direct product, abelian, monomial, 2-elementary
Aliases: C22×C78, SmallGroup(312,61)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C78 |
| C1 — C22×C78 |
| C1 — C22×C78 |
Generators and relations for C22×C78
G = < a,b,c | a2=b2=c78=1, ab=ba, ac=ca, bc=cb >
Subgroups: 64, all normal (8 characteristic)
C1, C2, C3, C22, C6, C23, C2×C6, C13, C22×C6, C26, C39, C2×C26, C78, C22×C26, C2×C78, C22×C78
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C13, C22×C6, C26, C39, C2×C26, C78, C22×C26, C2×C78, C22×C78
►Regular action on 312 points
Generators in S312
(1 164)(2 165)(3 166)(4 167)(5 168)(6 169)(7 170)(8 171)(9 172)(10 173)(11 174)(12 175)(13 176)(14 177)(15 178)(16 179)(17 180)(18 181)(19 182)(20 183)(21 184)(22 185)(23 186)(24 187)(25 188)(26 189)(27 190)(28 191)(29 192)(30 193)(31 194)(32 195)(33 196)(34 197)(35 198)(36 199)(37 200)(38 201)(39 202)(40 203)(41 204)(42 205)(43 206)(44 207)(45 208)(46 209)(47 210)(48 211)(49 212)(50 213)(51 214)(52 215)(53 216)(54 217)(55 218)(56 219)(57 220)(58 221)(59 222)(60 223)(61 224)(62 225)(63 226)(64 227)(65 228)(66 229)(67 230)(68 231)(69 232)(70 233)(71 234)(72 157)(73 158)(74 159)(75 160)(76 161)(77 162)(78 163)(79 247)(80 248)(81 249)(82 250)(83 251)(84 252)(85 253)(86 254)(87 255)(88 256)(89 257)(90 258)(91 259)(92 260)(93 261)(94 262)(95 263)(96 264)(97 265)(98 266)(99 267)(100 268)(101 269)(102 270)(103 271)(104 272)(105 273)(106 274)(107 275)(108 276)(109 277)(110 278)(111 279)(112 280)(113 281)(114 282)(115 283)(116 284)(117 285)(118 286)(119 287)(120 288)(121 289)(122 290)(123 291)(124 292)(125 293)(126 294)(127 295)(128 296)(129 297)(130 298)(131 299)(132 300)(133 301)(134 302)(135 303)(136 304)(137 305)(138 306)(139 307)(140 308)(141 309)(142 310)(143 311)(144 312)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(151 241)(152 242)(153 243)(154 244)(155 245)(156 246)
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 126)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 144)(34 145)(35 146)(36 147)(37 148)(38 149)(39 150)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 79)(47 80)(48 81)(49 82)(50 83)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 91)(59 92)(60 93)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 101)(69 102)(70 103)(71 104)(72 105)(73 106)(74 107)(75 108)(76 109)(77 110)(78 111)(157 273)(158 274)(159 275)(160 276)(161 277)(162 278)(163 279)(164 280)(165 281)(166 282)(167 283)(168 284)(169 285)(170 286)(171 287)(172 288)(173 289)(174 290)(175 291)(176 292)(177 293)(178 294)(179 295)(180 296)(181 297)(182 298)(183 299)(184 300)(185 301)(186 302)(187 303)(188 304)(189 305)(190 306)(191 307)(192 308)(193 309)(194 310)(195 311)(196 312)(197 235)(198 236)(199 237)(200 238)(201 239)(202 240)(203 241)(204 242)(205 243)(206 244)(207 245)(208 246)(209 247)(210 248)(211 249)(212 250)(213 251)(214 252)(215 253)(216 254)(217 255)(218 256)(219 257)(220 258)(221 259)(222 260)(223 261)(224 262)(225 263)(226 264)(227 265)(228 266)(229 267)(230 268)(231 269)(232 270)(233 271)(234 272)
(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)
G:=sub<Sym(312)| (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,220)(58,221)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,233)(71,234)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,247)(80,248)(81,249)(82,250)(83,251)(84,252)(85,253)(86,254)(87,255)(88,256)(89,257)(90,258)(91,259)(92,260)(93,261)(94,262)(95,263)(96,264)(97,265)(98,266)(99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)(144,312)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(151,241)(152,242)(153,243)(154,244)(155,245)(156,246), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(157,273)(158,274)(159,275)(160,276)(161,277)(162,278)(163,279)(164,280)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,235)(198,236)(199,237)(200,238)(201,239)(202,240)(203,241)(204,242)(205,243)(206,244)(207,245)(208,246)(209,247)(210,248)(211,249)(212,250)(213,251)(214,252)(215,253)(216,254)(217,255)(218,256)(219,257)(220,258)(221,259)(222,260)(223,261)(224,262)(225,263)(226,264)(227,265)(228,266)(229,267)(230,268)(231,269)(232,270)(233,271)(234,272), (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)>;
G:=Group( (1,164)(2,165)(3,166)(4,167)(5,168)(6,169)(7,170)(8,171)(9,172)(10,173)(11,174)(12,175)(13,176)(14,177)(15,178)(16,179)(17,180)(18,181)(19,182)(20,183)(21,184)(22,185)(23,186)(24,187)(25,188)(26,189)(27,190)(28,191)(29,192)(30,193)(31,194)(32,195)(33,196)(34,197)(35,198)(36,199)(37,200)(38,201)(39,202)(40,203)(41,204)(42,205)(43,206)(44,207)(45,208)(46,209)(47,210)(48,211)(49,212)(50,213)(51,214)(52,215)(53,216)(54,217)(55,218)(56,219)(57,220)(58,221)(59,222)(60,223)(61,224)(62,225)(63,226)(64,227)(65,228)(66,229)(67,230)(68,231)(69,232)(70,233)(71,234)(72,157)(73,158)(74,159)(75,160)(76,161)(77,162)(78,163)(79,247)(80,248)(81,249)(82,250)(83,251)(84,252)(85,253)(86,254)(87,255)(88,256)(89,257)(90,258)(91,259)(92,260)(93,261)(94,262)(95,263)(96,264)(97,265)(98,266)(99,267)(100,268)(101,269)(102,270)(103,271)(104,272)(105,273)(106,274)(107,275)(108,276)(109,277)(110,278)(111,279)(112,280)(113,281)(114,282)(115,283)(116,284)(117,285)(118,286)(119,287)(120,288)(121,289)(122,290)(123,291)(124,292)(125,293)(126,294)(127,295)(128,296)(129,297)(130,298)(131,299)(132,300)(133,301)(134,302)(135,303)(136,304)(137,305)(138,306)(139,307)(140,308)(141,309)(142,310)(143,311)(144,312)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(151,241)(152,242)(153,243)(154,244)(155,245)(156,246), (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,126)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,144)(34,145)(35,146)(36,147)(37,148)(38,149)(39,150)(40,151)(41,152)(42,153)(43,154)(44,155)(45,156)(46,79)(47,80)(48,81)(49,82)(50,83)(51,84)(52,85)(53,86)(54,87)(55,88)(56,89)(57,90)(58,91)(59,92)(60,93)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,101)(69,102)(70,103)(71,104)(72,105)(73,106)(74,107)(75,108)(76,109)(77,110)(78,111)(157,273)(158,274)(159,275)(160,276)(161,277)(162,278)(163,279)(164,280)(165,281)(166,282)(167,283)(168,284)(169,285)(170,286)(171,287)(172,288)(173,289)(174,290)(175,291)(176,292)(177,293)(178,294)(179,295)(180,296)(181,297)(182,298)(183,299)(184,300)(185,301)(186,302)(187,303)(188,304)(189,305)(190,306)(191,307)(192,308)(193,309)(194,310)(195,311)(196,312)(197,235)(198,236)(199,237)(200,238)(201,239)(202,240)(203,241)(204,242)(205,243)(206,244)(207,245)(208,246)(209,247)(210,248)(211,249)(212,250)(213,251)(214,252)(215,253)(216,254)(217,255)(218,256)(219,257)(220,258)(221,259)(222,260)(223,261)(224,262)(225,263)(226,264)(227,265)(228,266)(229,267)(230,268)(231,269)(232,270)(233,271)(234,272), (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) );
G=PermutationGroup([[(1,164),(2,165),(3,166),(4,167),(5,168),(6,169),(7,170),(8,171),(9,172),(10,173),(11,174),(12,175),(13,176),(14,177),(15,178),(16,179),(17,180),(18,181),(19,182),(20,183),(21,184),(22,185),(23,186),(24,187),(25,188),(26,189),(27,190),(28,191),(29,192),(30,193),(31,194),(32,195),(33,196),(34,197),(35,198),(36,199),(37,200),(38,201),(39,202),(40,203),(41,204),(42,205),(43,206),(44,207),(45,208),(46,209),(47,210),(48,211),(49,212),(50,213),(51,214),(52,215),(53,216),(54,217),(55,218),(56,219),(57,220),(58,221),(59,222),(60,223),(61,224),(62,225),(63,226),(64,227),(65,228),(66,229),(67,230),(68,231),(69,232),(70,233),(71,234),(72,157),(73,158),(74,159),(75,160),(76,161),(77,162),(78,163),(79,247),(80,248),(81,249),(82,250),(83,251),(84,252),(85,253),(86,254),(87,255),(88,256),(89,257),(90,258),(91,259),(92,260),(93,261),(94,262),(95,263),(96,264),(97,265),(98,266),(99,267),(100,268),(101,269),(102,270),(103,271),(104,272),(105,273),(106,274),(107,275),(108,276),(109,277),(110,278),(111,279),(112,280),(113,281),(114,282),(115,283),(116,284),(117,285),(118,286),(119,287),(120,288),(121,289),(122,290),(123,291),(124,292),(125,293),(126,294),(127,295),(128,296),(129,297),(130,298),(131,299),(132,300),(133,301),(134,302),(135,303),(136,304),(137,305),(138,306),(139,307),(140,308),(141,309),(142,310),(143,311),(144,312),(145,235),(146,236),(147,237),(148,238),(149,239),(150,240),(151,241),(152,242),(153,243),(154,244),(155,245),(156,246)], [(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,121),(11,122),(12,123),(13,124),(14,125),(15,126),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,144),(34,145),(35,146),(36,147),(37,148),(38,149),(39,150),(40,151),(41,152),(42,153),(43,154),(44,155),(45,156),(46,79),(47,80),(48,81),(49,82),(50,83),(51,84),(52,85),(53,86),(54,87),(55,88),(56,89),(57,90),(58,91),(59,92),(60,93),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,101),(69,102),(70,103),(71,104),(72,105),(73,106),(74,107),(75,108),(76,109),(77,110),(78,111),(157,273),(158,274),(159,275),(160,276),(161,277),(162,278),(163,279),(164,280),(165,281),(166,282),(167,283),(168,284),(169,285),(170,286),(171,287),(172,288),(173,289),(174,290),(175,291),(176,292),(177,293),(178,294),(179,295),(180,296),(181,297),(182,298),(183,299),(184,300),(185,301),(186,302),(187,303),(188,304),(189,305),(190,306),(191,307),(192,308),(193,309),(194,310),(195,311),(196,312),(197,235),(198,236),(199,237),(200,238),(201,239),(202,240),(203,241),(204,242),(205,243),(206,244),(207,245),(208,246),(209,247),(210,248),(211,249),(212,250),(213,251),(214,252),(215,253),(216,254),(217,255),(218,256),(219,257),(220,258),(221,259),(222,260),(223,261),(224,262),(225,263),(226,264),(227,265),(228,266),(229,267),(230,268),(231,269),(232,270),(233,271),(234,272)], [(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)]])
312 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 6A | ··· | 6N | 13A | ··· | 13L | 26A | ··· | 26CF | 39A | ··· | 39X | 78A | ··· | 78FL |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
312 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 |
| kernel | C22×C78 | C2×C78 | C22×C26 | C2×C26 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 7 | 2 | 14 | 12 | 84 | 24 | 168 |
Matrix representation of C22×C78 ►in GL3(𝔽79) generated by
| 1 | 0 | 0 |
| 0 | 78 | 0 |
| 0 | 0 | 78 |
| 1 | 0 | 0 |
| 0 | 78 | 0 |
| 0 | 0 | 1 |
| 28 | 0 | 0 |
| 0 | 28 | 0 |
| 0 | 0 | 5 |
G:=sub<GL(3,GF(79))| [1,0,0,0,78,0,0,0,78],[1,0,0,0,78,0,0,0,1],[28,0,0,0,28,0,0,0,5] >;
C22×C78 in GAP, Magma, Sage, TeX
C_2^2\times C_{78} % in TeX
G:=Group("C2^2xC78"); // GroupNames label
G:=SmallGroup(312,61);
// by ID
G=gap.SmallGroup(312,61);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^78=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations