Copied to
clipboard

G = C2×C178order 356 = 22·89

Abelian group of type [2,178]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C178, SmallGroup(356,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C178
C1C89C178 — C2×C178
C1 — C2×C178
C1 — C2×C178

Generators and relations for C2×C178
 G = < a,b | a2=b178=1, ab=ba >


Smallest permutation representation of C2×C178
Regular action on 356 points
Generators in S356
(1 355)(2 356)(3 179)(4 180)(5 181)(6 182)(7 183)(8 184)(9 185)(10 186)(11 187)(12 188)(13 189)(14 190)(15 191)(16 192)(17 193)(18 194)(19 195)(20 196)(21 197)(22 198)(23 199)(24 200)(25 201)(26 202)(27 203)(28 204)(29 205)(30 206)(31 207)(32 208)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 217)(42 218)(43 219)(44 220)(45 221)(46 222)(47 223)(48 224)(49 225)(50 226)(51 227)(52 228)(53 229)(54 230)(55 231)(56 232)(57 233)(58 234)(59 235)(60 236)(61 237)(62 238)(63 239)(64 240)(65 241)(66 242)(67 243)(68 244)(69 245)(70 246)(71 247)(72 248)(73 249)(74 250)(75 251)(76 252)(77 253)(78 254)(79 255)(80 256)(81 257)(82 258)(83 259)(84 260)(85 261)(86 262)(87 263)(88 264)(89 265)(90 266)(91 267)(92 268)(93 269)(94 270)(95 271)(96 272)(97 273)(98 274)(99 275)(100 276)(101 277)(102 278)(103 279)(104 280)(105 281)(106 282)(107 283)(108 284)(109 285)(110 286)(111 287)(112 288)(113 289)(114 290)(115 291)(116 292)(117 293)(118 294)(119 295)(120 296)(121 297)(122 298)(123 299)(124 300)(125 301)(126 302)(127 303)(128 304)(129 305)(130 306)(131 307)(132 308)(133 309)(134 310)(135 311)(136 312)(137 313)(138 314)(139 315)(140 316)(141 317)(142 318)(143 319)(144 320)(145 321)(146 322)(147 323)(148 324)(149 325)(150 326)(151 327)(152 328)(153 329)(154 330)(155 331)(156 332)(157 333)(158 334)(159 335)(160 336)(161 337)(162 338)(163 339)(164 340)(165 341)(166 342)(167 343)(168 344)(169 345)(170 346)(171 347)(172 348)(173 349)(174 350)(175 351)(176 352)(177 353)(178 354)
(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)(179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356)

G:=sub<Sym(356)| (1,355)(2,356)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,225)(50,226)(51,227)(52,228)(53,229)(54,230)(55,231)(56,232)(57,233)(58,234)(59,235)(60,236)(61,237)(62,238)(63,239)(64,240)(65,241)(66,242)(67,243)(68,244)(69,245)(70,246)(71,247)(72,248)(73,249)(74,250)(75,251)(76,252)(77,253)(78,254)(79,255)(80,256)(81,257)(82,258)(83,259)(84,260)(85,261)(86,262)(87,263)(88,264)(89,265)(90,266)(91,267)(92,268)(93,269)(94,270)(95,271)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,289)(114,290)(115,291)(116,292)(117,293)(118,294)(119,295)(120,296)(121,297)(122,298)(123,299)(124,300)(125,301)(126,302)(127,303)(128,304)(129,305)(130,306)(131,307)(132,308)(133,309)(134,310)(135,311)(136,312)(137,313)(138,314)(139,315)(140,316)(141,317)(142,318)(143,319)(144,320)(145,321)(146,322)(147,323)(148,324)(149,325)(150,326)(151,327)(152,328)(153,329)(154,330)(155,331)(156,332)(157,333)(158,334)(159,335)(160,336)(161,337)(162,338)(163,339)(164,340)(165,341)(166,342)(167,343)(168,344)(169,345)(170,346)(171,347)(172,348)(173,349)(174,350)(175,351)(176,352)(177,353)(178,354), (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)(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356)>;

