direct product, abelian, monomial, 2-elementary
Aliases: C22×C72, SmallGroup(288,179)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C72 |
| C1 — C22×C72 |
| C1 — C22×C72 |
Generators and relations for C22×C72
G = < a,b,c | a2=b2=c72=1, ab=ba, ac=ca, bc=cb >
Subgroups: 114, all normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C23, C9, C12, C12, C2×C6, C2×C8, C22×C4, C18, C18, C24, C2×C12, C22×C6, C22×C8, C36, C36, C2×C18, C2×C24, C22×C12, C72, C2×C36, C22×C18, C22×C24, C2×C72, C22×C36, C22×C72
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, C9, C12, C2×C6, C2×C8, C22×C4, C18, C24, C2×C12, C22×C6, C22×C8, C36, C2×C18, C2×C24, C22×C12, C72, C2×C36, C22×C18, C22×C24, C2×C72, C22×C36, C22×C72
►Regular action on 288 points
Generators in S288
(1 273)(2 274)(3 275)(4 276)(5 277)(6 278)(7 279)(8 280)(9 281)(10 282)(11 283)(12 284)(13 285)(14 286)(15 287)(16 288)(17 217)(18 218)(19 219)(20 220)(21 221)(22 222)(23 223)(24 224)(25 225)(26 226)(27 227)(28 228)(29 229)(30 230)(31 231)(32 232)(33 233)(34 234)(35 235)(36 236)(37 237)(38 238)(39 239)(40 240)(41 241)(42 242)(43 243)(44 244)(45 245)(46 246)(47 247)(48 248)(49 249)(50 250)(51 251)(52 252)(53 253)(54 254)(55 255)(56 256)(57 257)(58 258)(59 259)(60 260)(61 261)(62 262)(63 263)(64 264)(65 265)(66 266)(67 267)(68 268)(69 269)(70 270)(71 271)(72 272)(73 210)(74 211)(75 212)(76 213)(77 214)(78 215)(79 216)(80 145)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 161)(97 162)(98 163)(99 164)(100 165)(101 166)(102 167)(103 168)(104 169)(105 170)(106 171)(107 172)(108 173)(109 174)(110 175)(111 176)(112 177)(113 178)(114 179)(115 180)(116 181)(117 182)(118 183)(119 184)(120 185)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 193)(129 194)(130 195)(131 196)(132 197)(133 198)(134 199)(135 200)(136 201)(137 202)(138 203)(139 204)(140 205)(141 206)(142 207)(143 208)(144 209)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 135)(22 136)(23 137)(24 138)(25 139)(26 140)(27 141)(28 142)(29 143)(30 144)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(145 238)(146 239)(147 240)(148 241)(149 242)(150 243)(151 244)(152 245)(153 246)(154 247)(155 248)(156 249)(157 250)(158 251)(159 252)(160 253)(161 254)(162 255)(163 256)(164 257)(165 258)(166 259)(167 260)(168 261)(169 262)(170 263)(171 264)(172 265)(173 266)(174 267)(175 268)(176 269)(177 270)(178 271)(179 272)(180 273)(181 274)(182 275)(183 276)(184 277)(185 278)(186 279)(187 280)(188 281)(189 282)(190 283)(191 284)(192 285)(193 286)(194 287)(195 288)(196 217)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 225)(205 226)(206 227)(207 228)(208 229)(209 230)(210 231)(211 232)(212 233)(213 234)(214 235)(215 236)(216 237)
(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)
G:=sub<Sym(288)| (1,273)(2,274)(3,275)(4,276)(5,277)(6,278)(7,279)(8,280)(9,281)(10,282)(11,283)(12,284)(13,285)(14,286)(15,287)(16,288)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,225)(26,226)(27,227)(28,228)(29,229)(30,230)(31,231)(32,232)(33,233)(34,234)(35,235)(36,236)(37,237)(38,238)(39,239)(40,240)(41,241)(42,242)(43,243)(44,244)(45,245)(46,246)(47,247)(48,248)(49,249)(50,250)(51,251)(52,252)(53,253)(54,254)(55,255)(56,256)(57,257)(58,258)(59,259)(60,260)(61,261)(62,262)(63,263)(64,264)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,210)(74,211)(75,212)(76,213)(77,214)(78,215)(79,216)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,197)(133,198)(134,199)(135,200)(136,201)(137,202)(138,203)(139,204)(140,205)(141,206)(142,207)(143,208)(144,209), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(145,238)(146,239)(147,240)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(181,274)(182,275)(183,276)(184,277)(185,278)(186,279)(187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)(194,287)(195,288)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,225)(205,226)(206,227)(207,228)(208,229)(209,230)(210,231)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237), (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)>;
G:=Group( (1,273)(2,274)(3,275)(4,276)(5,277)(6,278)(7,279)(8,280)(9,281)(10,282)(11,283)(12,284)(13,285)(14,286)(15,287)(16,288)(17,217)(18,218)(19,219)(20,220)(21,221)(22,222)(23,223)(24,224)(25,225)(26,226)(27,227)(28,228)(29,229)(30,230)(31,231)(32,232)(33,233)(34,234)(35,235)(36,236)(37,237)(38,238)(39,239)(40,240)(41,241)(42,242)(43,243)(44,244)(45,245)(46,246)(47,247)(48,248)(49,249)(50,250)(51,251)(52,252)(53,253)(54,254)(55,255)(56,256)(57,257)(58,258)(59,259)(60,260)(61,261)(62,262)(63,263)(64,264)(65,265)(66,266)(67,267)(68,268)(69,269)(70,270)(71,271)(72,272)(73,210)(74,211)(75,212)(76,213)(77,214)(78,215)(79,216)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,161)(97,162)(98,163)(99,164)(100,165)(101,166)(102,167)(103,168)(104,169)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182)(118,183)(119,184)(120,185)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,193)(129,194)(130,195)(131,196)(132,197)(133,198)(134,199)(135,200)(136,201)(137,202)(138,203)(139,204)(140,205)(141,206)(142,207)(143,208)(144,209), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,133)(20,134)(21,135)(22,136)(23,137)(24,138)(25,139)(26,140)(27,141)(28,142)(29,143)(30,144)(31,73)(32,74)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,113)(72,114)(145,238)(146,239)(147,240)(148,241)(149,242)(150,243)(151,244)(152,245)(153,246)(154,247)(155,248)(156,249)(157,250)(158,251)(159,252)(160,253)(161,254)(162,255)(163,256)(164,257)(165,258)(166,259)(167,260)(168,261)(169,262)(170,263)(171,264)(172,265)(173,266)(174,267)(175,268)(176,269)(177,270)(178,271)(179,272)(180,273)(181,274)(182,275)(183,276)(184,277)(185,278)(186,279)(187,280)(188,281)(189,282)(190,283)(191,284)(192,285)(193,286)(194,287)(195,288)(196,217)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,225)(205,226)(206,227)(207,228)(208,229)(209,230)(210,231)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237), (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) );
G=PermutationGroup([[(1,273),(2,274),(3,275),(4,276),(5,277),(6,278),(7,279),(8,280),(9,281),(10,282),(11,283),(12,284),(13,285),(14,286),(15,287),(16,288),(17,217),(18,218),(19,219),(20,220),(21,221),(22,222),(23,223),(24,224),(25,225),(26,226),(27,227),(28,228),(29,229),(30,230),(31,231),(32,232),(33,233),(34,234),(35,235),(36,236),(37,237),(38,238),(39,239),(40,240),(41,241),(42,242),(43,243),(44,244),(45,245),(46,246),(47,247),(48,248),(49,249),(50,250),(51,251),(52,252),(53,253),(54,254),(55,255),(56,256),(57,257),(58,258),(59,259),(60,260),(61,261),(62,262),(63,263),(64,264),(65,265),(66,266),(67,267),(68,268),(69,269),(70,270),(71,271),(72,272),(73,210),(74,211),(75,212),(76,213),(77,214),(78,215),(79,216),(80,145),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,161),(97,162),(98,163),(99,164),(100,165),(101,166),(102,167),(103,168),(104,169),(105,170),(106,171),(107,172),(108,173),(109,174),(110,175),(111,176),(112,177),(113,178),(114,179),(115,180),(116,181),(117,182),(118,183),(119,184),(120,185),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,193),(129,194),(130,195),(131,196),(132,197),(133,198),(134,199),(135,200),(136,201),(137,202),(138,203),(139,204),(140,205),(141,206),(142,207),(143,208),(144,209)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,133),(20,134),(21,135),(22,136),(23,137),(24,138),(25,139),(26,140),(27,141),(28,142),(29,143),(30,144),(31,73),(32,74),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,81),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,113),(72,114),(145,238),(146,239),(147,240),(148,241),(149,242),(150,243),(151,244),(152,245),(153,246),(154,247),(155,248),(156,249),(157,250),(158,251),(159,252),(160,253),(161,254),(162,255),(163,256),(164,257),(165,258),(166,259),(167,260),(168,261),(169,262),(170,263),(171,264),(172,265),(173,266),(174,267),(175,268),(176,269),(177,270),(178,271),(179,272),(180,273),(181,274),(182,275),(183,276),(184,277),(185,278),(186,279),(187,280),(188,281),(189,282),(190,283),(191,284),(192,285),(193,286),(194,287),(195,288),(196,217),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,225),(205,226),(206,227),(207,228),(208,229),(209,230),(210,231),(211,232),(212,233),(213,234),(214,235),(215,236),(216,237)], [(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)]])
288 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 4A | ··· | 4H | 6A | ··· | 6N | 8A | ··· | 8P | 9A | ··· | 9F | 12A | ··· | 12P | 18A | ··· | 18AP | 24A | ··· | 24AF | 36A | ··· | 36AV | 72A | ··· | 72CR |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 24 | ··· | 24 | 36 | ··· | 36 | 72 | ··· | 72 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
288 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C9 | C12 | C12 | C18 | C18 | C24 | C36 | C36 | C72 |
| kernel | C22×C72 | C2×C72 | C22×C36 | C22×C24 | C2×C36 | C22×C18 | C2×C24 | C22×C12 | C2×C18 | C22×C8 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C2×C6 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 2 | 6 | 2 | 12 | 2 | 16 | 6 | 12 | 4 | 36 | 6 | 32 | 36 | 12 | 96 |
Matrix representation of C22×C72 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 1 |
| 60 | 0 | 0 |
| 0 | 43 | 0 |
| 0 | 0 | 61 |
G:=sub<GL(3,GF(73))| [1,0,0,0,72,0,0,0,72],[1,0,0,0,72,0,0,0,1],[60,0,0,0,43,0,0,0,61] >;
C22×C72 in GAP, Magma, Sage, TeX
C_2^2\times C_{72} % in TeX
G:=Group("C2^2xC72"); // GroupNames label
G:=SmallGroup(288,179);
// by ID
G=gap.SmallGroup(288,179);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-2,168,192,242]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^72=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations