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G = C2×C166order 332 = 22·83

Abelian group of type [2,166]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C166, SmallGroup(332,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C166
C1C83C166 — C2×C166
C1 — C2×C166
C1 — C2×C166

Generators and relations for C2×C166
 G = < a,b | a2=b166=1, ab=ba >


Smallest permutation representation of C2×C166
Regular action on 332 points
Generators in S332
(1 314)(2 315)(3 316)(4 317)(5 318)(6 319)(7 320)(8 321)(9 322)(10 323)(11 324)(12 325)(13 326)(14 327)(15 328)(16 329)(17 330)(18 331)(19 332)(20 167)(21 168)(22 169)(23 170)(24 171)(25 172)(26 173)(27 174)(28 175)(29 176)(30 177)(31 178)(32 179)(33 180)(34 181)(35 182)(36 183)(37 184)(38 185)(39 186)(40 187)(41 188)(42 189)(43 190)(44 191)(45 192)(46 193)(47 194)(48 195)(49 196)(50 197)(51 198)(52 199)(53 200)(54 201)(55 202)(56 203)(57 204)(58 205)(59 206)(60 207)(61 208)(62 209)(63 210)(64 211)(65 212)(66 213)(67 214)(68 215)(69 216)(70 217)(71 218)(72 219)(73 220)(74 221)(75 222)(76 223)(77 224)(78 225)(79 226)(80 227)(81 228)(82 229)(83 230)(84 231)(85 232)(86 233)(87 234)(88 235)(89 236)(90 237)(91 238)(92 239)(93 240)(94 241)(95 242)(96 243)(97 244)(98 245)(99 246)(100 247)(101 248)(102 249)(103 250)(104 251)(105 252)(106 253)(107 254)(108 255)(109 256)(110 257)(111 258)(112 259)(113 260)(114 261)(115 262)(116 263)(117 264)(118 265)(119 266)(120 267)(121 268)(122 269)(123 270)(124 271)(125 272)(126 273)(127 274)(128 275)(129 276)(130 277)(131 278)(132 279)(133 280)(134 281)(135 282)(136 283)(137 284)(138 285)(139 286)(140 287)(141 288)(142 289)(143 290)(144 291)(145 292)(146 293)(147 294)(148 295)(149 296)(150 297)(151 298)(152 299)(153 300)(154 301)(155 302)(156 303)(157 304)(158 305)(159 306)(160 307)(161 308)(162 309)(163 310)(164 311)(165 312)(166 313)
(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)

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

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

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

332 conjugacy classes

class 1 2A2B2C83A···83CD166A···166IL
order122283···83166···166
size11111···11···1

332 irreducible representations

dim1111
type++
imageC1C2C83C166
kernelC2×C166C166C22C2
# reps1382246

Matrix representation of C2×C166 in GL2(𝔽167) generated by

1660
0166
,
20
0138
G:=sub<GL(2,GF(167))| [166,0,0,166],[2,0,0,138] >;

C2×C166 in GAP, Magma, Sage, TeX

C_2\times C_{166}
% in TeX

G:=Group("C2xC166");
// GroupNames label

G:=SmallGroup(332,4);
// by ID

G=gap.SmallGroup(332,4);
# by ID

G:=PCGroup([3,-2,-2,-83]);
// Polycyclic

G:=Group<a,b|a^2=b^166=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C166 in TeX

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