direct product, abelian, monomial, 2-elementary
Aliases: C2×C178, SmallGroup(356,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C178 |
| C1 — C2×C178 |
| C1 — C2×C178 |
Generators and relations for C2×C178
G = < a,b | a2=b178=1, ab=ba >
Smallest permutation representation of C2×C178
►Regular action on 356 points
Generators in S356
(1 292)(2 293)(3 294)(4 295)(5 296)(6 297)(7 298)(8 299)(9 300)(10 301)(11 302)(12 303)(13 304)(14 305)(15 306)(16 307)(17 308)(18 309)(19 310)(20 311)(21 312)(22 313)(23 314)(24 315)(25 316)(26 317)(27 318)(28 319)(29 320)(30 321)(31 322)(32 323)(33 324)(34 325)(35 326)(36 327)(37 328)(38 329)(39 330)(40 331)(41 332)(42 333)(43 334)(44 335)(45 336)(46 337)(47 338)(48 339)(49 340)(50 341)(51 342)(52 343)(53 344)(54 345)(55 346)(56 347)(57 348)(58 349)(59 350)(60 351)(61 352)(62 353)(63 354)(64 355)(65 356)(66 179)(67 180)(68 181)(69 182)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(81 194)(82 195)(83 196)(84 197)(85 198)(86 199)(87 200)(88 201)(89 202)(90 203)(91 204)(92 205)(93 206)(94 207)(95 208)(96 209)(97 210)(98 211)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 219)(107 220)(108 221)(109 222)(110 223)(111 224)(112 225)(113 226)(114 227)(115 228)(116 229)(117 230)(118 231)(119 232)(120 233)(121 234)(122 235)(123 236)(124 237)(125 238)(126 239)(127 240)(128 241)(129 242)(130 243)(131 244)(132 245)(133 246)(134 247)(135 248)(136 249)(137 250)(138 251)(139 252)(140 253)(141 254)(142 255)(143 256)(144 257)(145 258)(146 259)(147 260)(148 261)(149 262)(150 263)(151 264)(152 265)(153 266)(154 267)(155 268)(156 269)(157 270)(158 271)(159 272)(160 273)(161 274)(162 275)(163 276)(164 277)(165 278)(166 279)(167 280)(168 281)(169 282)(170 283)(171 284)(172 285)(173 286)(174 287)(175 288)(176 289)(177 290)(178 291)
(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)(179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356)
G:=sub<Sym(356)| (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,305)(15,306)(16,307)(17,308)(18,309)(19,310)(20,311)(21,312)(22,313)(23,314)(24,315)(25,316)(26,317)(27,318)(28,319)(29,320)(30,321)(31,322)(32,323)(33,324)(34,325)(35,326)(36,327)(37,328)(38,329)(39,330)(40,331)(41,332)(42,333)(43,334)(44,335)(45,336)(46,337)(47,338)(48,339)(49,340)(50,341)(51,342)(52,343)(53,344)(54,345)(55,346)(56,347)(57,348)(58,349)(59,350)(60,351)(61,352)(62,353)(63,354)(64,355)(65,356)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,241)(129,242)(130,243)(131,244)(132,245)(133,246)(134,247)(135,248)(136,249)(137,250)(138,251)(139,252)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281)(169,282)(170,283)(171,284)(172,285)(173,286)(174,287)(175,288)(176,289)(177,290)(178,291), (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)(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356)>;
G:=Group( (1,292)(2,293)(3,294)(4,295)(5,296)(6,297)(7,298)(8,299)(9,300)(10,301)(11,302)(12,303)(13,304)(14,305)(15,306)(16,307)(17,308)(18,309)(19,310)(20,311)(21,312)(22,313)(23,314)(24,315)(25,316)(26,317)(27,318)(28,319)(29,320)(30,321)(31,322)(32,323)(33,324)(34,325)(35,326)(36,327)(37,328)(38,329)(39,330)(40,331)(41,332)(42,333)(43,334)(44,335)(45,336)(46,337)(47,338)(48,339)(49,340)(50,341)(51,342)(52,343)(53,344)(54,345)(55,346)(56,347)(57,348)(58,349)(59,350)(60,351)(61,352)(62,353)(63,354)(64,355)(65,356)(66,179)(67,180)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(81,194)(82,195)(83,196)(84,197)(85,198)(86,199)(87,200)(88,201)(89,202)(90,203)(91,204)(92,205)(93,206)(94,207)(95,208)(96,209)(97,210)(98,211)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,225)(113,226)(114,227)(115,228)(116,229)(117,230)(118,231)(119,232)(120,233)(121,234)(122,235)(123,236)(124,237)(125,238)(126,239)(127,240)(128,241)(129,242)(130,243)(131,244)(132,245)(133,246)(134,247)(135,248)(136,249)(137,250)(138,251)(139,252)(140,253)(141,254)(142,255)(143,256)(144,257)(145,258)(146,259)(147,260)(148,261)(149,262)(150,263)(151,264)(152,265)(153,266)(154,267)(155,268)(156,269)(157,270)(158,271)(159,272)(160,273)(161,274)(162,275)(163,276)(164,277)(165,278)(166,279)(167,280)(168,281)(169,282)(170,283)(171,284)(172,285)(173,286)(174,287)(175,288)(176,289)(177,290)(178,291), (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)(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356) );
G=PermutationGroup([[(1,292),(2,293),(3,294),(4,295),(5,296),(6,297),(7,298),(8,299),(9,300),(10,301),(11,302),(12,303),(13,304),(14,305),(15,306),(16,307),(17,308),(18,309),(19,310),(20,311),(21,312),(22,313),(23,314),(24,315),(25,316),(26,317),(27,318),(28,319),(29,320),(30,321),(31,322),(32,323),(33,324),(34,325),(35,326),(36,327),(37,328),(38,329),(39,330),(40,331),(41,332),(42,333),(43,334),(44,335),(45,336),(46,337),(47,338),(48,339),(49,340),(50,341),(51,342),(52,343),(53,344),(54,345),(55,346),(56,347),(57,348),(58,349),(59,350),(60,351),(61,352),(62,353),(63,354),(64,355),(65,356),(66,179),(67,180),(68,181),(69,182),(70,183),(71,184),(72,185),(73,186),(74,187),(75,188),(76,189),(77,190),(78,191),(79,192),(80,193),(81,194),(82,195),(83,196),(84,197),(85,198),(86,199),(87,200),(88,201),(89,202),(90,203),(91,204),(92,205),(93,206),(94,207),(95,208),(96,209),(97,210),(98,211),(99,212),(100,213),(101,214),(102,215),(103,216),(104,217),(105,218),(106,219),(107,220),(108,221),(109,222),(110,223),(111,224),(112,225),(113,226),(114,227),(115,228),(116,229),(117,230),(118,231),(119,232),(120,233),(121,234),(122,235),(123,236),(124,237),(125,238),(126,239),(127,240),(128,241),(129,242),(130,243),(131,244),(132,245),(133,246),(134,247),(135,248),(136,249),(137,250),(138,251),(139,252),(140,253),(141,254),(142,255),(143,256),(144,257),(145,258),(146,259),(147,260),(148,261),(149,262),(150,263),(151,264),(152,265),(153,266),(154,267),(155,268),(156,269),(157,270),(158,271),(159,272),(160,273),(161,274),(162,275),(163,276),(164,277),(165,278),(166,279),(167,280),(168,281),(169,282),(170,283),(171,284),(172,285),(173,286),(174,287),(175,288),(176,289),(177,290),(178,291)], [(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),(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356)]])
356 conjugacy classes
| class | 1 | 2A | 2B | 2C | 89A | ··· | 89CJ | 178A | ··· | 178JD |
| order | 1 | 2 | 2 | 2 | 89 | ··· | 89 | 178 | ··· | 178 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
356 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C89 | C178 |
| kernel | C2×C178 | C178 | C22 | C2 |
| # reps | 1 | 3 | 88 | 264 |
Matrix representation of C2×C178 ►in GL2(𝔽179) generated by
| 178 | 0 |
| 0 | 1 |
| 78 | 0 |
| 0 | 6 |
G:=sub<GL(2,GF(179))| [178,0,0,1],[78,0,0,6] >;
C2×C178 in GAP, Magma, Sage, TeX
C_2\times C_{178} % in TeX
G:=Group("C2xC178"); // GroupNames label
G:=SmallGroup(356,5);
// by ID
G=gap.SmallGroup(356,5);
# by ID
G:=PCGroup([3,-2,-2,-89]);
// Polycyclic
G:=Group<a,b|a^2=b^178=1,a*b=b*a>;
// generators/relations
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