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## G = C22×C78order 312 = 23·3·13

### Abelian group of type [2,2,78]

Aliases: C22×C78, SmallGroup(312,61)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C78
 Chief series C1 — C13 — C39 — C78 — C2×C78 — C22×C78
 Lower central C1 — C22×C78
 Upper central 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 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C13, C22×C6, C26 [×7], C39, C2×C26 [×7], C78 [×7], C22×C26, C2×C78 [×7], C22×C78
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C13, C22×C6, C26 [×7], C39, C2×C26 [×7], C78 [×7], C22×C26, C2×C78 [×7], C22×C78

Smallest permutation representation of C22×C78
Regular action on 312 points
Generators in S312
(1 204)(2 205)(3 206)(4 207)(5 208)(6 209)(7 210)(8 211)(9 212)(10 213)(11 214)(12 215)(13 216)(14 217)(15 218)(16 219)(17 220)(18 221)(19 222)(20 223)(21 224)(22 225)(23 226)(24 227)(25 228)(26 229)(27 230)(28 231)(29 232)(30 233)(31 234)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 196)(72 197)(73 198)(74 199)(75 200)(76 201)(77 202)(78 203)(79 285)(80 286)(81 287)(82 288)(83 289)(84 290)(85 291)(86 292)(87 293)(88 294)(89 295)(90 296)(91 297)(92 298)(93 299)(94 300)(95 301)(96 302)(97 303)(98 304)(99 305)(100 306)(101 307)(102 308)(103 309)(104 310)(105 311)(106 312)(107 235)(108 236)(109 237)(110 238)(111 239)(112 240)(113 241)(114 242)(115 243)(116 244)(117 245)(118 246)(119 247)(120 248)(121 249)(122 250)(123 251)(124 252)(125 253)(126 254)(127 255)(128 256)(129 257)(130 258)(131 259)(132 260)(133 261)(134 262)(135 263)(136 264)(137 265)(138 266)(139 267)(140 268)(141 269)(142 270)(143 271)(144 272)(145 273)(146 274)(147 275)(148 276)(149 277)(150 278)(151 279)(152 280)(153 281)(154 282)(155 283)(156 284)
(1 127)(2 128)(3 129)(4 130)(5 131)(6 132)(7 133)(8 134)(9 135)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 155)(30 156)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(157 286)(158 287)(159 288)(160 289)(161 290)(162 291)(163 292)(164 293)(165 294)(166 295)(167 296)(168 297)(169 298)(170 299)(171 300)(172 301)(173 302)(174 303)(175 304)(176 305)(177 306)(178 307)(179 308)(180 309)(181 310)(182 311)(183 312)(184 235)(185 236)(186 237)(187 238)(188 239)(189 240)(190 241)(191 242)(192 243)(193 244)(194 245)(195 246)(196 247)(197 248)(198 249)(199 250)(200 251)(201 252)(202 253)(203 254)(204 255)(205 256)(206 257)(207 258)(208 259)(209 260)(210 261)(211 262)(212 263)(213 264)(214 265)(215 266)(216 267)(217 268)(218 269)(219 270)(220 271)(221 272)(222 273)(223 274)(224 275)(225 276)(226 277)(227 278)(228 279)(229 280)(230 281)(231 282)(232 283)(233 284)(234 285)
(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)| (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,228)(26,229)(27,230)(28,231)(29,232)(30,233)(31,234)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,285)(80,286)(81,287)(82,288)(83,289)(84,290)(85,291)(86,292)(87,293)(88,294)(89,295)(90,296)(91,297)(92,298)(93,299)(94,300)(95,301)(96,302)(97,303)(98,304)(99,305)(100,306)(101,307)(102,308)(103,309)(104,310)(105,311)(106,312)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(113,241)(114,242)(115,243)(116,244)(117,245)(118,246)(119,247)(120,248)(121,249)(122,250)(123,251)(124,252)(125,253)(126,254)(127,255)(128,256)(129,257)(130,258)(131,259)(132,260)(133,261)(134,262)(135,263)(136,264)(137,265)(138,266)(139,267)(140,268)(141,269)(142,270)(143,271)(144,272)(145,273)(146,274)(147,275)(148,276)(149,277)(150,278)(151,279)(152,280)(153,281)(154,282)(155,283)(156,284), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(157,286)(158,287)(159,288)(160,289)(161,290)(162,291)(163,292)(164,293)(165,294)(166,295)(167,296)(168,297)(169,298)(170,299)(171,300)(172,301)(173,302)(174,303)(175,304)(176,305)(177,306)(178,307)(179,308)(180,309)(181,310)(182,311)(183,312)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255)(205,256)(206,257)(207,258)(208,259)(209,260)(210,261)(211,262)(212,263)(213,264)(214,265)(215,266)(216,267)(217,268)(218,269)(219,270)(220,271)(221,272)(222,273)(223,274)(224,275)(225,276)(226,277)(227,278)(228,279)(229,280)(230,281)(231,282)(232,283)(233,284)(234,285), (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( (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,217)(15,218)(16,219)(17,220)(18,221)(19,222)(20,223)(21,224)(22,225)(23,226)(24,227)(25,228)(26,229)(27,230)(28,231)(29,232)(30,233)(31,234)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,176)(52,177)(53,178)(54,179)(55,180