direct product, abelian, monomial, 2-elementary
Aliases: C2×C4×C36, SmallGroup(288,164)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C4×C36 |
| C1 — C2×C4×C36 |
| C1 — C2×C4×C36 |
Generators and relations for C2×C4×C36
G = < a,b,c | a2=b4=c36=1, ab=ba, ac=ca, bc=cb >
Subgroups: 162, all normal (12 characteristic)
C1, C2, C3, C4, C22, C22, C6, C2×C4, C23, C9, C12, C2×C6, C2×C6, C42, C22×C4, C18, C2×C12, C22×C6, C2×C42, C36, C2×C18, C2×C18, C4×C12, C22×C12, C2×C36, C22×C18, C2×C4×C12, C4×C36, C22×C36, C2×C4×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C9, C12, C2×C6, C42, C22×C4, C18, C2×C12, C22×C6, C2×C42, C36, C2×C18, C4×C12, C22×C12, C2×C36, C22×C18, C2×C4×C12, C4×C36, C22×C36, C2×C4×C36
►Regular action on 288 points
Generators in S288
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 199)(15 200)(16 201)(17 202)(18 203)(19 204)(20 205)(21 206)(22 207)(23 208)(24 209)(25 210)(26 211)(27 212)(28 213)(29 214)(30 215)(31 216)(32 181)(33 182)(34 183)(35 184)(36 185)(37 231)(38 232)(39 233)(40 234)(41 235)(42 236)(43 237)(44 238)(45 239)(46 240)(47 241)(48 242)(49 243)(50 244)(51 245)(52 246)(53 247)(54 248)(55 249)(56 250)(57 251)(58 252)(59 217)(60 218)(61 219)(62 220)(63 221)(64 222)(65 223)(66 224)(67 225)(68 226)(69 227)(70 228)(71 229)(72 230)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(145 283)(146 284)(147 285)(148 286)(149 287)(150 288)(151 253)(152 254)(153 255)(154 256)(155 257)(156 258)(157 259)(158 260)(159 261)(160 262)(161 263)(162 264)(163 265)(164 266)(165 267)(166 268)(167 269)(168 270)(169 271)(170 272)(171 273)(172 274)(173 275)(174 276)(175 277)(176 278)(177 279)(178 280)(179 281)(180 282)
(1 176 104 238)(2 177 105 239)(3 178 106 240)(4 179 107 241)(5 180 108 242)(6 145 73 243)(7 146 74 244)(8 147 75 245)(9 148 76 246)(10 149 77 247)(11 150 78 248)(12 151 79 249)(13 152 80 250)(14 153 81 251)(15 154 82 252)(16 155 83 217)(17 156 84 218)(18 157 85 219)(19 158 86 220)(20 159 87 221)(21 160 88 222)(22 161 89 223)(23 162 90 224)(24 163 91 225)(25 164 92 226)(26 165 93 227)(27 166 94 228)(28 167 95 229)(29 168 96 230)(30 169 97 231)(31 170 98 232)(32 171 99 233)(33 172 100 234)(34 173 101 235)(35 174 102 236)(36 175 103 237)(37 215 271 116)(38 216 272 117)(39 181 273 118)(40 182 274 119)(41 183 275 120)(42 184 276 121)(43 185 277 122)(44 186 278 123)(45 187 279 124)(46 188 280 125)(47 189 281 126)(48 190 282 127)(49 191 283 128)(50 192 284 129)(51 193 285 130)(52 194 286 131)(53 195 287 132)(54 196 288 133)(55 197 253 134)(56 198 254 135)(57 199 255 136)(58 200 256 137)(59 201 257 138)(60 202 258 139)(61 203 259 140)(62 204 260 141)(63 205 261 142)(64 206 262 143)(65 207 263 144)(66 208 264 109)(67 209 265 110)(68 210 266 111)(69 211 267 112)(70 212 268 113)(71 213 269 114)(72 214 270 115)
(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)| (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,211)(27,212)(28,213)(29,214)(30,215)(31,216)(32,181)(33,182)(34,183)(35,184)(36,185)(37,231)(38,232)(39,233)(40,234)(41,235)(42,236)(43,237)(44,238)(45,239)(46,240)(47,241)(48,242)(49,243)(50,244)(51,245)(52,246)(53,247)(54,248)(55,249)(56,250)(57,251)(58,252)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(145,283)(146,284)(147,285)(148,286)(149,287)(150,288)(151,253)(152,254)(153,255)(154,256)(155,257)(156,258)(157,259)(158,260)(159,261)(160,262)(161,263)(162,264)(163,265)(164,266)(165,267)(166,268)(167,269)(168,270)(169,271)(170,272)(171,273)(172