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G = C2×C198order 396 = 22·32·11

Abelian group of type [2,198]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C198, SmallGroup(396,10)

Series: Derived Chief Lower central Upper central

C1 — C2×C198
C1C3C33C99C198 — C2×C198
C1 — C2×C198
C1 — C2×C198

Generators and relations for C2×C198
 G = < a,b | a2=b198=1, ab=ba >


Smallest permutation representation of C2×C198
Regular action on 396 points
Generators in S396
(1 206)(2 207)(3 208)(4 209)(5 210)(6 211)(7 212)(8 213)(9 214)(10 215)(11 216)(12 217)(13 218)(14 219)(15 220)(16 221)(17 222)(18 223)(19 224)(20 225)(21 226)(22 227)(23 228)(24 229)(25 230)(26 231)(27 232)(28 233)(29 234)(30 235)(31 236)(32 237)(33 238)(34 239)(35 240)(36 241)(37 242)(38 243)(39 244)(40 245)(41 246)(42 247)(43 248)(44 249)(45 250)(46 251)(47 252)(48 253)(49 254)(50 255)(51 256)(52 257)(53 258)(54 259)(55 260)(56 261)(57 262)(58 263)(59 264)(60 265)(61 266)(62 267)(63 268)(64 269)(65 270)(66 271)(67 272)(68 273)(69 274)(70 275)(71 276)(72 277)(73 278)(74 279)(75 280)(76 281)(77 282)(78 283)(79 284)(80 285)(81 286)(82 287)(83 288)(84 289)(85 290)(86 291)(87 292)(88 293)(89 294)(90 295)(91 296)(92 297)(93 298)(94 299)(95 300)(96 301)(97 302)(98 303)(99 304)(100 305)(101 306)(102 307)(103 308)(104 309)(105 310)(106 311)(107 312)(108 313)(109 314)(110 315)(111 316)(112 317)(113 318)(114 319)(115 320)(116 321)(117 322)(118 323)(119 324)(120 325)(121 326)(122 327)(123 328)(124 329)(125 330)(126 331)(127 332)(128 333)(129 334)(130 335)(131 336)(132 337)(133 338)(134 339)(135 340)(136 341)(137 342)(138 343)(139 344)(140 345)(141 346)(142 347)(143 348)(144 349)(145 350)(146 351)(147 352)(148 353)(149 354)(150 355)(151 356)(152 357)(153 358)(154 359)(155 360)(156 361)(157 362)(158 363)(159 364)(160 365)(161 366)(162 367)(163 368)(164 369)(165 370)(166 371)(167 372)(168 373)(169 374)(170 375)(171 376)(172 377)(173 378)(174 379)(175 380)(176 381)(177 382)(178 383)(179 384)(180 385)(181 386)(182 387)(183 388)(184 389)(185 390)(186 391)(187 392)(188 393)(189 394)(190 395)(191 396)(192 199)(193 200)(194 201)(195 202)(196 203)(197 204)(198 205)
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)

G:=sub<Sym(396)| (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,273)(69,274)(70,275)(71,276)(72,277)(73,278)(74,279)(75,280)(76,281)(77,282)(78,283)(79,284)(80,285)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,301)(97,302)(98,303)(99,304)(100,305)(101,306)(102,307)(103,308)(104,309)(105,310)(106,311)(107,312)(108,313)(109,314)(110,315)(111,316)(112,317)(113,318)(114,319)(115,320)(116,321)(117,322)(118,323)(119,324)(120,325)(121,326)(122,327)(123,328)(124,329)(125,330)(126,331)(127,332)(128,333)(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)(136,341)(137,342)(138,343)(139,344)(140,345)(141,346)(142,347)(143,348)(144,349)(145,350)(146,351)(147,352)(148,353)(149,354)(150,355)(151,356)(152,357)(153,358)(154,359)(155,360)(156,361)(157,362)(158,363)(159,364)(160,365)(161,366)(162,367)(163,368)(164,369)(165,370)(166,371)(167,372)(168,373)(169,374)(170,375)(171,376)(172,377)(173,378)(174,379)(175,380)(176,381)(177,382)(178,383)(179,384)(180,385)(181,386)(182,387)(183,388)(184,389)(185,390)(186,391)(187,392)(188,393)(189,394)(190,395)(191,396)(192,199)(193,200)(194,201)(195,202)(196,203)(197,204)(198,205), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)>;

