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G = C2×C172order 344 = 23·43

Abelian group of type [2,172]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C172, SmallGroup(344,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C172
C1C2C86C172 — 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
(1 215)(2 216)(3 217)(4 218)(5 219)(6 220)(7 221)(8 222)(9 223)(10 224)(11 225)(12 226)(13 227)(14 228)(15 229)(16 230)(17 231)(18 232)(19 233)(20 234)(21 235)(22 236)(23 237)(24 238)(25 239)(26 240)(27 241)(28 242)(29 243)(30 244)(31 245)(32 246)(33 247)(34 248)(35 249)(36 250)(37 251)(38 252)(39 253)(40 254)(41 255)(42 256)(43 257)(44 258)(45 259)(46 260)(47 261)(48 262)(49 263)(50 264)(51 265)(52 266)(53 267)(54 268)(55 269)(56 270)(57 271)(58 272)(59 273)(60 274)(61 275)(62 276)(63 277)(64 278)(65 279)(66 280)(67 281)(68 282)(69 283)(70 284)(71 285)(72 286)(73 287)(74 288)(75 289)(76 290)(77 291)(78 292)(79 293)(80 294)(81 295)(82 296)(83 297)(84 298)(85 299)(86 300)(87 301)(88 302)(89 303)(90 304)(91 305)(92 306)(93 307)(94 308)(95 309)(96 310)(97 311)(98 312)(99 313)(100 314)(101 315)(102 316)(103 317)(104 318)(105 319)(106 320)(107 321)(108 322)(109 323)(110 324)(111 325)(112 326)(113 327)(114 328)(115 329)(116 330)(117 331)(118 332)(119 333)(120 334)(121 335)(122 336)(123 337)(124 338)(125 339)(126 340)(127 341)(128 342)(129 343)(130 344)(131 173)(132 174)(133 175)(134 176)(135 177)(136 178)(137 179)(138 180)(139 181)(140 182)(141 183)(142 184)(143 185)(144 186)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 193)(152 194)(153 195)(154 196)(155 197)(156 198)(157 199)(158 200)(159 201)(160 202)(161 203)(162 204)(163 205)(164 206)(165 207)(166 208)(167 209)(168 210)(169 211)(170 212)(171 213)(172 214)
(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)| (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,221)(8,222)(9,223)(10,224)(11,225)(12,226)(13,227)(14,228)(15,229)(16,230)(17,231)(18,232)(19,233)(20,234)(21,235)(22,236)(23,237)(24,238)(25,239)(26,240)(27,241)(28,242)(29,243)(30,244)(31,245)(32,246)(33,247)(34,248)(35,249)(36,250)(37,251)(38,252)(39,253)(40,254)(41,255)(42,256)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,263)(50,264)(51,265)(52,266)(53,267)(54,268)(55,269)(56,270)(57,271)(58,272)(59,273)(60,274)(61,275)(62,276)(63,277)(64,278)(65,279)(66,280)(67,281)(68,282)(69,283)(70,284)(71,285)(72,286)(73,287)(74,288)(75,289)(76,290)(77,291)(78,292)(79,293)(80,294)(81,295)(82,296)(83,297)(84,298)(85,299)(86,300)(87,301)(88,302)(89,303)(90,304)(91,305)(92,306)(93,307)(94,308)(95,309)(96,310)(97,311)(98,312)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,321)(108,322)(109,323)(110,324)(111,325)(112,326)(113,327)(114,328)(115,329)(116,330)(117,331)(118,332)(119,333)(120,334)(121,335)(122,336)(123,337)(124,338)(125,339)(126,340)(127,341)(128,342)(129,343)(130,344)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,193)(152,194)(153,195)(154,196)(155,197)(156,198)(157,199)(158,200)(159,201)(160,202)(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210)(169,211)(170,212)(171,213)(172,214), (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( (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,221)(8,222)(9,223)(10,224)(11,225)(12,226)(13,227)(14,228)(15,229)(16,230)(17,231)(18,232)(19,233)(20,234)(21,235)(22,236)(23,237)(24,238)(25,239)(26,240)(27,241)(28,242)(29,243)(30,244)(31,245)(32,246)(33,247)(34,248)(35,249)(36,250)(37,251)(38,252)(39,253)(40,254)(41,255)(42,256)(43,257)(44,258)(45,259)(46,260)(47,261)(48,262)(49,263)(50,264)(51,265)(52,266)(53,267)(54,268)(55,269)(56,270)(57,271)(58,272)(59,273)(60,274)(61,275)(62,276)(63,277)(64,278)(65,279)(66,280)(67,281)(68,282)(69,283)(70,284)(71,285)(72,286)(73,287)(74,288)(75,289)(76,290)(77,291)(78,292)(79,293)(80,294)(81,295)(82,296)(83,297)(84,298)(85,299)(86,300)(87,301)(88,302)(89,303)(90,304)(91,305)(92,306)(93,307)(94,308)(95,309)(96,310)(97,311)(98,312)(99,313)(100,314)(101,315)(102,316)(103,317)(104,318)(105,319)(106,320)(107,321)(108,322)(109,323)(110,324)(111,325)(112,326)(113,327)(114,328)(115,329)(116,330)(117,331)(118,332)(119,333)(120,334)(121,335)(122,336)(123,337)(124,338)(125,339)(126,340)(127,341)(128,342)(129,343)(130,344)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(141,183)(142,184)(143,185)(144,186)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,193)(152,194)(153,195)(154,196)(155,197)(156,198)(157,199)(158,200)(159,201)(160,202)(161,203)(162,204)(163,205)(164,206)(165,207)(166,208)(167,209)(168,210)(169,211)(170,212)(171,213)(172,214), (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([(1,215),(2,216),(3,217),(4,218),(5,219),(6,220),(7,221),(8,222),(9,223),(10,224),(11,225),(12,226),(13,227),(14,228),(15,229),(16,230),(17,231),(18,232),(19,233),(20,234),(21,235),(22,236),(23,237),(24,238),(25,239),(26,240),(27,241),(28,242),(29,243),(30,244),(31,245),(32,246),(33,247),(34,248),(35,249),(36,250),(37,251),(38,252),(39,253),(40,254),(41,255),(42,256),(43,257),(44,258),(45,259),(46,260),(47,261),(48,262),(49,263),(50,264),(51,265),(52,266),(53,267),(54,268),(55,269),(56,270),(57,271),(58,272),(59,273),(60,274),(61,275),(62,276),(63,277),(64,278),(65,279),(66,280),(67,281),(68,282),(69,283),(70,284),(71,285),(72,286),(73,287),(74,288),(75,289),(76,290),(77,291),(78,292),(79,293),(80,294),(81,295),(82,296),(83,297),(84,298),(85,299),(86,300),(87,301),(88,302),(89,303),(90,304),(91,305),(92,306),(93,307),(94,308),(95,309),(96,310),(97,311),(98,312),(99,313),(100,314),(101,315),(102,316),(103,317),(104,318),(105,319),(106,320),(107,321),(108,322),(109,323),(110,324),(111,325),(112,326),(113,327),(114,328),(115,329),(116,330),(117,331),(118,332),(119,333),(120,334),(121,335),(122,336),(123,337),(124,338),(125,339),(126,340),(127,341),(128,342),(129,343),(130,344),(131,173),(132,174),(133,175),(134,176),(135,177),(136,178),(137,179),(138,180),(139,181),(140,182),(141,183),(142,184),(143,185),(144,186),(145,187),(146,188),(147,189),(148,190),(149,191),(150,192),(151,193),(152,194),(153,195),(154,196),(155,197),(156,198),(157,199),(158,200),(159,201),(160,202),(161,203),(162,204),(163,205),(164,206),(165,207),(166,208),(167,209),(168,210),(169,211),(170,212),(171,213),(172,214)], [(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 2A2B2C4A4B4C4D43A···43AP86A···86DV172A···172FL
order1222444443···4386···86172···172
size111111111···11···11···1

344 irreducible representations

dim11111111
type+++
imageC1C2C2C4C43C86C86C172
kernelC2×C172C172C2×C86C86C2×C4C4C22C2
# reps1214428442168

Matrix representation of C2×C172 in GL2(𝔽173) generated by

1720
0172
,
1720
018
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

Export

Subgroup lattice of C2×C172 in TeX

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