direct product, abelian, monomial, 2-elementary
Aliases: C22×C76, SmallGroup(304,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C76 |
| C1 — C22×C76 |
| C1 — C22×C76 |
Generators and relations for C22×C76
G = < a,b,c | a2=b2=c76=1, ab=ba, ac=ca, bc=cb >
Subgroups: 54, all normal (8 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C19, C38, C38, C76, C2×C38, C2×C76, C22×C38, C22×C76
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C19, C38, C76, C2×C38, C2×C76, C22×C38, C22×C76
►Regular action on 304 points
Generators in S304
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(153 264)(154 265)(155 266)(156 267)(157 268)(158 269)(159 270)(160 271)(161 272)(162 273)(163 274)(164 275)(165 276)(166 277)(167 278)(168 279)(169 280)(170 281)(171 282)(172 283)(173 284)(174 285)(175 286)(176 287)(177 288)(178 289)(179 290)(180 291)(181 292)(182 293)(183 294)(184 295)(185 296)(186 297)(187 298)(188 299)(189 300)(190 301)(191 302)(192 303)(193 304)(194 229)(195 230)(196 231)(197 232)(198 233)(199 234)(200 235)(201 236)(202 237)(203 238)(204 239)(205 240)(206 241)(207 242)(208 243)(209 244)(210 245)(211 246)(212 247)(213 248)(214 249)(215 250)(216 251)(217 252)(218 253)(219 254)(220 255)(221 256)(222 257)(223 258)(224 259)(225 260)(226 261)(227 262)(228 263)
(1 287)(2 288)(3 289)(4 290)(5 291)(6 292)(7 293)(8 294)(9 295)(10 296)(11 297)(12 298)(13 299)(14 300)(15 301)(16 302)(17 303)(18 304)(19 229)(20 230)(21 231)(22 232)(23 233)(24 234)(25 235)(26 236)(27 237)(28 238)(29 239)(30 240)(31 241)(32 242)(33 243)(34 244)(35 245)(36 246)(37 247)(38 248)(39 249)(40 250)(41 251)(42 252)(43 253)(44 254)(45 255)(46 256)(47 257)(48 258)(49 259)(50 260)(51 261)(52 262)(53 263)(54 264)(55 265)(56 266)(57 267)(58 268)(59 269)(60 270)(61 271)(62 272)(63 273)(64 274)(65 275)(66 276)(67 277)(68 278)(69 279)(70 280)(71 281)(72 282)(73 283)(74 284)(75 285)(76 286)(77 166)(78 167)(79 168)(80 169)(81 170)(82 171)(83 172)(84 173)(85 174)(86 175)(87 176)(88 177)(89 178)(90 179)(91 180)(92 181)(93 182)(94 183)(95 184)(96 185)(97 186)(98 187)(99 188)(100 189)(101 190)(102 191)(103 192)(104 193)(105 194)(106 195)(107 196)(108 197)(109 198)(110 199)(111 200)(112 201)(113 202)(114 203)(115 204)(116 205)(117 206)(118 207)(119 208)(120 209)(121 210)(122 211)(123 212)(124 213)(125 214)(126 215)(127 216)(128 217)(129 218)(130 219)(131 220)(132 221)(133 222)(134 223)(135 224)(136 225)(137 226)(138 227)(139 228)(140 153)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 161)(149 162)(150 163)(151 164)(152 165)
(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)
G:=sub<Sym(304)| (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(153,264)(154,265)(155,266)(156,267)(157,268)(158,269)(159,270)(160,271)(161,272)(162,273)(163,274)(164,275)(165,276)(166,277)(167,278)(168,279)(169,280)(170,281)(171,282)(172,283)(173,284)(174,285)(175,286)(176,287)(177,288)(178,289)(179,290)(180,291)(181,292)(182,293)(183,294)(184,295)(185,296)(186,297)(187,298)(188,299)(189,300)(190,301)(191,302)(192,303)(193,304)(194,229)(195,230)(196,231)(197,232)(198,233)(199,234)(200,235)(201,236)(202,237)(203,238)(204,239)(205,240)(206,241)(207,242)(208,243)(209,244)(210,245)(211,246)(212,247)(213,248)(214,249)(215,250)(216,251)(217,252)(218,253)(219,254)(220,255)(221,256)(222,257)(223,258)(224,259)(225,260)(226,261)(227,262)(228,263), (1,287)(2,288)(3,289)(4,290)(5,291)(6,292)(7,293)(8,294)(9,295)(10,296)(11,297)(12,298)(13,299)(14,300)(15,301)(16,302)(17,303)(18,304)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,241)(32,242)(33,243)(34,244)(35,245)(36,246)(37,247)(38,248)(39,249)(40,250)(41,251)(42,252)(43,253)(44,254)(45,255)(46,256)(47,257)(48,258)(49,259)(50,260)(51,261)(52,262)(53,263)(54,264)(55,265)(56,266)(57,267)(58,268)(59,269)(60,270)(61,271)(62,272)(63,273)(64,274)(65,275)(66,276)(67,277)(68,278)(69,279)(70,280)(71,281)(72,282)(73,283)(74,284)(75,285)(76,286)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(109,198)(110,199)(111,200)(112,201)(113,202)(114,203)(115,204)(116,205)(117,206)(118,207)(119,208)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,225)(137,226)(138,227)(139,228)(140,153)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,161)(149,162)(150,163)(151,164)(152,165), (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)>;
G:=Group( (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,127)(42,128)(43,129)(44,130)(45,131)(46,132)(47,133)(48,134)(49,135)(50,136)(51,137)(52,138)(53,139)(54,140)(55,141)(56,142)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,77)(68,78)(69,79)(70,80)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(153,264)(154,265)(155,266)(156,267)(157,268)(158,269)(159,270)(160,271)(161,272)(162,273)(163,274)(164,275)(165,276)(166,277)(167,278)(168,279)(169,280)(170,281)(171,282)(172,283)(173,284)(174,285)(175,286)(176,287)(177,288)(178,289)(179,290)(180,291)(181,292)(182,293)(183,294)(184,295)(185,296)(186,297)(187,298)(188,299)(189,300)(190,301)(191,302)(192,303)(193,304)(194,229)(195,230)(196,231)(197,232)(198,233)(199,234)(200,235)(201,236)(202,237)(203,238)(204,239)(205,240)(206,241)(207,242)(208,243)(209,244)(210,245)(211,246)(212,247)(213,248)(214,249)(215,250)(216,251)(217,252)(218,253)(219,254)(220,255)(221,256)(222,257)(223,258)(224,259)(225,260)(226,261)(227,262)(228,263), (1,287)(2,288)(3,289)(4,290)(5,291)(6,292)(7,293)(8,294)(9,295)(10,296)(11,297)(12,298)(13,299)(14,300)(15,301)(16,302)(17,303)(18,304)(19,229)(20,230)(21,231)(22,232)(23,233)(24,234)(25,235)(26,236)(27,237)(28,238)(29,239)(30,240)(31,241)(32,242)(33,243)(34,244)(35,245)(36,246)(37,247)(38,248)(39,249)(40,250)(41,251)(42,252)(43,253)(44,254)(45,255)(46,256)(47,257)(48,258)(49,259)(50,260)(51,261)(52,262)(53,263)(54,264)(55,265)(56,266)(57,267)(58,268)(59,269)(60,270)(61,271)(62,272)(63,273)(64,274)(65,275)(66,276)(67,277)(68,278)(69,279)(70,280)(71,281)(72,282)(73,283)(74,284)(75,285)(76,286)(77,166)(78,167)(79,168)(80,169)(81,170)(82,171)(83,172)(84,173)(85,174)(86,175)(87,176)(88,177)(89,178)(90,179)(91,180)(92,181)(93,182)(94,183)(95,184)(96,185)(97,186)(98,187)(99,188)(100,189)(101,190)(102,191)(103,192)(104,193)(105,194)(106,195)(107,196)(108,197)(109,198)(110,199)(111,200)(112,201)(113,202)(114,203)(115,204)(116,205)(117,206)(118,207)(119,208)(120,209)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,217)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,225)(137,226)(138,227)(139,228)(140,153)(141,154)(142,155)(143,156)(144,157)(145,158)(146,159)(147,160)(148,161)(149,162)(150,163)(151,164)(152,165), (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) );
G=PermutationGroup([[(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,127),(42,128),(43,129),(44,130),(45,131),(46,132),(47,133),(48,134),(49,135),(50,136),(51,137),(52,138),(53,139),(54,140),(55,141),(56,142),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,77),(68,78),(69,79),(70,80),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(153,264),(154,265),(155,266),(156,267),(157,268),(158,269),(159,270),(160,271),(161,272),(162,273),(163,274),(164,275),(165,276),(166,277),(167,278),(168,279),(169,280),(170,281),(171,282),(172,283),(173,284),(174,285),(175,286),(176,287),(177,288),(178,289),(179,290),(180,291),(181,292),(182,293),(183,294),(184,295),(185,296),(186,297),(187,298),(188,299),(189,300),(190,301),(191,302),(192,303),(193,304),(194,229),(195,230),(196,231),(197,232),(198,233),(199,234),(200,235),(201,236),(202,237),(203,238),(204,239),(205,240),(206,241),(207,242),(208,243),(209,244),(210,245),(211,246),(212,247),(213,248),(214,249),(215,250),(216,251),(217,252),(218,253),(219,254),(220,255),(221,256),(222,257),(223,258),(224,259),(225,260),(226,261),(227,262),(228,263)], [(1,287),(2,288),(3,289),(4,290),(5,291),(6,292),(7,293),(8,294),(9,295),(10,296),(11,297),(12,298),(13,299),(14,300),(15,301),(16,302),(17,303),(18,304),(19,229),(20,230),(21,231),(22,232),(23,233),(24,234),(25,235),(26,236),(27,237),(28,238),(29,239),(30,240),(31,241),(32,242),(33,243),(34,244),(35,245),(36,246),(37,247),(38,248),(39,249),(40,250),(41,251),(42,252),(43,253),(44,254),(45,255),(46,256),(47,257),(48,258),(49,259),(50,260),(51,261),(52,262),(53,263),(54,264),(55,265),(56,266),(57,267),(58,268),(59,269),(60,270),(61,271),(62,272),(63,273),(64,274),(65,275),(66,276),(67,277),(68,278),(69,279),(70,280),(71,281),(72,282),(73,283),(74,284),(75,285),(76,286),(77,166),(78,167),(79,168),(80,169),(81,170),(82,171),(83,172),(84,173),(85,174),(86,175),(87,176),(88,177),(89,178),(90,179),(91,180),(92,181),(93,182),(94,183),(95,184),(96,185),(97,186),(98,187),(99,188),(100,189),(101,190),(102,191),(103,192),(104,193),(105,194),(106,195),(107,196),(108,197),(109,198),(110,199),(111,200),(112,201),(113,202),(114,203),(115,204),(116,205),(117,206),(118,207),(119,208),(120,209),(121,210),(122,211),(123,212),(124,213),(125,214),(126,215),(127,216),(128,217),(129,218),(130,219),(131,220),(132,221),(133,222),(134,223),(135,224),(136,225),(137,226),(138,227),(139,228),(140,153),(141,154),(142,155),(143,156),(144,157),(145,158),(146,159),(147,160),(148,161),(149,162),(150,163),(151,164),(152,165)], [(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)]])
304 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 19A | ··· | 19R | 38A | ··· | 38DV | 76A | ··· | 76EN |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
304 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C19 | C38 | C38 | C76 |
| kernel | C22×C76 | C2×C76 | C22×C38 | C2×C38 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 8 | 18 | 108 | 18 | 144 |
Matrix representation of C22×C76 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 228 |
| 34 | 0 | 0 |
| 0 | 213 | 0 |
| 0 | 0 | 225 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,1,0,0,0,228],[34,0,0,0,213,0,0,0,225] >;
C22×C76 in GAP, Magma, Sage, TeX
C_2^2\times C_{76} % in TeX
G:=Group("C2^2xC76"); // GroupNames label
G:=SmallGroup(304,37);
// by ID
G=gap.SmallGroup(304,37);
# by ID
G:=PCGroup([5,-2,-2,-2,-19,-2,760]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^76=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations