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G = C2×C162order 324 = 22·34

Abelian group of type [2,162]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C162, SmallGroup(324,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C162
C1C3C9C27C81C162 — C2×C162
C1 — C2×C162
C1 — C2×C162

Generators and relations for C2×C162
 G = < a,b | a2=b162=1, ab=ba >


Smallest permutation representation of C2×C162
Regular action on 324 points
Generators in S324
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 196)(24 197)(25 198)(26 199)(27 200)(28 201)(29 202)(30 203)(31 204)(32 205)(33 206)(34 207)(35 208)(36 209)(37 210)(38 211)(39 212)(40 213)(41 214)(42 215)(43 216)(44 217)(45 218)(46 219)(47 220)(48 221)(49 222)(50 223)(51 224)(52 225)(53 226)(54 227)(55 228)(56 229)(57 230)(58 231)(59 232)(60 233)(61 234)(62 235)(63 236)(64 237)(65 238)(66 239)(67 240)(68 241)(69 242)(70 243)(71 244)(72 245)(73 246)(74 247)(75 248)(76 249)(77 250)(78 251)(79 252)(80 253)(81 254)(82 255)(83 256)(84 257)(85 258)(86 259)(87 260)(88 261)(89 262)(90 263)(91 264)(92 265)(93 266)(94 267)(95 268)(96 269)(97 270)(98 271)(99 272)(100 273)(101 274)(102 275)(103 276)(104 277)(105 278)(106 279)(107 280)(108 281)(109 282)(110 283)(111 284)(112 285)(113 286)(114 287)(115 288)(116 289)(117 290)(118 291)(119 292)(120 293)(121 294)(122 295)(123 296)(124 297)(125 298)(126 299)(127 300)(128 301)(129 302)(130 303)(131 304)(132 305)(133 306)(134 307)(135 308)(136 309)(137 310)(138 311)(139 312)(140 313)(141 314)(142 315)(143 316)(144 317)(145 318)(146 319)(147 320)(148 321)(149 322)(150 323)(151 324)(152 163)(153 164)(154 165)(155 166)(156 167)(157 168)(158 169)(159 170)(160 171)(161 172)(162 173)
(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)

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

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

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

324 conjugacy classes

class 1 2A2B2C3A3B6A···6F9A···9F18A···18R27A···27R54A···54BB81A···81BB162A···162FF
order1222336···69···918···1827···2754···5481···81162···162
size1111111···11···11···11···11···11···11···1

324 irreducible representations

dim1111111111
type++
imageC1C2C3C6C9C18C27C54C81C162
kernelC2×C162C162C2×C54C54C2×C18C18C2×C6C6C22C2
# reps1326618185454162

Matrix representation of C2×C162 in GL2(𝔽163) generated by

1620
01
,
1200
092
G:=sub<GL(2,GF(163))| [162,0,0,1],[120,0,0,92] >;

C2×C162 in GAP, Magma, Sage, TeX

C_2\times C_{162}
% in TeX

G:=Group("C2xC162");
// GroupNames label

G:=SmallGroup(324,5);
// by ID

G=gap.SmallGroup(324,5);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,93,118]);
// Polycyclic

G:=Group<a,b|a^2=b^162=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C162 in TeX

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