direct product, abelian, monomial, 2-elementary
Aliases: C22×C74, SmallGroup(296,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C74 |
| C1 — C22×C74 |
| C1 — C22×C74 |
Generators and relations for C22×C74
G = < a,b,c | a2=b2=c74=1, ab=ba, ac=ca, bc=cb >
Smallest permutation representation of C22×C74
►Regular action on 296 points
Generators in S296
(1 219)(2 220)(3 221)(4 222)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 155)(12 156)(13 157)(14 158)(15 159)(16 160)(17 161)(18 162)(19 163)(20 164)(21 165)(22 166)(23 167)(24 168)(25 169)(26 170)(27 171)(28 172)(29 173)(30 174)(31 175)(32 176)(33 177)(34 178)(35 179)(36 180)(37 181)(38 182)(39 183)(40 184)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 191)(48 192)(49 193)(50 194)(51 195)(52 196)(53 197)(54 198)(55 199)(56 200)(57 201)(58 202)(59 203)(60 204)(61 205)(62 206)(63 207)(64 208)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(73 217)(74 218)(75 230)(76 231)(77 232)(78 233)(79 234)(80 235)(81 236)(82 237)(83 238)(84 239)(85 240)(86 241)(87 242)(88 243)(89 244)(90 245)(91 246)(92 247)(93 248)(94 249)(95 250)(96 251)(97 252)(98 253)(99 254)(100 255)(101 256)(102 257)(103 258)(104 259)(105 260)(106 261)(107 262)(108 263)(109 264)(110 265)(111 266)(112 267)(113 268)(114 269)(115 270)(116 271)(117 272)(118 273)(119 274)(120 275)(121 276)(122 277)(123 278)(124 279)(125 280)(126 281)(127 282)(128 283)(129 284)(130 285)(131 286)(132 287)(133 288)(134 289)(135 290)(136 291)(137 292)(138 293)(139 294)(140 295)(141 296)(142 223)(143 224)(144 225)(145 226)(146 227)(147 228)(148 229)
(1 128)(2 129)(3 130)(4 131)(5 132)(6 133)(7 134)(8 135)(9 136)(10 137)(11 138)(12 139)(13 140)(14 141)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 148)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 81)(29 82)(30 83)(31 84)(32 85)(33 86)(34 87)(35 88)(36 89)(37 90)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(46 99)(47 100)(48 101)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(149 287)(150 288)(151 289)(152 290)(153 291)(154 292)(155 293)(156 294)(157 295)(158 296)(159 223)(160 224)(161 225)(162 226)(163 227)(164 228)(165 229)(166 230)(167 231)(168 232)(169 233)(170 234)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 241)(178 242)(179 243)(180 244)(181 245)(182 246)(183 247)(184 248)(185 249)(186 250)(187 251)(188 252)(189 253)(190 254)(191 255)(192 256)(193 257)(194 258)(195 259)(196 260)(197 261)(198 262)(199 263)(200 264)(201 265)(202 266)(203 267)(204 268)(205 269)(206 270)(207 271)(208 272)(209 273)(210 274)(211 275)(212 276)(213 277)(214 278)(215 279)(216 280)(217 281)(218 282)(219 283)(220 284)(221 285)(222 286)
(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)
G:=sub<Sym(296)| (1,219)(2,220)(3,221)(4,222)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,161)(18,162)(19,163)(20,164)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,201)(58,202)(59,203)(60,204)(61,205)(62,206)(63,207)(64,208)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,217)(74,218)(75,230)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,241)(87,242)(88,243)(89,244)(90,245)(91,246)(92,247)(93,248)(94,249)(95,250)(96,251)(97,252)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,268)(114,269)(115,270)(116,271)(117,272)(118,273)(119,274)(120,275)(121,276)(122,277)(123