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G = C2×C176order 352 = 25·11

Abelian group of type [2,176]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C176, SmallGroup(352,58)

Series: Derived Chief Lower central Upper central

C1 — C2×C176
C1C2C4C8C88C176 — C2×C176
C1 — C2×C176
C1 — C2×C176

Generators and relations for C2×C176
 G = < a,b | a2=b176=1, ab=ba >


Smallest permutation representation of C2×C176
Regular action on 352 points
Generators in S352
(1 297)(2 298)(3 299)(4 300)(5 301)(6 302)(7 303)(8 304)(9 305)(10 306)(11 307)(12 308)(13 309)(14 310)(15 311)(16 312)(17 313)(18 314)(19 315)(20 316)(21 317)(22 318)(23 319)(24 320)(25 321)(26 322)(27 323)(28 324)(29 325)(30 326)(31 327)(32 328)(33 329)(34 330)(35 331)(36 332)(37 333)(38 334)(39 335)(40 336)(41 337)(42 338)(43 339)(44 340)(45 341)(46 342)(47 343)(48 344)(49 345)(50 346)(51 347)(52 348)(53 349)(54 350)(55 351)(56 352)(57 177)(58 178)(59 179)(60 180)(61 181)(62 182)(63 183)(64 184)(65 185)(66 186)(67 187)(68 188)(69 189)(70 190)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(89 209)(90 210)(91 211)(92 212)(93 213)(94 214)(95 215)(96 216)(97 217)(98 218)(99 219)(100 220)(101 221)(102 222)(103 223)(104 224)(105 225)(106 226)(107 227)(108 228)(109 229)(110 230)(111 231)(112 232)(113 233)(114 234)(115 235)(116 236)(117 237)(118 238)(119 239)(120 240)(121 241)(122 242)(123 243)(124 244)(125 245)(126 246)(127 247)(128 248)(129 249)(130 250)(131 251)(132 252)(133 253)(134 254)(135 255)(136 256)(137 257)(138 258)(139 259)(140 260)(141 261)(142 262)(143 263)(144 264)(145 265)(146 266)(147 267)(148 268)(149 269)(150 270)(151 271)(152 272)(153 273)(154 274)(155 275)(156 276)(157 277)(158 278)(159 279)(160 280)(161 281)(162 282)(163 283)(164 284)(165 285)(166 286)(167 287)(168 288)(169 289)(170 290)(171 291)(172 292)(173 293)(174 294)(175 295)(176 296)
(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)

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

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

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

352 conjugacy classes

class 1 2A2B2C4A4B4C4D8A···8H11A···11J16A···16P22A···22AD44A···44AN88A···88CB176A···176FD
order122244448···811···1116···1622···2244···4488···88176···176
size111111111···11···11···11···11···11···11···1

352 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C4C4C8C8C11C16C22C22C44C44C88C88C176
kernelC2×C176C176C2×C88C88C2×C44C44C2×C22C2×C16C22C16C2×C8C8C2×C4C4C22C2
# reps12122441016201020204040160

Matrix representation of C2×C176 in GL2(𝔽353) generated by

3520
0352
,
2700
092
G:=sub<GL(2,GF(353))| [352,0,0,352],[270,0,0,92] >;

C2×C176 in GAP, Magma, Sage, TeX

C_2\times C_{176}
% in TeX

G:=Group("C2xC176");
// GroupNames label

G:=SmallGroup(352,58);
// by ID

G=gap.SmallGroup(352,58);
# by ID

G:=PCGroup([6,-2,-2,-11,-2,-2,-2,264,69,88]);
// Polycyclic

G:=Group<a,b|a^2=b^176=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C176 in TeX

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