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## G = C22×C84order 336 = 24·3·7

### Abelian group of type [2,2,84]

Aliases: C22×C84, SmallGroup(336,204)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C84
 Chief series C1 — C2 — C14 — C42 — C84 — C2×C84 — C22×C84
 Lower central C1 — C22×C84
 Upper central C1 — C22×C84

Generators and relations for C22×C84
G = < a,b,c | a2=b2=c84=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C6, C6, C7, C2×C4, C23, C12, C2×C6, C14, C14, C22×C4, C21, C2×C12, C22×C6, C28, C2×C14, C42, C42, C22×C12, C2×C28, C22×C14, C84, C2×C42, C22×C28, C2×C84, C22×C42, C22×C84
Quotients: C1, C2, C3, C4, C22, C6, C7, C2×C4, C23, C12, C2×C6, C14, C22×C4, C21, C2×C12, C22×C6, C28, C2×C14, C42, C22×C12, C2×C28, C22×C14, C84, C2×C42, C22×C28, C2×C84, C22×C42, C22×C84

Smallest permutation representation of C22×C84
Regular action on 336 points
Generators in S336
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)(169 319)(170 320)(171 321)(172 322)(173 323)(174 324)(175 325)(176 326)(177 327)(178 328)(179 329)(180 330)(181 331)(182 332)(183 333)(184 334)(185 335)(186 336)(187 253)(188 254)(189 255)(190 256)(191 257)(192 258)(193 259)(194 260)(195 261)(196 262)(197 263)(198 264)(199 265)(200 266)(201 267)(202 268)(203 269)(204 270)(205 271)(206 272)(207 273)(208 274)(209 275)(210 276)(211 277)(212 278)(213 279)(214 280)(215 281)(216 282)(217 283)(218 284)(219 285)(220 286)(221 287)(222 288)(223 289)(224 290)(225 291)(226 292)(227 293)(228 294)(229 295)(230 296)(231 297)(232 298)(233 299)(234 300)(235 301)(236 302)(237 303)(238 304)(239 305)(240 306)(241 307)(242 308)(243 309)(244 310)(245 311)(246 312)(247 313)(248 314)(249 315)(250 316)(251 317)(252 318)
(1 235)(2 236)(3 237)(4 238)(5 239)(6 240)(7 241)(8 242)(9 243)(10 244)(11 245)(12 246)(13 247)(14 248)(15 249)(16 250)(17 251)(18 252)(19 169)(20 170)(21 171)(22 172)(23 173)(24 174)(25 175)(26 176)(27 177)(28 178)(29 179)(30 180)(31 181)(32 182)(33 183)(34 184)(35 185)(36 186)(37 187)(38 188)(39 189)(40 190)(41 191)(42 192)(43 193)(44 194)(45 195)(46 196)(47 197)(48 198)(49 199)(50 200)(51 201)(52 202)(53 203)(54 204)(55 205)(56 206)(57 207)(58 208)(59 209)(60 210)(61 211)(62 212)(63 213)(64 214)(65 215)(66 216)(67 217)(68 218)(69 219)(70 220)(71 221)(72 222)(73 223)(74 224)(75 225)(76 226)(77 227)(78 228)(79 229)(80 230)(81 231)(82 232)(83 233)(84 234)(85 301)(86 302)(87 303)(88 304)(89 305)(90 306)(91 307)(92 308)(93 309)(94 310)(95 311)(96 312)(97 313)(98 314)(99 315)(100 316)(101 317)(102 318)(103 319)(104 320)(105 321)(106 322)(107 323)(108 324)(109 325)(110 326)(111 327)(112 328)(113 329)(114 330)(115 331)(116 332)(117 333)(118 334)(119 335)(120 336)(121 253)(122 254)(123 255)(124 256)(125 257)(126 258)(127 259)(128 260)(129 261)(130 262)(131 263)(132 264)(133 265)(134 266)(135 267)(136 268)(137 269)(138 270)(139 271)(140 272)(141 273)(142 274)(143 275)(144 276)(145 277)(146 278)(147 279)(148 280)(149 281)(150 282)(151 283)(152 284)(153 285)(154 286)(155 287)(156 288)(157 289)(158 290)(159 291)(160 292)(161 293)(162 294)(163 295)(164 296)(165 297)(166 298)(167 299)(168 300)
(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)

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

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

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

336 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4H 6A ··· 6N 7A ··· 7F 12A ··· 12P 14A ··· 14AP 21A ··· 21L 28A ··· 28AV 42A ··· 42CF 84A ··· 84CR order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 7 ··· 7 12 ··· 12 14 ··· 14 21 ··· 21 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

336 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C7 C12 C14 C14 C21 C28 C42 C42 C84 kernel C22×C84 C2×C84 C22×C42 C22×C28 C2×C42 C2×C28 C22×C14 C22×C12 C2×C14 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 6 1 2 8 12 2 6 16 36 6 12 48 72 12 96

Matrix representation of C22×C84 in GL3(𝔽337) generated by

 336 0 0 0 336 0 0 0 1
,
 336 0 0 0 336 0 0 0 336
,
 144 0 0 0 26 0 0 0 206
G:=sub<GL(3,GF(337))| [336,0,0,0,336,0,0,0,1],[336,0,0,0,336,0,0,0,336],[144,0,0,0,26,0,0,0,206] >;

C22×C84 in GAP, Magma, Sage, TeX

C_2^2\times C_{84}
% in TeX

G:=Group("C2^2xC84");
// GroupNames label

G:=SmallGroup(336,204);
// by ID

G=gap.SmallGroup(336,204);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-7,-2,1008]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^84=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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