G:=Group( (1,355)(2,356)(3,179)(4,180)(5,181)(6,182)(7,183)(8,184)(9,185)(10,186)(11,187)(12,188)(13,189)(14,190)(15,191)(16,192)(17,193)(18,194)(19,195)(20,196)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,205)(30,206)(31,207)(32,208)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,217)(42,218)(43,219)(44,220)(45,221)(46,222)(47,223)(48,224)(49,225)(50,226)(51,227)(52,228)(53,229)(54,230)(55,231)(56,232)(57,233)(58,234)(59,235)(60,236)(61,237)(62,238)(63,239)(64,240)(65,241)(66,242)(67,243)(68,244)(69,245)(70,246)(71,247)(72,248)(73,249)(74,250)(75,251)(76,252)(77,253)(78,254)(79,255)(80,256)(81,257)(82,258)(83,259)(84,260)(85,261)(86,262)(87,263)(88,264)(89,265)(90,266)(91,267)(92,268)(93,269)(94,270)(95,271)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,286)(111,287)(112,288)(113,289)(114,290)(115,291)(116,292)(117,293)(118,294)(119,295)(120,296)(121,297)(122,298)(123,299)(124,300)(125,301)(126,302)(127,303)(128,304)(129,305)(130,306)(131,307)(132,308)(133,309)(134,310)(135,311)(136,312)(137,313)(138,314)(139,315)(140,316)(141,317)(142,318)(143,319)(144,320)(145,321)(146,322)(147,323)(148,324)(149,325)(150,326)(151,327)(152,328)(153,329)(154,330)(155,331)(156,332)(157,333)(158,334)(159,335)(160,336)(161,337)(162,338)(163,339)(164,340)(165,341)(166,342)(167,343)(168,344)(169,345)(170,346)(171,347)(172,348)(173,349)(174,350)(175,351)(176,352)(177,353)(178,354), (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)(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356) );

G=PermutationGroup([(1,355),(2,356),(3,179),(4,180),(5,181),(6,182),(7,183),(8,184),(9,185),(10,186),(11,187),(12,188),(13,189),(14,190),(15,191),(16,192),(17,193),(18,194),(19,195),(20,196),(21,197),(22,198),(23,199),(24,200),(25,201),(26,202),(27,203),(28,204),(29,205),(30,206),(31,207),(32,208),(33,209),(34,210),(35,211),(36,212),(37,213),(38,214),(39,215),(40,216),(41,217),(42,218),(43,219),(44,220),(45,221),(46,222),(47,223),(48,224),(49,225),(50,226),(51,227),(52,228),(53,229),(54,230),(55,231),(56,232),(57,233),(58,234),(59,235),(60,236),(61,237),(62,238),(63,239),(64,240),(65,241),(66,242),(67,243),(68,244),(69,245),(70,246),(71,247),(72,248),(73,249),(74,250),(75,251),(76,252),(77,253),(78,254),(79,255),(80,256),(81,257),(82,258),(83,259),(84,260),(85,261),(86,262),(87,263),(88,264),(89,265),(90,266),(91,267),(92,268),(93,269),(94,270),(95,271),(96,272),(97,273),(98,274),(99,275),(100,276),(101,277),(102,278),(103,279),(104,280),(105,281),(106,282),(107,283),(108,284),(109,285),(110,286),(111,287),(112,288),(113,289),(114,290),(115,291),(116,292),(117,293),(118,294),(119,295),(120,296),(121,297),(122,298),(123,299),(124,300),(125,301),(126,302),(127,303),(128,304),(129,305),(130,306),(131,307),(132,308),(133,309),(134,310),(135,311),(136,312),(137,313),(138,314),(139,315),(140,316),(141,317),(142,318),(143,319),(144,320),(145,321),(146,322),(147,323),(148,324),(149,325),(150,326),(151,327),(152,328),(153,329),(154,330),(155,331),(156,332),(157,333),(158,334),(159,335),(160,336),(161,337),(162,338),(163,339),(164,340),(165,341),(166,342),(167,343),(168,344),(169,345),(170,346),(171,347),(172,348),(173,349),(174,350),(175,351),(176,352),(177,353),(178,354)], [(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),(179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356)])

356 conjugacy classes

class 1 2A2B2C89A···89CJ178A···178JD
order122289···89178···178
size11111···11···1

356 irreducible representations

dim1111
type++
imageC1C2C89C178
kernelC2×C178C178C22C2
# reps1388264

Matrix representation of C2×C178 in GL2(𝔽179) generated by

1780
01
,
780
06
G:=sub<GL(2,GF(179))| [178,0,0,1],[78,0,0,6] >;

C2×C178 in GAP, Magma, Sage, TeX

C_2\times C_{178}
% in TeX

G:=Group("C2xC178");
// GroupNames label

G:=SmallGroup(356,5);
// by ID

G=gap.SmallGroup(356,5);
# by ID

G:=PCGroup([3,-2,-2,-89]);
// Polycyclic

G:=Group<a,b|a^2=b^178=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C178 in TeX

׿
×
𝔽