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G = C2×C194order 388 = 22·97

Abelian group of type [2,194]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C194, SmallGroup(388,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C194
C1C97C194 — C2×C194
C1 — C2×C194
C1 — C2×C194

Generators and relations for C2×C194
 G = < a,b | a2=b194=1, ab=ba >


Smallest permutation representation of C2×C194
Regular action on 388 points
Generators in S388
(1 329)(2 330)(3 331)(4 332)(5 333)(6 334)(7 335)(8 336)(9 337)(10 338)(11 339)(12 340)(13 341)(14 342)(15 343)(16 344)(17 345)(18 346)(19 347)(20 348)(21 349)(22 350)(23 351)(24 352)(25 353)(26 354)(27 355)(28 356)(29 357)(30 358)(31 359)(32 360)(33 361)(34 362)(35 363)(36 364)(37 365)(38 366)(39 367)(40 368)(41 369)(42 370)(43 371)(44 372)(45 373)(46 374)(47 375)(48 376)(49 377)(50 378)(51 379)(52 380)(53 381)(54 382)(55 383)(56 384)(57 385)(58 386)(59 387)(60 388)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 205)(72 206)(73 207)(74 208)(75 209)(76 210)(77 211)(78 212)(79 213)(80 214)(81 215)(82 216)(83 217)(84 218)(85 219)(86 220)(87 221)(88 222)(89 223)(90 224)(91 225)(92 226)(93 227)(94 228)(95 229)(96 230)(97 231)(98 232)(99 233)(100 234)(101 235)(102 236)(103 237)(104 238)(105 239)(106 240)(107 241)(108 242)(109 243)(110 244)(111 245)(112 246)(113 247)(114 248)(115 249)(116 250)(117 251)(118 252)(119 253)(120 254)(121 255)(122 256)(123 257)(124 258)(125 259)(126 260)(127 261)(128 262)(129 263)(130 264)(131 265)(132 266)(133 267)(134 268)(135 269)(136 270)(137 271)(138 272)(139 273)(140 274)(141 275)(142 276)(143 277)(144 278)(145 279)(146 280)(147 281)(148 282)(149 283)(150 284)(151 285)(152 286)(153 287)(154 288)(155 289)(156 290)(157 291)(158 292)(159 293)(160 294)(161 295)(162 296)(163 297)(164 298)(165 299)(166 300)(167 301)(168 302)(169 303)(170 304)(171 305)(172 306)(173 307)(174 308)(175 309)(176 310)(177 311)(178 312)(179 313)(180 314)(181 315)(182 316)(183 317)(184 318)(185 319)(186 320)(187 321)(188 322)(189 323)(190 324)(191 325)(192 326)(193 327)(194 328)
(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)(195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388)

G:=sub<Sym(388)| (1,329)(2,330)(3,331)(4,332)(5,333)(6,334)(7,335)(8,336)(9,337)(10,338)(11,339)(12,340)(13,341)(14,342)(15,343)(16,344)(17,345)(18,346)(19,347)(20,348)(21,349)(22,350)(23,351)(24,352)(25,353)(26,354)(27,355)(28,356)(29,357)(30,358)(31,359)(32,360)(33,361)(34,362)(35,363)(36,364)(37,365)(38,366)(39,367)(40,368)(41,369)(42,370)(43,371)(44,372)(45,373)(46,374)(47,375)(48,376)(49,377)(50,378)(51,379)(52,380)(53,381)(54,382)(55,383)(56,384)(57,385)(58,386)(59,387)(60,388)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,221)(88,222)(89,223)(90,224)(91,225)(92,226)(93,227)(94,228)(95,229)(96,230)(97,231)(98,232)(99,233)(100,234)(101,235)(102,236)(103,237)(104,238)(105,239)(106,240)(107,241)(108,242)(109,243)(110,244)(111,245)(112,246)(113,247)(114,248)(115,249)(116,250)(117,251)(118,252)(119,253)(120,254)(121,255)(122,256)(123,257)(124,258)(125,259)(126,260)(127,261)(128,262)(129,263)(130,264)(131,265)(132,266)(133,267)(134,268)(135,269)(136,270)(137,271)(138,272)(139,273)(140,274)(141,275)(142,276)(143,277)(144,278)(145,279)(146,280)(147,281)(148,282)(149,283)(150,284)(151,285)(152,286)(153,287)(154,288)(155,289)(156,290)(157,291)(158,292)(159,293)(160,294)(161,295)(162,296)(163,297)(164,298)(165,299)(166,300)(167,301)(168,302)(169,303)(170,304)(171,305)(172,306)(173,307)(174,308)(175,309)(176,310)(177,311)(178,312)(179,313)(180,314)(181,315)(182,316)(183,317)(184,318)(185,319)(186,320)(187,321)(188,322)(189,323)(190,324)(191,325)(192,326)(193,327)(194,328), (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)(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388)>;