G:=Group( (1,206)(2,207)(3,208)(4,209)(5,210)(6,211)(7,212)(8,213)(9,214)(10,215)(11,216)(12,217)(13,218)(14,219)(15,220)(16,221)(17,222)(18,223)(19,224)(20,225)(21,226)(22,227)(23,228)(24,229)(25,230)(26,231)(27,232)(28,233)(29,234)(30,235)(31,236)(32,237)(33,238)(34,239)(35,240)(36,241)(37,242)(38,243)(39,244)(40,245)(41,246)(42,247)(43,248)(44,249)(45,250)(46,251)(47,252)(48,253)(49,254)(50,255)(51,256)(52,257)(53,258)(54,259)(55,260)(56,261)(57,262)(58,263)(59,264)(60,265)(61,266)(62,267)(63,268)(64,269)(65,270)(66,271)(67,272)(68,273)(69,274)(70,275)(71,276)(72,277)(73,278)(74,279)(75,280)(76,281)(77,282)(78,283)(79,284)(80,285)(81,286)(82,287)(83,288)(84,289)(85,290)(86,291)(87,292)(88,293)(89,294)(90,295)(91,296)(92,297)(93,298)(94,299)(95,300)(96,301)(97,302)(98,303)(99,304)(100,305)(101,306)(102,307)(103,308)(104,309)(105,310)(106,311)(107,312)(108,313)(109,314)(110,315)(111,316)(112,317)(113,318)(114,319)(115,320)(116,321)(117,322)(118,323)(119,324)(120,325)(121,326)(122,327)(123,328)(124,329)(125,330)(126,331)(127,332)(128,333)(129,334)(130,335)(131,336)(132,337)(133,338)(134,339)(135,340)(136,341)(137,342)(138,343)(139,344)(140,345)(141,346)(142,347)(143,348)(144,349)(145,350)(146,351)(147,352)(148,353)(149,354)(150,355)(151,356)(152,357)(153,358)(154,359)(155,360)(156,361)(157,362)(158,363)(159,364)(160,365)(161,366)(162,367)(163,368)(164,369)(165,370)(166,371)(167,372)(168,373)(169,374)(170,375)(171,376)(172,377)(173,378)(174,379)(175,380)(176,381)(177,382)(178,383)(179,384)(180,385)(181,386)(182,387)(183,388)(184,389)(185,390)(186,391)(187,392)(188,393)(189,394)(190,395)(191,396)(192,199)(193,200)(194,201)(195,202)(196,203)(197,204)(198,205), (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396) );

G=PermutationGroup([(1,206),(2,207),(3,208),(4,209),(5,210),(6,211),(7,212),(8,213),(9,214),(10,215),(11,216),(12,217),(13,218),(14,219),(15,220),(16,221),(17,222),(18,223),(19,224),(20,225),(21,226),(22,227),(23,228),(24,229),(25,230),(26,231),(27,232),(28,233),(29,234),(30,235),(31,236),(32,237),(33,238),(34,239),(35,240),(36,241),(37,242),(38,243),(39,244),(40,245),(41,246),(42,247),(43,248),(44,249),(45,250),(46,251),(47,252),(48,253),(49,254),(50,255),(51,256),(52,257),(53,258),(54,259),(55,260),(56,261),(57,262),(58,263),(59,264),(60,265),(61,266),(62,267),(63,268),(64,269),(65,270),(66,271),(67,272),(68,273),(69,274),(70,275),(71,276),(72,277),(73,278),(74,279),(75,280),(76,281),(77,282),(78,283),(79,284),(80,285),(81,286),(82,287),(83,288),(84,289),(85,290),(86,291),(87,292),(88,293),(89,294),(90,295),(91,296),(92,297),(93,298),(94,299),(95,300),(96,301),(97,302),(98,303),(99,304),(100,305),(101,306),(102,307),(103,308),(104,309),(105,310),(106,311),(107,312),(108,313),(109,314),(110,315),(111,316),(112,317),(113,318),(114,319),(115,320),(116,321),(117,322),(118,323),(119,324),(120,325),(121,326),(122,327),(123,328),(124,329),(125,330),(126,331),(127,332),(128,333),(129,334),(130,335),(131,336),(132,337),(133,338),(134,339),(135,340),(136,341),(137,342),(138,343),(139,344),(140,345),(141,346),(142,347),(143,348),(144,349),(145,350),(146,351),(147,352),(148,353),(149,354),(150,355),(151,356),(152,357),(153,358),(154,359),(155,360),(156,361),(157,362),(158,363),(159,364),(160,365),(161,366),(162,367),(163,368),(164,369),(165,370),(166,371),(167,372),(168,373),(169,374),(170,375),(171,376),(172,377),(173,378),(174,379),(175,380),(176,381),(177,382),(178,383),(179,384),(180,385),(181,386),(182,387),(183,388),(184,389),(185,390),(186,391),(187,392),(188,393),(189,394),(190,395),(191,396),(192,199),(193,200),(194,201),(195,202),(196,203),(197,204),(198,205)], [(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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)])

396 conjugacy classes

class 1 2A2B2C3A3B6A···6F9A···9F11A···11J18A···18R22A···22AD33A···33T66A···66BH99A···99BH198A···198FX
order1222336···69···911···1118···1822···2233···3366···6699···99198···198
size1111111···11···11···11···11···11···11···11···11···1

396 irreducible representations

dim111111111111
type++
imageC1C2C3C6C9C11C18C22C33C66C99C198
kernelC2×C198C198C2×C66C66C2×C22C2×C18C22C18C2×C6C6C22C2
# reps13266101830206060180

Matrix representation of C2×C198 in GL2(𝔽199) generated by

1980
0198
,
740
051
G:=sub<GL(2,GF(199))| [198,0,0,198],[74,0,0,51] >;

C2×C198 in GAP, Magma, Sage, TeX

C_2\times C_{198}
% in TeX

G:=Group("C2xC198");
// GroupNames label

G:=SmallGroup(396,10);
// by ID

G=gap.SmallGroup(396,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-3,507]);
// Polycyclic

G:=Group<a,b|a^2=b^198=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C198 in TeX

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