35,281)(36,282)(37,283)(38,284)(39,285)(40,286)(41,287)(42,288)(43,289)(44,290)(45,291)(46,292)(47,293)(48,294)(49,295)(50,296)(51,297)(52,298)(53,299)(54,300)(55,301)(56,302)(57,303)(58,304)(59,305)(60,306)(61,307)(62,308)(63,309)(64,310)(65,311)(66,312)(67,313)(68,314)(69,315)(70,316)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176)(89,177)(90,178)(91,179)(92,180)(93,181)(94,182)(95,183)(96,184)(97,185)(98,186)(99,187)(100,188)(101,189)(102,190)(103,191)(104,192)(105,193)(106,194)(107,195)(108,196)(109,197)(110,198)(111,199)(112,200)(113,201)(114,202)(115,203)(116,204)(117,205)(118,206)(119,207)(120,208)(121,209)(122,210)(123,211)(124,212)(125,213)(126,214)(127,215)(128,216)(129,217)(130,218)(131,219)(132,220)(133,221)(134,222)(135,223)(136,224)(137,225)(138,226)(139,227)(140,228)(141,229)(142,230)(143,231)(144,232)(145,233)(146,234)(147,235)(148,236)(149,237)(150,238)(151,239)(152,240)(153,241)(154,242)(155,243)(156,244)(157,245)(158,246), (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) );
G=PermutationGroup([[(1,247),(2,248),(3,249),(4,250),(5,251),(6,252),(7,253),(8,254),(9,255),(10,256),(11,257),(12,258),(13,259),(14,260),(15,261),(16,262),(17,263),(18,264),(19,265),(20,266),(21,267),(22,268),(23,269),(24,270),(25,271),(26,272),(27,273),(28,274),(29,275),(30,276),(31,277),(32,278),(33,279),(34,280),(35,281),(36,282),(37,283),(38,284),(39,285),(40,286),(41,287),(42,288),(43,289),(44,290),(45,291),(46,292),(47,293),(48,294),(49,295),(50,296),(51,297),(52,298),(53,299),(54,300),(55,301),(56,302),(57,303),(58,304),(59,305),(60,306),(61,307),(62,308),(63,309),(64,310),(65,311),(66,312),(67,313),(68,314),(69,315),(70,316),(71,159),(72,160),(73,161),(74,162),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176),(89,177),(90,178),(91,179),(92,180),(93,181),(94,182),(95,183),(96,184),(97,185),(98,186),(99,187),(100,188),(101,189),(102,190),(103,191),(104,192),(105,193),(106,194),(107,195),(108,196),(109,197),(110,198),(111,199),(112,200),(113,201),(114,202),(115,203),(116,204),(117,205),(118,206),(119,207),(120,208),(121,209),(122,210),(123,211),(124,212),(125,213),(126,214),(127,215),(128,216),(129,217),(130,218),(131,219),(132,220),(133,221),(134,222),(135,223),(136,224),(137,225),(138,226),(139,227),(140,228),(141,229),(142,230),(143,231),(144,232),(145,233),(146,234),(147,235),(148,236),(149,237),(150,238),(151,239),(152,240),(153,241),(154,242),(155,243),(156,244),(157,245),(158,246)], [(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)]])
316 conjugacy classes
| class | 1 | 2A | 2B | 2C | 79A | ··· | 79BZ | 158A | ··· | 158HZ |
| order | 1 | 2 | 2 | 2 | 79 | ··· | 79 | 158 | ··· | 158 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
316 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C79 | C158 |
| kernel | C2×C158 | C158 | C22 | C2 |
| # reps | 1 | 3 | 78 | 234 |
Matrix representation of C2×C158 ►in GL2(𝔽317) generated by
| 316 | 0 |
| 0 | 316 |
| 205 | 0 |
| 0 | 96 |
G:=sub<GL(2,GF(317))| [316,0,0,316],[205,0,0,96] >;
C2×C158 in GAP, Magma, Sage, TeX
C_2\times C_{158} % in TeX
G:=Group("C2xC158"); // GroupNames label
G:=SmallGroup(316,4);
// by ID
G=gap.SmallGroup(316,4);
# by ID
G:=PCGroup([3,-2,-2,-79]);
// Polycyclic
G:=Group<a,b|a^2=b^158=1,a*b=b*a>;
// generators/relations
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