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G = C2×C160order 320 = 26·5

Abelian group of type [2,160]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C160, SmallGroup(320,174)

Series: Derived Chief Lower central Upper central

C1 — C2×C160
C1C2C4C8C16C80C160 — C2×C160
C1 — C2×C160
C1 — C2×C160

Generators and relations for C2×C160
 G = < a,b | a2=b160=1, ab=ba >


Smallest permutation representation of C2×C160
Regular action on 320 points
Generators in S320
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 221)(9 222)(10 223)(11 224)(12 225)(13 226)(14 227)(15 228)(16 229)(17 230)(18 231)(19 232)(20 233)(21 234)(22 235)(23 236)(24 237)(25 238)(26 239)(27 240)(28 241)(29 242)(30 243)(31 244)(32 245)(33 246)(34 247)(35 248)(36 249)(37 250)(38 251)(39 252)(40 253)(41 254)(42 255)(43 256)(44 257)(45 258)(46 259)(47 260)(48 261)(49 262)(50 263)(51 264)(52 265)(53 266)(54 267)(55 268)(56 269)(57 270)(58 271)(59 272)(60 273)(61 274)(62 275)(63 276)(64 277)(65 278)(66 279)(67 280)(68 281)(69 282)(70 283)(71 284)(72 285)(73 286)(74 287)(75 288)(76 289)(77 290)(78 291)(79 292)(80 293)(81 294)(82 295)(83 296)(84 297)(85 298)(86 299)(87 300)(88 301)(89 302)(90 303)(91 304)(92 305)(93 306)(94 307)(95 308)(96 309)(97 310)(98 311)(99 312)(100 313)(101 314)(102 315)(103 316)(104 317)(105 318)(106 319)(107 320)(108 161)(109 162)(110 163)(111 164)(112 165)(113 166)(114 167)(115 168)(116 169)(117 170)(118 171)(119 172)(120 173)(121 174)(122 175)(123 176)(124 177)(125 178)(126 179)(127 180)(128 181)(129 182)(130 183)(131 184)(132 185)(133 186)(134 187)(135 188)(136 189)(137 190)(138 191)(139 192)(140 193)(141 194)(142 195)(143 196)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)(155 208)(156 209)(157 210)(158 211)(159 212)(160 213)
(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)

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

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

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

320 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D8A···8H10A···10L16A···16P20A···20P32A···32AF40A···40AF80A···80BL160A···160DX
order1222444455558···810···1016···1620···2032···3240···4080···80160···160
size1111111111111···11···11···11···11···11···11···11···1

320 irreducible representations

dim11111111111111111111
type+++
imageC1C2C2C4C4C5C8C8C10C10C16C16C20C20C32C40C40C80C80C160
kernelC2×C160C160C2×C80C80C2×C40C2×C32C40C2×C20C32C2×C16C20C2×C10C16C2×C8C10C8C2×C4C4C22C2
# reps121224448488883216163232128

Matrix representation of C2×C160 in GL2(𝔽641) generated by

10
0640
,
2820
0353
G:=sub<GL(2,GF(641))| [1,0,0,640],[282,0,0,353] >;

C2×C160 in GAP, Magma, Sage, TeX

C_2\times C_{160}
% in TeX

G:=Group("C2xC160");
// GroupNames label

G:=SmallGroup(320,174);
// by ID

G=gap.SmallGroup(320,174);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,80,102,124]);
// Polycyclic

G:=Group<a,b|a^2=b^160=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C160 in TeX

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