,274)(173,275)(174,276)(175,277)(176,278)(177,279)(178,280)(179,281)(180,282), (1,176,104,238)(2,177,105,239)(3,178,106,240)(4,179,107,241)(5,180,108,242)(6,145,73,243)(7,146,74,244)(8,147,75,245)(9,148,76,246)(10,149,77,247)(11,150,78,248)(12,151,79,249)(13,152,80,250)(14,153,81,251)(15,154,82,252)(16,155,83,217)(17,156,84,218)(18,157,85,219)(19,158,86,220)(20,159,87,221)(21,160,88,222)(22,161,89,223)(23,162,90,224)(24,163,91,225)(25,164,92,226)(26,165,93,227)(27,166,94,228)(28,167,95,229)(29,168,96,230)(30,169,97,231)(31,170,98,232)(32,171,99,233)(33,172,100,234)(34,173,101,235)(35,174,102,236)(36,175,103,237)(37,215,271,116)(38,216,272,117)(39,181,273,118)(40,182,274,119)(41,183,275,120)(42,184,276,121)(43,185,277,122)(44,186,278,123)(45,187,279,124)(46,188,280,125)(47,189,281,126)(48,190,282,127)(49,191,283,128)(50,192,284,129)(51,193,285,130)(52,194,286,131)(53,195,287,132)(54,196,288,133)(55,197,253,134)(56,198,254,135)(57,199,255,136)(58,200,256,137)(59,201,257,138)(60,202,258,139)(61,203,259,140)(62,204,260,141)(63,205,261,142)(64,206,262,143)(65,207,263,144)(66,208,264,109)(67,209,265,110)(68,210,266,111)(69,211,267,112)(70,212,268,113)(71,213,269,114)(72,214,270,115), (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( (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,197)(13,198)(14,199)(15,200)(16,201)(17,202)(18,203)(19,204)(20,205)(21,206)(22,207)(23,208)(24,209)(25,210)(26,211)(27,212)(28,213)(29,214)(30,215)(31,216)(32,181)(33,182)(34,183)(35,184)(36,185)(37,231)(38,232)(39,233)(40,234)(41,235)(42,236)(43,237)(44,238)(45,239)(46,240)(47,241)(48,242)(49,243)(50,244)(51,245)(52,246)(53,247)(54,248)(55,249)(56,250)(57,251)(58,252)(59,217)(60,218)(61,219)(62,220)(63,221)(64,222)(65,223)(66,224)(67,225)(68,226)(69,227)(70,228)(71,229)(72,230)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(89,144)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(145,283)(146,284)(147,285)(148,286)(149,287)(150,288)(151,253)(152,254)(153,255)(154,256)(155,257)(156,258)(157,259)(158,260)(159,261)(160,262)(161,263)(162,264)(163,265)(164,266)(165,267)(166,268)(167,269)(168,270)(169,271)(170,272)(171,273)(172,274)(173,275)(174,276)(175,277)(176,278)(177,279)(178,280)(179,281)(180,282), (1,176,104,238)(2,177,105,239)(3,178,106,240)(4,179,107,241)(5,180,108,242)(6,145,73,243)(7,146,74,244)(8,147,75,245)(9,148,76,246)(10,149,77,247)(11,150,78,248)(12,151,79,249)(13,152,80,250)(14,153,81,251)(15,154,82,252)(16,155,83,217)(17,156,84,218)(18,157,85,219)(19,158,86,220)(20,159,87,221)(21,160,88,222)(22,161,89,223)(23,162,90,224)(24,163,91,225)(25,164,92,226)(26,165,93,227)(27,166,94,228)(28,167,95,229)(29,168,96,230)(30,169,97,231)(31,170,98,232)(32,171,99,233)(33,172,100,234)(34,173,101,235)(35,174,102,236)(36,175,103,237)(37,215,271,116)(38,216,272,117)(39,181,273,118)(40,182,274,119)(41,183,275,120)(42,184,276,121)(43,185,277,122)(44,186,278,123)(45,187,279,124)(46,188,280,125)(47,189,281,126)(48,190,282,127)(49,191,283,128)(50,192,284,129)(51,193,285,130)(52,194,286,131)(53,195,287,132)(54,196,288,133)(55,197,253,134)(56,198,254,135)(57,199,255,136)(58,200,256,137)(59,201,257,138)(60,202,258,139)(61,203,259,140)(62,204,260,141)(63,205,261,142)(64,206,262,143)(65,207,263,144)(66,208,264,109)(67,209,265,110)(68,210,266,111)(69,211,267,112)(70,212,268,113)(71,213,269,114)(72,214,270,115), (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([[(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