G:=Group( (1,329)(2,330)(3,331)(4,332)(5,333)(6,334)(7,335)(8,336)(9,337)(10,338)(11,339)(12,340)(13,341)(14,342)(15,343)(16,344)(17,345)(18,346)(19,347)(20,348)(21,349)(22,350)(23,351)(24,352)(25,353)(26,354)(27,355)(28,356)(29,357)(30,358)(31,359)(32,360)(33,361)(34,362)(35,363)(36,364)(37,365)(38,366)(39,367)(40,368)(41,369)(42,370)(43,371)(44,372)(45,373)(46,374)(47,375)(48,376)(49,377)(50,378)(51,379)(52,380)(53,381)(54,382)(55,383)(56,384)(57,385)(58,386)(59,387)(60,388)(61,195)(62,196)(63,197)(64,198)(65,199)(66,200)(67,201)(68,202)(69,203)(70,204)(71,205)(72,206)(73,207)(74,208)(75,209)(76,210)(77,211)(78,212)(79,213)(80,214)(81,215)(82,216)(83,217)(84,218)(85,219)(86,220)(87,221)(88,222)(89,223)(90,224)(91,225)(92,226)(93,227)(94,228)(95,229)(96,230)(97,231)(98,232)(99,233)(100,234)(101,235)(102,236)(103,237)(104,238)(105,239)(106,240)(107,241)(108,242)(109,243)(110,244)(111,245)(112,246)(113,247)(114,248)(115,249)(116,250)(117,251)(118,252)(119,253)(120,254)(121,255)(122,256)(123,257)(124,258)(125,259)(126,260)(127,261)(128,262)(129,263)(130,264)(131,265)(132,266)(133,267)(134,268)(135,269)(136,270)(137,271)(138,272)(139,273)(140,274)(141,275)(142,276)(143,277)(144,278)(145,279)(146,280)(147,281)(148,282)(149,283)(150,284)(151,285)(152,286)(153,287)(154,288)(155,289)(156,290)(157,291)(158,292)(159,293)(160,294)(161,295)(162,296)(163,297)(164,298)(165,299)(166,300)(167,301)(168,302)(169,303)(170,304)(171,305)(172,306)(173,307)(174,308)(175,309)(176,310)(177,311)(178,312)(179,313)(180,314)(181,315)(182,316)(183,317)(184,318)(185,319)(186,320)(187,321)(188,322)(189,323)(190,324)(191,325)(192,326)(193,327)(194,328), (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)(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388) );

G=PermutationGroup([(1,329),(2,330),(3,331),(4,332),(5,333),(6,334),(7,335),(8,336),(9,337),(10,338),(11,339),(12,340),(13,341),(14,342),(15,343),(16,344),(17,345),(18,346),(19,347),(20,348),(21,349),(22,350),(23,351),(24,352),(25,353),(26,354),(27,355),(28,356),(29,357),(30,358),(31,359),(32,360),(33,361),(34,362),(35,363),(36,364),(37,365),(38,366),(39,367),(40,368),(41,369),(42,370),(43,371),(44,372),(45,373),(46,374),(47,375),(48,376),(49,377),(50,378),(51,379),(52,380),(53,381),(54,382),(55,383),(56,384),(57,385),(58,386),(59,387),(60,388),(61,195),(62,196),(63,197),(64,198),(65,199),(66,200),(67,201),(68,202),(69,203),(70,204),(71,205),(72,206),(73,207),(74,208),(75,209),(76,210),(77,211),(78,212),(79,213),(80,214),(81,215),(82,216),(83,217),(84,218),(85,219),(86,220),(87,221),(88,222),(89,223),(90,224),(91,225),(92,226),(93,227),(94,228),(95,229),(96,230),(97,231),(98,232),(99,233),(100,234),(101,235),(102,236),(103,237),(104,238),(105,239),(106,240),(107,241),(108,242),(109,243),(110,244),(111,245),(112,246),(113,247),(114,248),(115,249),(116,250),(117,251),(118,252),(119,253),(120,254),(121,255),(122,256),(123,257),(124,258),(125,259),(126,260),(127,261),(128,262),(129,263),(130,264),(131,265),(132,266),(133,267),(134,268),(135,269),(136,270),(137,271),(138,272),(139,273),(140,274),(141,275),(142,276),(143,277),(144,278),(145,279),(146,280),(147,281),(148,282),(149,283),(150,284),(151,285),(152,286),(153,287),(154,288),(155,289),(156,290),(157,291),(158,292),(159,293),(160,294),(161,295),(162,296),(163,297),(164,298),(165,299),(166,300),(167,301),(168,302),(169,303),(170,304),(171,305),(172,306),(173,307),(174,308),(175,309),(176,310),(177,311),(178,312),(179,313),(180,314),(181,315),(182,316),(183,317),(184,318),(185,319),(186,320),(187,321),(188,322),(189,323),(190,324),(191,325),(192,326),(193,327),(194,328)], [(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),(195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388)])

388 conjugacy classes

class 1 2A2B2C97A···97CR194A···194KB
order122297···97194···194
size11111···11···1

388 irreducible representations

dim1111
type++
imageC1C2C97C194
kernelC2×C194C194C22C2
# reps1396288

Matrix representation of C2×C194 in GL2(𝔽389) generated by

3880
0388
,
3760
0142
G:=sub<GL(2,GF(389))| [388,0,0,388],[376,0,0,142] >;

C2×C194 in GAP, Magma, Sage, TeX

C_2\times C_{194}
% in TeX

G:=Group("C2xC194");
// GroupNames label

G:=SmallGroup(388,5);
// by ID

G=gap.SmallGroup(388,5);
# by ID

G:=PCGroup([3,-2,-2,-97]);
// Polycyclic

G:=Group<a,b|a^2=b^194=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C194 in TeX

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