direct product, abelian, monomial, 2-elementary
Aliases: C2×C172, SmallGroup(344,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C172 |
| C1 — C2×C172 |
| C1 — C2×C172 |
Generators and relations for C2×C172
G = < a,b | a2=b172=1, ab=ba >
Smallest permutation representation of C2×C172
►Regular action on 344 points
Generators in S344
(1 299)(2 300)(3 301)(4 302)(5 303)(6 304)(7 305)(8 306)(9 307)(10 308)(11 309)(12 310)(13 311)(14 312)(15 313)(16 314)(17 315)(18 316)(19 317)(20 318)(21 319)(22 320)(23 321)(24 322)(25 323)(26 324)(27 325)(28 326)(29 327)(30 328)(31 329)(32 330)(33 331)(34 332)(35 333)(36 334)(37 335)(38 336)(39 337)(40 338)(41 339)(42 340)(43 341)(44 342)(45 343)(46 344)(47 173)(48 174)(49 175)(50 176)(51 177)(52 178)(53 179)(54 180)(55 181)(56 182)(57 183)(58 184)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(65 191)(66 192)(67 193)(68 194)(69 195)(70 196)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 211)(86 212)(87 213)(88 214)(89 215)(90 216)(91 217)(92 218)(93 219)(94 220)(95 221)(96 222)(97 223)(98 224)(99 225)(100 226)(101 227)(102 228)(103 229)(104 230)(105 231)(106 232)(107 233)(108 234)(109 235)(110 236)(111 237)(112 238)(113 239)(114 240)(115 241)(116 242)(117 243)(118 244)(119 245)(120 246)(121 247)(122 248)(123 249)(124 250)(125 251)(126 252)(127 253)(128 254)(129 255)(130 256)(131 257)(132 258)(133 259)(134 260)(135 261)(136 262)(137 263)(138 264)(139 265)(140 266)(141 267)(142 268)(143 269)(144 270)(145 271)(146 272)(147 273)(148 274)(149 275)(150 276)(151 277)(152 278)(153 279)(154 280)(155 281)(156 282)(157 283)(158 284)(159 285)(160 286)(161 287)(162 288)(163 289)(164 290)(165 291)(166 292)(167 293)(168 294)(169 295)(170 296)(171 297)(172 298)
(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)
G:=sub<Sym(344)| (1,299)(2,300)(3,301)(4,302)(5,303)(6,304)(7,305)(8,306)(9,307)(10,308)(11,309)(12,310)(13,311)(14,312)(15,313)(16,314)(17,315)(18,316)(19,317)(20,318)(21,319)(22,320)(23,321)(24,322)(25,323)(26,324)(27,325)(28,326)(29,327)(30,328)(31,329)(32,330)(33,331)(34,332)(35,333)(36,334)(37,335)(38,336)(39,337)(40,338)(41,339)(42,340)(43,341)(44,342)(45,343)(46,344)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,196)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)(136,262)(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)(144,270)(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)(152,278)(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)(169,295)(170,296)(171,297)(172,298), (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)>;
G:=Group( (1,299)(2,300)(3,301)(4,302)(5,303)(6,304)(7,305)(8,306)(9,307)(10,308)(11,309)(12,310)(13,311)(14,312)(15,313)(16,314)(17,315)(18,316)(19,317)(20,318)(21,319)(22,320)(23,321)(24,322)(25,323)(26,324)(27,325)(28,326)(29,327)(30,328)(31,329)(32,330)(33,331)(34,332)(35,333)(36,334)(37,335)(38,336)(39,337)(40,338)(41,339)(42,340)(43,341)(44,342)(45,343)(46,344)(47,173)(48,174)(49,175)(50,176)(51,177)(52,178)(53,179)(54,180)(55,181)(56,182)(57,183)(58,184)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(65,191)(66,192)(67,193)(68,194)(69,195)(70,196)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,211)(86,212)(87,213)(88,214)(89,215)(90,216)(91,217)(92,218)(93,219)(94,220)(95,221)(96,222)(97,223)(98,224)(99,225)(100,226)(101,227)(102,228)(103,229)(104,230)(105,231)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)(136,262)(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)(144,270)(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)(152,278)(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)(169,295)(170,296)(171,297)(172,298), (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) );
G=PermutationGroup([[(1,299),(2,300),(3,301),(4,302),(5,303),(6,304),(7,305),(8,306),(9,307),(10,308),(11,309),(12,310),(13,311),(14,312),(15,313),(16,314),(17,315),(18,316),(19,317),(20,318),(21,319),(22,320),(23,321),(24,322),(25,323),(26,324),(27,325),(28,326),(29,327),(30,328),(31,329),(32,330),(33,331),(34,332),(35,333),(36,334),(37,335),(38,336),(39,337),(40,338),(41,339),(42,340),(43,341),(44,342),(45,343),(46,344),(47,173),(48,174),(49,175),(50,176),(51,177),(52,178),(53,179),(54,180),(55,181),(56,182),(57,183),(58,184),(59,185),(60,186),(61,187),(62,188),(63,189),(64,190),(65,191),(66,192),(67,193),(68,194),(69,195),(70,196),(71,197),(72,198),(73,199),(74,200),(75,201),(76,202),(77,203),(78,204),(79,205),(80,206),(81,207),(82,208),(83,209),(84,210),(85,211),(86,212),(87,213),(88,214),(89,215),(90,216),(91,217),(92,218),(93,219),(94,220),(95,221),(96,222),(97,223),(98,224),(99,225),(100,226),(101,227),(102,228),(103,229),(104,230),(105,231),(106,232),(107,233),(108,234),(109,235),(110,236),(111,237),(112,238),(113,239),(114,240),(115,241),(116,242),(117,243),(118,244),(119,245),(120,246),(121,247),(122,248),(123,249),(124,250),(125,251),(126,252),(127,253),(128,254),(129,255),(130,256),(131,257),(132,258),(133,259),(134,260),(135,261),(136,262),(137,263),(138,264),(139,265),(140,266),(141,267),(142,268),(143,269),(144,270),(145,271),(146,272),(147,273),(148,274),(149,275),(150,276),(151,277),(152,278),(153,279),(154,280),(155,281),(156,282),(157,283),(158,284),(159,285),(160,286),(161,287),(162,288),(163,289),(164,290),(165,291),(166,292),(167,293),(168,294),(169,295),(170,296),(171,297),(172,298)], [(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)]])
344 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 43A | ··· | 43AP | 86A | ··· | 86DV | 172A | ··· | 172FL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
344 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C43 | C86 | C86 | C172 |
| kernel | C2×C172 | C172 | C2×C86 | C86 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 42 | 84 | 42 | 168 |
Matrix representation of C2×C172 ►in GL2(𝔽173) generated by
| 172 | 0 |
| 0 | 172 |
| 172 | 0 |
| 0 | 18 |
G:=sub<GL(2,GF(173))| [172,0,0,172],[172,0,0,18] >;
C2×C172 in GAP, Magma, Sage, TeX
C_2\times C_{172} % in TeX
G:=Group("C2xC172"); // GroupNames label
G:=SmallGroup(344,8);
// by ID
G=gap.SmallGroup(344,8);
# by ID
G:=PCGroup([4,-2,-2,-43,-2,688]);
// Polycyclic
G:=Group<a,b|a^2=b^172=1,a*b=b*a>;
// generators/relations
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