Copied to
clipboard

G = C2×C196order 392 = 23·72

Abelian group of type [2,196]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C196, SmallGroup(392,8)

Series: Derived Chief Lower central Upper central

C1 — C2×C196
C1C7C14C98C196 — C2×C196
C1 — C2×C196
C1 — C2×C196

Generators and relations for C2×C196
 G = < a,b | a2=b196=1, ab=ba >


Smallest permutation representation of C2×C196
Regular action on 392 points
Generators in S392
(1 247)(2 248)(3 249)(4 250)(5 251)(6 252)(7 253)(8 254)(9 255)(10 256)(11 257)(12 258)(13 259)(14 260)(15 261)(16 262)(17 263)(18 264)(19 265)(20 266)(21 267)(22 268)(23 269)(24 270)(25 271)(26 272)(27 273)(28 274)(29 275)(30 276)(31 277)(32 278)(33 279)(34 280)(35 281)(36 282)(37 283)(38 284)(39 285)(40 286)(41 287)(42 288)(43 289)(44 290)(45 291)(46 292)(47 293)(48 294)(49 295)(50 296)(51 297)(52 298)(53 299)(54 300)(55 301)(56 302)(57 303)(58 304)(59 305)(60 306)(61 307)(62 308)(63 309)(64 310)(65 311)(66 312)(67 313)(68 314)(69 315)(70 316)(71 317)(72 318)(73 319)(74 320)(75 321)(76 322)(77 323)(78 324)(79 325)(80 326)(81 327)(82 328)(83 329)(84 330)(85 331)(86 332)(87 333)(88 334)(89 335)(90 336)(91 337)(92 338)(93 339)(94 340)(95 341)(96 342)(97 343)(98 344)(99 345)(100 346)(101 347)(102 348)(103 349)(104 350)(105 351)(106 352)(107 353)(108 354)(109 355)(110 356)(111 357)(112 358)(113 359)(114 360)(115 361)(116 362)(117 363)(118 364)(119 365)(120 366)(121 367)(122 368)(123 369)(124 370)(125 371)(126 372)(127 373)(128 374)(129 375)(130 376)(131 377)(132 378)(133 379)(134 380)(135 381)(136 382)(137 383)(138 384)(139 385)(140 386)(141 387)(142 388)(143 389)(144 390)(145 391)(146 392)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 211)(162 212)(163 213)(164 214)(165 215)(166 216)(167 217)(168 218)(169 219)(170 220)(171 221)(172 222)(173 223)(174 224)(175 225)(176 226)(177 227)(178 228)(179 229)(180 230)(181 231)(182 232)(183 233)(184 234)(185 235)(186 236)(187 237)(188 238)(189 239)(190 240)(191 241)(192 242)(193 243)(194 244)(195 245)(196 246)
(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)

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

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

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

392 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F14A···14R28A···28X49A···49AP98A···98DV196A···196FL
order122244447···714···1428···2849···4998···98196···196
size111111111···11···11···11···11···11···1

392 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C7C14C14C28C49C98C98C196
kernelC2×C196C196C2×C98C98C2×C28C28C2×C14C14C2×C4C4C22C2
# reps1214612624428442168

Matrix representation of C2×C196 in GL2(𝔽197) generated by

1960
01
,
1570
0103
G:=sub<GL(2,GF(197))| [196,0,0,1],[157,0,0,103] >;

C2×C196 in GAP, Magma, Sage, TeX

C_2\times C_{196}
% in TeX

G:=Group("C2xC196");
// GroupNames label

G:=SmallGroup(392,8);
// by ID

G=gap.SmallGroup(392,8);
# by ID

G:=PCGroup([5,-2,-2,-7,-2,-7,140,222]);
// Polycyclic

G:=Group<a,b|a^2=b^196=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C2×C196 in TeX

׿
×
𝔽