,278)(124,279)(125,280)(126,281)(127,282)(128,283)(129,284)(130,285)(131,286)(132,287)(133,288)(134,289)(135,290)(136,291)(137,292)(138,293)(139,294)(140,295)(141,296)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(149,287)(150,288)(151,289)(152,290)(153,291)(154,292)(155,293)(156,294)(157,295)(158,296)(159,223)(160,224)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)(193,257)(194,258)(195,259)(196,260)(197,261)(198,262)(199,263)(200,264)(201,265)(202,266)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(211,275)(212,276)(213,277)(214,278)(215,279)(216,280)(217,281)(218,282)(219,283)(220,284)(221,285)(222,286), (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)>;
G:=Group( (1,219)(2,220)(3,221)(4,222)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,155)(12,156)(13,157)(14,158)(15,159)(16,160)(17,161)(18,162)(19,163)(20,164)(21,165)(22,166)(23,167)(24,168)(25,169)(26,170)(27,171)(28,172)(29,173)(30,174)(31,175)(32,176)(33,177)(34,178)(35,179)(36,180)(37,181)(38,182)(39,183)(40,184)(41,185)(42,186)(43,187)(44,188)(45,189)(46,190)(47,191)(48,192)(49,193)(50,194)(51,195)(52,196)(53,197)(54,198)(55,199)(56,200)(57,201)(58,202)(59,203)(60,204)(61,205)(62,206)(63,207)(64,208)(65,209)(66,210)(67,211)(68,212)(69,213)(70,214)(71,215)(72,216)(73,217)(74,218)(75,230)(76,231)(77,232)(78,233)(79,234)(80,235)(81,236)(82,237)(83,238)(84,239)(85,240)(86,241)(87,242)(88,243)(89,244)(90,245)(91,246)(92,247)(93,248)(94,249)(95,250)(96,251)(97,252)(98,253)(99,254)(100,255)(101,256)(102,257)(103,258)(104,259)(105,260)(106,261)(107,262)(108,263)(109,264)(110,265)(111,266)(112,267)(113,268)(114,269)(115,270)(116,271)(117,272)(118,273)(119,274)(120,275)(121,276)(122,277)(123,278)(124,279)(125,280)(126,281)(127,282)(128,283)(129,284)(130,285)(131,286)(132,287)(133,288)(134,289)(135,290)(136,291)(137,292)(138,293)(139,294)(140,295)(141,296)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229), (1,128)(2,129)(3,130)(4,131)(5,132)(6,133)(7,134)(8,135)(9,136)(10,137)(11,138)(12,139)(13,140)(14,141)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,148)(22,75)(23,76)(24,77)(25,78)(26,79)(27,80)(28,81)(29,82)(30,83)(31,84)(32,85)(33,86)(34,87)(35,88)(36,89)(37,90)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(46,99)(47,100)(48,101)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(149,287)(150,288)(151,289)(152,290)(153,291)(154,292)(155,293)(156,294)(157,295)(158,296)(159,223)(160,224)(161,225)(162,226)(163,227)(164,228)(165,229)(166,230)(167,231)(168,232)(169,233)(170,234)(171,235)(172,236)(173,237)(174,238)(175,239)(176,240)(177,241)(178,242)(179,243)(180,244)(181,245)(182,246)(183,247)(184,248)(185,249)(186,250)(187,251)(188,252)(189,253)(190,254)(191,255)(192,256)(193,257)(194,258)(195,259)(196,260)(197,261)(198,262)(199,263)(200,264)(201,265)(202,266)(203,267)(204,268)(205,269)(206,270)(207,271)(208,272)(209,273)(210,274)(211,275)(212,276)(213,277)(214,278)(215,279)(216,280)(217,281)(218,282)(219,283)(220,284)(221,285)(222,286), (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) );