)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,201)(77,202)(78,203)(79,285)(80,286)(81,287)(82,288)(83,289)(84,290)(85,291)(86,292)(87,293)(88,294)(89,295)(90,296)(91,297)(92,298)(93,299)(94,300)(95,301)(96,302)(97,303)(98,304)(99,305)(100,306)(101,307)(102,308)(103,309)(104,310)(105,311)(106,312)(107,235)(108,236)(109,237)(110,238)(111,239)(112,240)(113,241)(114,242)(115,243)(116,244)(117,245)(118,246)(119,247)(120,248)(121,249)(122,250)(123,251)(124,252)(125,253)(126,254)(127,255)(128,256)(129,257)(130,258)(131,259)(132,260)(133,261)(134,262)(135,263)(136,264)(137,265)(138,266)(139,267)(140,268)(141,269)(142,270)(143,271)(144,272)(145,273)(146,274)(147,275)(148,276)(149,277)(150,278)(151,279)(152,280)(153,281)(154,282)(155,283)(156,284), (1,127)(2,128)(3,129)(4,130)(5,131)(6,132)(7,133)(8,134)(9,135)(10,136)(11,137)(12,138)(13,139)(14,140)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,155)(30,156)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96)(49,97)(50,98)(51,99)(52,100)(53,101)(54,102)(55,103)(56,104)(57,105)(58,106)(59,107)(60,108)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(157,286)(158,287)(159,288)(160,289)(161,290)(162,291)(163,292)(164,293)(165,294)(166,295)(167,296)(168,297)(169,298)(170,299)(171,300)(172,301)(173,302)(174,303)(175,304)(176,305)(177,306)(178,307)(179,308)(180,309)(181,310)(182,311)(183,312)(184,235)(185,236)(186,237)(187,238)(188,239)(189,240)(190,241)(191,242)(192,243)(193,244)(194,245)(195,246)(196,247)(197,248)(198,249)(199,250)(200,251)(201,252)(202,253)(203,254)(204,255)(205,256)(206,257)(207,258)(208,259)(209,260)(210,261)(211,262)(212,263)(213,264)(214,265)(215,266)(216,267)(217,268)(218,269)(219,270)(220,271)(221,272)(222,273)(223,274)(224,275)(225,276)(226,277)(227,278)(228,279)(229,280)(230,281)(231,282)(232,283)(233,284)(234,285), (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([(1,204),(2,205),(3,206),(4,207),(5,208),(6,209),(7,210),(8,211),(9,212),(10,213),(11,214),(12,215),(13,216),(14,217),(15,218),(16,219),(17,220),(18,221),(19,222),(20,223),(21,224),(22,225),(23,226),(24,227),(25,228),(26,229),(27,230),(28,231),(29,232),(30,233),(31,234),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,166),(42,167),(43,168),(44,169),(45,170),(46,171),(47,172),(48,173),(49,174),(50,175),(51,176),(52,177),(53,178),(54,179),(55,180),(56,181),(57,182),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,194),(70,195),(71,196),(72,197),(73,198),(74,199),(75,200),(76,201),(77,202),(78,203),(79,285),(80,286),(81,287),(82,288),(83,289),(84,290),(85,291),(86,292),(87,293),(88,294),(89,295),(90,296),(91,297),(92,298),(93,299),(94,300),(95,301),(96,302),(97,303),(98,304),(99,305),(100,306),(101,307),(102,308),(103,309),(104,310),(105,311),(106,312),(107,235),(108,236),(109,237),(110,238),(111,239),(112,240),(113,241),(114,242),(115,243),(116,244),(117,245),(118,246),(119,247),(120,248),(121,249),(122,250),(123,251),(124,252),(125,253),(126,254),(127,255),(128,256),(129,257),(130,258),(131,259),(132,260),(133,261),(134,262),(135,263),(136,264),(137,265),(138,266),(139,267),(140,268),(141,269),(142,270),(143,271),(144,272),(145,273),(146,274),(147,275),(148,276),(149,277),(150,278),(151,279),(152,280),(153,281),(154,282),(155,283),(156,284)], [(1,127),(2,128),(3,129),(4,130),(5,131),(6,132),(7,133),(8,134),(9,135),(10,136),(11,137),(12,138),(13,139),(14,140),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,155),(30,156),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96),(49,97),(50,98),(51,99),(52,100),(53,101),(54,102),(55,103),(56,104),(57,105),(58,106),(59,107),(60,108),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(157,286),(158,287),(159,288),(160,289),(161,290),(162,291),(163,292),(164,293),(165,294),(166,295),(167,296),(168,297),(169,298),(170,299),(171,300),(172,301),(173,302),(174,303),(175,304),(176,305),(177,306),(178,307),(179,308),(180,309),(181,310),(182,311),(183,312),(184,235),(185,236),(186,237),(187,238),(188,239),(189,240),(190,241),(191,242),(192,243),(193,244),(194,245),(195,246),(196,247),(197,248),(198,249),(199,250),(200,251),(201,252),(202,253),(203,254),(204,255),(205,256),(206,257),(207,258),(208,259),(209,260),(210,261),(211,262),(212,263),(213,264),(214,265),(215,266),(216,267),(217,268),(218,269),(219,270),(220,271),(221,272),(222,273),(223,274),(224,275),(225,276),(226,277),(227,278),(228,279),(229,280),(230,281),(231,282),(232,283),(233,284),(234,285)], [(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

׿
×
𝔽