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G = C2×C186order 372 = 22·3·31

Abelian group of type [2,186]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C186, SmallGroup(372,15)

Series: Derived Chief Lower central Upper central

C1 — C2×C186
C1C31C93C186 — C2×C186
C1 — C2×C186
C1 — C2×C186

Generators and relations for C2×C186
 G = < a,b | a2=b186=1, ab=ba >


Smallest permutation representation of C2×C186
Regular action on 372 points
Generators in S372
(1 258)(2 259)(3 260)(4 261)(5 262)(6 263)(7 264)(8 265)(9 266)(10 267)(11 268)(12 269)(13 270)(14 271)(15 272)(16 273)(17 274)(18 275)(19 276)(20 277)(21 278)(22 279)(23 280)(24 281)(25 282)(26 283)(27 284)(28 285)(29 286)(30 287)(31 288)(32 289)(33 290)(34 291)(35 292)(36 293)(37 294)(38 295)(39 296)(40 297)(41 298)(42 299)(43 300)(44 301)(45 302)(46 303)(47 304)(48 305)(49 306)(50 307)(51 308)(52 309)(53 310)(54 311)(55 312)(56 313)(57 314)(58 315)(59 316)(60 317)(61 318)(62 319)(63 320)(64 321)(65 322)(66 323)(67 324)(68 325)(69 326)(70 327)(71 328)(72 329)(73 330)(74 331)(75 332)(76 333)(77 334)(78 335)(79 336)(80 337)(81 338)(82 339)(83 340)(84 341)(85 342)(86 343)(87 344)(88 345)(89 346)(90 347)(91 348)(92 349)(93 350)(94 351)(95 352)(96 353)(97 354)(98 355)(99 356)(100 357)(101 358)(102 359)(103 360)(104 361)(105 362)(106 363)(107 364)(108 365)(109 366)(110 367)(111 368)(112 369)(113 370)(114 371)(115 372)(116 187)(117 188)(118 189)(119 190)(120 191)(121 192)(122 193)(123 194)(124 195)(125 196)(126 197)(127 198)(128 199)(129 200)(130 201)(131 202)(132 203)(133 204)(134 205)(135 206)(136 207)(137 208)(138 209)(139 210)(140 211)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 225)(155 226)(156 227)(157 228)(158 229)(159 230)(160 231)(161 232)(162 233)(163 234)(164 235)(165 236)(166 237)(167 238)(168 239)(169 240)(170 241)(171 242)(172 243)(173 244)(174 245)(175 246)(176 247)(177 248)(178 249)(179 250)(180 251)(181 252)(182 253)(183 254)(184 255)(185 256)(186 257)
(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)(187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372)

G:=sub<Sym(372)| (1,258)(2,259)(3,260)(4,261)(5,262)(6,263)(7,264)(8,265)(9,266)(10,267)(11,268)(12,269)(13,270)(14,271)(15,272)(16,273)(17,274)(18,275)(19,276)(20,277)(21,278)(22,279)(23,280)(24,281)(25,282)(26,283)(27,284)(28,285)(29,286)(30,287)(31,288)(32,289)(33,290)(34,291)(35,292)(36,293)(37,294)(38,295)(39,296)(40,297)(41,298)(42,299)(43,300)(44,301)(45,302)(46,303)(47,304)(48,305)(49,306)(50,307)(51,308)(52,309)(53,310)(54,311)(55,312)(56,313)(57,314)(58,315)(59,316)(60,317)(61,318)(62,319)(63,320)(64,321)(65,322)(66,323)(67,324)(68,325)(69,326)(70,327)(71,328)(72,329)(73,330)(74,331)(75,332)(76,333)(77,334)(78,335)(79,336)(80,337)(81,338)(82,339)(83,340)(84,341)(85,342)(86,343)(87,344)(88,345)(89,346)(90,347)(91,348)(92,349)(93,350)(94,351)(95,352)(96,353)(97,354)(98,355)(99,356)(100,357)(101,358)(102,359)(103,360)(104,361)(105,362)(106,363)(107,364)(108,365)(109,366)(110,367)(111,368)(112,369)(113,370)(114,371)(115,372)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,232)(162,233)(163,234)(164,235)(165,236)(166,237)(167,238)(168,239)(169,240)(170,241)(171,242)(172,243)(173,244)(174,245)(175,246)(176,247)(177,248)(178,249)(179,250)(180,251)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257), (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)(187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372)>;

G:=Group( (1,258)(2,259)(3,260)(4,261)(5,262)(6,263)(7,264)(8,265)(9,266)(10,267)(11,268)(12,269)(13,270)(14,271)(15,272)(16,273)(17,274)(18,275)(19,276)(20,277)(21,278)(22,279)(23,280)(24,281)(25,282)(26,283)(27,284)(28,285)(29,286)(30,287)(31,288)(32,289)(33,290)(34,291)(35,292)(36,293)(37,294)(38,295)(39,296)(40,297)(41,298)(42,299)(43,300)(44,301)(45,302)(46,303)(47,304)(48,305)(49,306)(50,307)(51,308)(52,309)(53,310)(54,311)(55,312)(56,313)(57,314)(58,315)(59,316)(60,317)(61,318)(62,319)(63,320)(64,321)(65,322)(66,323)(67,324)(68,325)(69,326)(70,327)(71,328)(72,329)(73,330)(74,331)(75,332)(76,333)(77,334)(78,335)(79,336)(80,337)(81,338)(82,339)(83,340)(84,341)(85,342)(86,343)(87,344)(88,345)(89,346)(90,347)(91,348)(92,349)(93,350)(94,351)(95,352)(96,353)(97,354)(98,355)(99,356)(100,357)(101,358)(102,359)(103,360)(104,361)(105,362)(106,363)(107,364)(108,365)(109,366)(110,367)(111,368)(112,369)(113,370)(114,371)(115,372)(116,187)(117,188)(118,189)(119,190)(120,191)(121,192)(122,193)(123,194)(124,195)(125,196)(126,197)(127,198)(128,199)(129,200)(130,201)(131,202)(132,203)(133,204)(134,205)(135,206)(136,207)(137,208)(138,209)(139,210)(140,211)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,225)(155,226)(156,227)(157,228)(158,229)(159,230)(160,231)(161,232)(162,233)(163,234)(164,235)(165,236)(166,237)(167,238)(168,239)(169,240)(170,241)(171,242)(172,243)(173,244)(174,245)(175,246)(176,247)(177,248)(178,249)(179,250)(180,251)(181,252)(182,253)(183,254)(184,255)(185,256)(186,257), (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)(187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372) );

G=PermutationGroup([(1,258),(2,259),(3,260),(4,261),(5,262),(6,263),(7,264),(8,265),(9,266),(10,267),(11,268),(12,269),(13,270),(14,271),(15,272),(16,273),(17,274),(18,275),(19,276),(20,277),(21,278),(22,279),(23,280),(24,281),(25,282),(26,283),(27,284),(28,285),(29,286),(30,287),(31,288),(32,289),(33,290),(34,291),(35,292),(36,293),(37,294),(38,295),(39,296),(40,297),(41,298),(42,299),(43,300),(44,301),(45,302),(46,303),(47,304),(48,305),(49,306),(50,307),(51,308),(52,309),(53,310),(54,311),(55,312),(56,313),(57,314),(58,315),(59,316),(60,317),(61,318),(62,319),(63,320),(64,321),(65,322),(66,323),(67,324),(68,325),(69,326),(70,327),(71,328),(72,329),(73,330),(74,331),(75,332),(76,333),(77,334),(78,335),(79,336),(80,337),(81,338),(82,339),(83,340),(84,341),(85,342),(86,343),(87,344),(88,345),(89,346),(90,347),(91,348),(92,349),(93,350),(94,351),(95,352),(96,353),(97,354),(98,355),(99,356),(100,357),(101,358),(102,359),(103,360),(104,361),(105,362),(106,363),(107,364),(108,365),(109,366),(110,367),(111,368),(112,369),(113,370),(114,371),(115,372),(116,187),(117,188),(118,189),(119,190),(120,191),(121,192),(122,193),(123,194),(124,195),(125,196),(126,197),(127,198),(128,199),(129,200),(130,201),(131,202),(132,203),(133,204),(134,205),(135,206),(136,207),(137,208),(138,209),(139,210),(140,211),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,225),(155,226),(156,227),(157,228),(158,229),(159,230),(160,231),(161,232),(162,233),(163,234),(164,235),(165,236),(166,237),(167,238),(168,239),(169,240),(170,241),(171,242),(172,243),(173,244),(174,245),(175,246),(176,247),(177,248),(178,249),(179,250),(180,251),(181,252),(182,253),(183,254),(184,255),(185,256),(186,257)], [(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),(187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372)])

372 conjugacy classes

class 1 2A2B2C3A3B6A···6F31A···31AD62A···62CL93A···93BH186A···186FX
order1222336···631···3162···6293···93186···186
size1111111···11···11···11···11···1

372 irreducible representations

dim11111111
type++
imageC1C2C3C6C31C62C93C186
kernelC2×C186C186C2×C62C62C2×C6C6C22C2
# reps1326309060180

Matrix representation of C2×C186 in GL2(𝔽373) generated by

3720
0372
,
3720
09
G:=sub<GL(2,GF(373))| [372,0,0,372],[372,0,0,9] >;

C2×C186 in GAP, Magma, Sage, TeX

C_2\times C_{186}
% in TeX

G:=Group("C2xC186");
// GroupNames label

G:=SmallGroup(372,15);
// by ID

G=gap.SmallGroup(372,15);
# by ID

G:=PCGroup([4,-2,-2,-3,-31]);
// Polycyclic

G:=Group<a,b|a^2=b^186=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C186 in TeX

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