direct product, abelian, monomial, 2-elementary
Aliases: C2×C176, SmallGroup(352,58)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C176 |
| C1 — C2×C176 |
| C1 — C2×C176 |
Generators and relations for C2×C176
G = < a,b | a2=b176=1, ab=ba >
Smallest permutation representation of C2×C176
►Regular action on 352 points
Generators in S352
(1 308)(2 309)(3 310)(4 311)(5 312)(6 313)(7 314)(8 315)(9 316)(10 317)(11 318)(12 319)(13 320)(14 321)(15 322)(16 323)(17 324)(18 325)(19 326)(20 327)(21 328)(22 329)(23 330)(24 331)(25 332)(26 333)(27 334)(28 335)(29 336)(30 337)(31 338)(32 339)(33 340)(34 341)(35 342)(36 343)(37 344)(38 345)(39 346)(40 347)(41 348)(42 349)(43 350)(44 351)(45 352)(46 177)(47 178)(48 179)(49 180)(50 181)(51 182)(52 183)(53 184)(54 185)(55 186)(56 187)(57 188)(58 189)(59 190)(60 191)(61 192)(62 193)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 201)(71 202)(72 203)(73 204)(74 205)(75 206)(76 207)(77 208)(78 209)(79 210)(80 211)(81 212)(82 213)(83 214)(84 215)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 225)(95 226)(96 227)(97 228)(98 229)(99 230)(100 231)(101 232)(102 233)(103 234)(104 235)(105 236)(106 237)(107 238)(108 239)(109 240)(110 241)(111 242)(112 243)(113 244)(114 245)(115 246)(116 247)(117 248)(118 249)(119 250)(120 251)(121 252)(122 253)(123 254)(124 255)(125 256)(126 257)(127 258)(128 259)(129 260)(130 261)(131 262)(132 263)(133 264)(134 265)(135 266)(136 267)(137 268)(138 269)(139 270)(140 271)(141 272)(142 273)(143 274)(144 275)(145 276)(146 277)(147 278)(148 279)(149 280)(150 281)(151 282)(152 283)(153 284)(154 285)(155 286)(156 287)(157 288)(158 289)(159 290)(160 291)(161 292)(162 293)(163 294)(164 295)(165 296)(166 297)(167 298)(168 299)(169 300)(170 301)(171 302)(172 303)(173 304)(174 305)(175 306)(176 307)
(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)
G:=sub<Sym(352)| (1,308)(2,309)(3,310)(4,311)(5,312)(6,313)(7,314)(8,315)(9,316)(10,317)(11,318)(12,319)(13,320)(14,321)(15,322)(16,323)(17,324)(18,325)(19,326)(20,327)(21,328)(22,329)(23,330)(24,331)(25,332)(26,333)(27,334)(28,335)(29,336)(30,337)(31,338)(32,339)(33,340)(34,341)(35,342)(36,343)(37,344)(38,345)(39,346)(40,347)(41,348)(42,349)(43,350)(44,351)(45,352)(46,177)(47,178)(48,179)(49,180)(50,181)(51,182)(52,183)(53,184)(54,185)(55,186)(56,187)(57,188)(58,189)(59,190)(60,191)(61,192)(62,193)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(101,232)(102,233)(103,234)(104,235)(105,236)(106,237)(107,238)(108,239)(109,240)(110,241)(111,242)(112,243)(113,244)(114,245)(115,246)(116,247)(117,248)(118,249)(119,250)(120,251)(121,252)(122,253)(123,254)(124,255)(125,256)(126,257)(127,258)(128,259)(129,260)(130,261)(131,262)(132,263)(133,264)(134,265)(135,266)(136,267)(137,268)(138,269)(139,270)(140,271)(141,272)(142,273)(143,274)(144,275)(145,276)(146,277)(147,278)(148,279)(149,280)(150,281)(151,282)(152,283)(153,284)(154,285)(155,286)(156,287)(157,288)(158,289)(159,290)(160,291)(161,292)(162,293)(163,294)(164,295)(165,296)(166,297)(167,298)(168,299)(169,300)(170,301)(171,302)(172,303)(173,304)(174,305)(175,306)(176,307), (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)>;
G:=Group( (1,308)(2,309)(3,310)(4,311)(5,312)(6,313)(7,314)(8,315)(9,316)(10,317)(11,318)(12,319)(13,320)(14,321)(15,322)(16,323)(17,324)(18,325)(19,326)(20,327)(21,328)(22,329)(23,330)(24,331)(25,332)(26,333)(27,334)(28,335)(29,336)(30,337)(31,338)(32,339)(33,340)(34,341)(35,342)(36,343)(37,344)(38,345)(39,346)(40,347)(41,348)(42,349)(43,350)(44,351)(45,352)(46,177)(47,178)(48,179)(49,180)(50,181)(51,182)(52,183)(53,184)(54,185)(55,186)(56,187)(57,188)(58,189)(59,190)(60,191)(61,192)(62,193)(63,194)(64,195)(65,196)(66,197)(67,198)(68,199)(69,200)(70,201)(71,202)(72,203)(73,204)(74,205)(75,206)(76,207)(77,208)(78,209)(79,210)(80,211)(81,212)(82,213)(83,214)(84,215)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,225)(95,226)(96,227)(97,228)(98,229)(99,230)(100,231)(101,232)(102,233)(103,234)(104,235)(105,236)(106,237)(107,238)(108,239)(109,240)(110,241)(111,242)(112,243)(113,244)(114,245)(115,246)(116,247)(117,248)(118,249)(119,250)(120,251)(121,252)(122,253)(123,254)(124,255)(125,256)(126,257)(127,258)(128,259)(129,260)(130,261)(131,262)(132,263)(133,264)(134,265)(135,266)(136,267)(137,268)(138,269)(139,270)(140,271)(141,272)(142,273)(143,274)(144,275)(145,276)(146,277)(147,278)(148,279)(149,280)(150,281)(151,282)(152,283)(153,284)(154,285)(155,286)(156,287)(157,288)(158,289)(159,290)(160,291)(161,292)(162,293)(163,294)(164,295)(165,296)(166,297)(167,298)(168,299)(169,300)(170,301)(171,302)(172,303)(173,304)(174,305)(175,306)(176,307), (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) );
G=PermutationGroup([[(1,308),(2,309),(3,310),(4,311),(5,312),(6,313),(7,314),(8,315),(9,316),(10,317),(11,318),(12,319),(13,320),(14,321),(15,322),(16,323),(17,324),(18,325),(19,326),(20,327),(21,328),(22,329),(23,330),(24,331),(25,332),(26,333),(27,334),(28,335),(29,336),(30,337),(31,338),(32,339),(33,340),(34,341),(35,342),(36,343),(37,344),(38,345),(39,346),(40,347),(41,348),(42,349),(43,350),(44,351),(45,352),(46,177),(47,178),(48,179),(49,180),(50,181),(51,182),(52,183),(53,184),(54,185),(55,186),(56,187),(57,188),(58,189),(59,190),(60,191),(61,192),(62,193),(63,194),(64,195),(65,196),(66,197),(67,198),(68,199),(69,200),(70,201),(71,202),(72,203),(73,204),(74,205),(75,206),(76,207),(77,208),(78,209),(79,210),(80,211),(81,212),(82,213),(83,214),(84,215),(85,216),(86,217),(87,218),(88,219),(89,220),(90,221),(91,222),(92,223),(93,224),(94,225),(95,226),(96,227),(97,228),(98,229),(99,230),(100,231),(101,232),(102,233),(103,234),(104,235),(105,236),(106,237),(107,238),(108,239),(109,240),(110,241),(111,242),(112,243),(113,244),(114,245),(115,246),(116,247),(117,248),(118,249),(119,250),(120,251),(121,252),(122,253),(123,254),(124,255),(125,256),(126,257),(127,258),(128,259),(129,260),(130,261),(131,262),(132,263),(133,264),(134,265),(135,266),(136,267),(137,268),(138,269),(139,270),(140,271),(141,272),(142,273),(143,274),(144,275),(145,276),(146,277),(147,278),(148,279),(149,280),(150,281),(151,282),(152,283),(153,284),(154,285),(155,286),(156,287),(157,288),(158,289),(159,290),(160,291),(161,292),(162,293),(163,294),(164,295),(165,296),(166,297),(167,298),(168,299),(169,300),(170,301),(171,302),(172,303),(173,304),(174,305),(175,306),(176,307)], [(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)]])
352 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 11A | ··· | 11J | 16A | ··· | 16P | 22A | ··· | 22AD | 44A | ··· | 44AN | 88A | ··· | 88CB | 176A | ··· | 176FD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 16 | ··· | 16 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 | 176 | ··· | 176 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
352 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C11 | C16 | C22 | C22 | C44 | C44 | C88 | C88 | C176 |
| kernel | C2×C176 | C176 | C2×C88 | C88 | C2×C44 | C44 | C2×C22 | C2×C16 | C22 | C16 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 10 | 16 | 20 | 10 | 20 | 20 | 40 | 40 | 160 |
Matrix representation of C2×C176 ►in GL2(𝔽353) generated by
| 352 | 0 |
| 0 | 352 |
| 270 | 0 |
| 0 | 92 |
G:=sub<GL(2,GF(353))| [352,0,0,352],[270,0,0,92] >;
C2×C176 in GAP, Magma, Sage, TeX
C_2\times C_{176} % in TeX
G:=Group("C2xC176"); // GroupNames label
G:=SmallGroup(352,58);
// by ID
G=gap.SmallGroup(352,58);
# by ID
G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,69,88]);
// Polycyclic
G:=Group<a,b|a^2=b^176=1,a*b=b*a>;
// generators/relations
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