G=PermutationGroup([[(1,219),(2,220),(3,221),(4,222),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,155),(12,156),(13,157),(14,158),(15,159),(16,160),(17,161),(18,162),(19,163),(20,164),(21,165),(22,166),(23,167),(24,168),(25,169),(26,170),(27,171),(28,172),(29,173),(30,174),(31,175),(32,176),(33,177),(34,178),(35,179),(36,180),(37,181),(38,182),(39,183),(40,184),(41,185),(42,186),(43,187),(44,188),(45,189),(46,190),(47,191),(48,192),(49,193),(50,194),(51,195),(52,196),(53,197),(54,198),(55,199),(56,200),(57,201),(58,202),(59,203),(60,204),(61,205),(62,206),(63,207),(64,208),(65,209),(66,210),(67,211),(68,212),(69,213),(70,214),(71,215),(72,216),(73,217),(74,218),(75,230),(76,231),(77,232),(78,233),(79,234),(80,235),(81,236),(82,237),(83,238),(84,239),(85,240),(86,241),(87,242),(88,243),(89,244),(90,245),(91,246),(92,247),(93,248),(94,249),(95,250),(96,251),(97,252),(98,253),(99,254),(100,255),(101,256),(102,257),(103,258),(104,259),(105,260),(106,261),(107,262),(108,263),(109,264),(110,265),(111,266),(112,267),(113,268),(114,269),(115,270),(116,271),(117,272),(118,273),(119,274),(120,275),(121,276),(122,277),(123,278),(124,279),(125,280),(126,281),(127,282),(128,283),(129,284),(130,285),(131,286),(132,287),(133,288),(134,289),(135,290),(136,291),(137,292),(138,293),(139,294),(140,295),(141,296),(142,223),(143,224),(144,225),(145,226),(146,227),(147,228),(148,229)], [(1,128),(2,129),(3,130),(4,131),(5,132),(6,133),(7,134),(8,135),(9,136),(10,137),(11,138),(12,139),(13,140),(14,141),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,148),(22,75),(23,76),(24,77),(25,78),(26,79),(27,80),(28,81),(29,82),(30,83),(31,84),(32,85),(33,86),(34,87),(35,88),(36,89),(37,90),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(46,99),(47,100),(48,101),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(149,287),(150,288),(151,289),(152,290),(153,291),(154,292),(155,293),(156,294),(157,295),(158,296),(159,223),(160,224),(161,225),(162,226),(163,227),(164,228),(165,229),(166,230),(167,231),(168,232),(169,233),(170,234),(171,235),(172,236),(173,237),(174,238),(175,239),(176,240),(177,241),(178,242),(179,243),(180,244),(181,245),(182,246),(183,247),(184,248),(185,249),(186,250),(187,251),(188,252),(189,253),(190,254),(191,255),(192,256),(193,257),(194,258),(195,259),(196,260),(197,261),(198,262),(199,263),(200,264),(201,265),(202,266),(203,267),(204,268),(205,269),(206,270),(207,271),(208,272),(209,273),(210,274),(211,275),(212,276),(213,277),(214,278),(215,279),(216,280),(217,281),(218,282),(219,283),(220,284),(221,285),(222,286)], [(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)]])
296 conjugacy classes
| class | 1 | 2A | ··· | 2G | 37A | ··· | 37AJ | 74A | ··· | 74IR |
| order | 1 | 2 | ··· | 2 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
296 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C37 | C74 |
| kernel | C22×C74 | C2×C74 | C23 | C22 |
| # reps | 1 | 7 | 36 | 252 |
Matrix representation of C22×C74 ►in GL3(𝔽149) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 148 |
| 1 | 0 | 0 |
| 0 | 148 | 0 |
| 0 | 0 | 1 |
| 110 | 0 | 0 |
| 0 | 123 | 0 |
| 0 | 0 | 142 |
G:=sub<GL(3,GF(149))| [1,0,0,0,1,0,0,0,148],[1,0,0,0,148,0,0,0,1],[110,0,0,0,123,0,0,0,142] >;
C22×C74 in GAP, Magma, Sage, TeX
C_2^2\times C_{74} % in TeX
G:=Group("C2^2xC74"); // GroupNames label
G:=SmallGroup(296,14);
// by ID
G=gap.SmallGroup(296,14);
# by ID
G:=PCGroup([4,-2,-2,-2,-37]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^74=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations
Export