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G = C2×C158order 316 = 22·79

Abelian group of type [2,158]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C158, SmallGroup(316,4)

Series: Derived Chief Lower central Upper central

C1 — C2×C158
C1C79C158 — C2×C158
C1 — C2×C158
C1 — C2×C158

Generators and relations for C2×C158
 G = < a,b | a2=b158=1, ab=ba >


Smallest permutation representation of C2×C158
Regular action on 316 points
Generators in S316
(1 315)(2 316)(3 159)(4 160)(5 161)(6 162)(7 163)(8 164)(9 165)(10 166)(11 167)(12 168)(13 169)(14 170)(15 171)(16 172)(17 173)(18 174)(19 175)(20 176)(21 177)(22 178)(23 179)(24 180)(25 181)(26 182)(27 183)(28 184)(29 185)(30 186)(31 187)(32 188)(33 189)(34 190)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 211)(56 212)(57 213)(58 214)(59 215)(60 216)(61 217)(62 218)(63 219)(64 220)(65 221)(66 222)(67 223)(68 224)(69 225)(70 226)(71 227)(72 228)(73 229)(74 230)(75 231)(76 232)(77 233)(78 234)(79 235)(80 236)(81 237)(82 238)(83 239)(84 240)(85 241)(86 242)(87 243)(88 244)(89 245)(90 246)(91 247)(92 248)(93 249)(94 250)(95 251)(96 252)(97 253)(98 254)(99 255)(100 256)(101 257)(102 258)(103 259)(104 260)(105 261)(106 262)(107 263)(108 264)(109 265)(110 266)(111 267)(112 268)(113 269)(114 270)(115 271)(116 272)(117 273)(118 274)(119 275)(120 276)(121 277)(122 278)(123 279)(124 280)(125 281)(126 282)(127 283)(128 284)(129 285)(130 286)(131 287)(132 288)(133 289)(134 290)(135 291)(136 292)(137 293)(138 294)(139 295)(140 296)(141 297)(142 298)(143 299)(144 300)(145 301)(146 302)(147 303)(148 304)(149 305)(150 306)(151 307)(152 308)(153 309)(154 310)(155 311)(156 312)(157 313)(158 314)
(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)

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

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

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

316 conjugacy classes

class 1 2A2B2C79A···79BZ158A···158HZ
order122279···79158···158
size11111···11···1

316 irreducible representations

dim1111
type++
imageC1C2C79C158
kernelC2×C158C158C22C2
# reps1378234

Matrix representation of C2×C158 in GL2(𝔽317) generated by

3160
0316
,
2050
096
G:=sub<GL(2,GF(317))| [316,0,0,316],[205,0,0,96] >;

C2×C158 in GAP, Magma, Sage, TeX

C_2\times C_{158}
% in TeX

G:=Group("C2xC158");
// GroupNames label

G:=SmallGroup(316,4);
// by ID

G=gap.SmallGroup(316,4);
# by ID

G:=PCGroup([3,-2,-2,-79]);
// Polycyclic

G:=Group<a,b|a^2=b^158=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C158 in TeX

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