,193),(9,194),(10,195),(11,196),(12,197),(13,198),(14,199),(15,200),(16,201),(17,202),(18,203),(19,204),(20,205),(21,206),(22,207),(23,208),(24,209),(25,210),(26,211),(27,212),(28,213),(29,214),(30,215),(31,216),(32,181),(33,182),(34,183),(35,184),(36,185),(37,231),(38,232),(39,233),(40,234),(41,235),(42,236),(43,237),(44,238),(45,239),(46,240),(47,241),(48,242),(49,243),(50,244),(51,245),(52,246),(53,247),(54,248),(55,249),(56,250),(57,251),(58,252),(59,217),(60,218),(61,219),(62,220),(63,221),(64,222),(65,223),(66,224),(67,225),(68,226),(69,227),(70,228),(71,229),(72,230),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(89,144),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124),(106,125),(107,126),(108,127),(145,283),(146,284),(147,285),(148,286),(149,287),(150,288),(151,253),(152,254),(153,255),(154,256),(155,257),(156,258),(157,259),(158,260),(159,261),(160,262),(161,263),(162,264),(163,265),(164,266),(165,267),(166,268),(167,269),(168,270),(169,271),(170,272),(171,273),(172,274),(173,275),(174,276),(175,277),(176,278),(177,279),(178,280),(179,281),(180,282)], [(1,176,104,238),(2,177,105,239),(3,178,106,240),(4,179,107,241),(5,180,108,242),(6,145,73,243),(7,146,74,244),(8,147,75,245),(9,148,76,246),(10,149,77,247),(11,150,78,248),(12,151,79,249),(13,152,80,250),(14,153,81,251),(15,154,82,252),(16,155,83,217),(17,156,84,218),(18,157,85,219),(19,158,86,220),(20,159,87,221),(21,160,88,222),(22,161,89,223),(23,162,90,224),(24,163,91,225),(25,164,92,226),(26,165,93,227),(27,166,94,228),(28,167,95,229),(29,168,96,230),(30,169,97,231),(31,170,98,232),(32,171,99,233),(33,172,100,234),(34,173,101,235),(35,174,102,236),(36,175,103,237),(37,215,271,116),(38,216,272,117),(39,181,273,118),(40,182,274,119),(41,183,275,120),(42,184,276,121),(43,185,277,122),(44,186,278,123),(45,187,279,124),(46,188,280,125),(47,189,281,126),(48,190,282,127),(49,191,283,128),(50,192,284,129),(51,193,285,130),(52,194,286,131),(53,195,287,132),(54,196,288,133),(55,197,253,134),(56,198,254,135),(57,199,255,136),(58,200,256,137),(59,201,257,138),(60,202,258,139),(61,203,259,140),(62,204,260,141),(63,205,261,142),(64,206,262,143),(65,207,263,144),(66,208,264,109),(67,209,265,110),(68,210,266,111),(69,211,267,112),(70,212,268,113),(71,213,269,114),(72,214,270,115)], [(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 | ··· | 4X | 6A | ··· | 6N | 9A | ··· | 9F | 12A | ··· | 12AV | 18A | ··· | 18AP | 36A | ··· | 36EN |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 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 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C36 |
| kernel | C2×C4×C36 | C4×C36 | C22×C36 | C2×C4×C12 | C2×C36 | C4×C12 | C22×C12 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C4 |
| # reps | 1 | 4 | 3 | 2 | 24 | 8 | 6 | 6 | 48 | 24 | 18 | 144 |
Matrix representation of C2×C4×C36 ►in GL3(𝔽37) generated by
| 36 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 29 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 33 |
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[6,0,0,0,6,0,0,0,6],[29,0,0,0,5,0,0,0,33] >;
C2×C4×C36 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_{36} % in TeX
G:=Group("C2xC4xC36"); // GroupNames label
G:=SmallGroup(288,164);
// by ID
G=gap.SmallGroup(288,164);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,168,344,360]);
// Polycyclic
G:=Group<a,b,c|a^2=b^4=c^36=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations