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## G = C32×C36order 324 = 22·34

### Abelian group of type [3,3,36]

Aliases: C32×C36, SmallGroup(324,105)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C36
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — C32×C18 — C32×C36
 Lower central C1 — C32×C36
 Upper central C1 — C32×C36

Generators and relations for C32×C36
G = < a,b,c | a3=b3=c36=1, ab=ba, ac=ca, bc=cb >

Subgroups: 150, all normal (12 characteristic)
C1, C2, C3, C3, C4, C6, C6, C9, C32, C12, C12, C18, C3×C6, C3×C9, C33, C36, C3×C12, C3×C18, C32×C6, C32×C9, C3×C36, C32×C12, C32×C18, C32×C36
Quotients: C1, C2, C3, C4, C6, C9, C32, C12, C18, C3×C6, C3×C9, C33, C36, C3×C12, C3×C18, C32×C6, C32×C9, C3×C36, C32×C12, C32×C18, C32×C36

Smallest permutation representation of C32×C36
Regular action on 324 points
Generators in S324
(1 314 236)(2 315 237)(3 316 238)(4 317 239)(5 318 240)(6 319 241)(7 320 242)(8 321 243)(9 322 244)(10 323 245)(11 324 246)(12 289 247)(13 290 248)(14 291 249)(15 292 250)(16 293 251)(17 294 252)(18 295 217)(19 296 218)(20 297 219)(21 298 220)(22 299 221)(23 300 222)(24 301 223)(25 302 224)(26 303 225)(27 304 226)(28 305 227)(29 306 228)(30 307 229)(31 308 230)(32 309 231)(33 310 232)(34 311 233)(35 312 234)(36 313 235)(37 285 104)(38 286 105)(39 287 106)(40 288 107)(41 253 108)(42 254 73)(43 255 74)(44 256 75)(45 257 76)(46 258 77)(47 259 78)(48 260 79)(49 261 80)(50 262 81)(51 263 82)(52 264 83)(53 265 84)(54 266 85)(55 267 86)(56 268 87)(57 269 88)(58 270 89)(59 271 90)(60 272 91)(61 273 92)(62 274 93)(63 275 94)(64 276 95)(65 277 96)(66 278 97)(67 279 98)(68 280 99)(69 281 100)(70 282 101)(71 283 102)(72 284 103)(109 181 159)(110 182 160)(111 183 161)(112 184 162)(113 185 163)(114 186 164)(115 187 165)(116 188 166)(117 189 167)(118 190 168)(119 191 169)(120 192 170)(121 193 171)(122 194 172)(123 195 173)(124 196 174)(125 197 175)(126 198 176)(127 199 177)(128 200 178)(129 201 179)(130 202 180)(131 203 145)(132 204 146)(133 205 147)(134 206 148)(135 207 149)(136 208 150)(137 209 151)(138 210 152)(139 211 153)(140 212 154)(141 213 155)(142 214 156)(143 215 157)(144 216 158)
(1 206 60)(2 207 61)(3 208 62)(4 209 63)(5 210 64)(6 211 65)(7 212 66)(8 213 67)(9 214 68)(10 215 69)(11 216 70)(12 181 71)(13 182 72)(14 183 37)(15 184 38)(16 185 39)(17 186 40)(18 187 41)(19 188 42)(20 189 43)(21 190 44)(22 191 45)(23 192 46)(24 193 47)(25 194 48)(26 195 49)(27 196 50)(28 197 51)(29 198 52)(30 199 53)(31 200 54)(32 201 55)(33 202 56)(34 203 57)(35 204 58)(36 205 59)(73 218 116)(74 219 117)(75 220 118)(76 221 119)(77 222 120)(78 223 121)(79 224 122)(80 225 123)(81 226 124)(82 227 125)(83 228 126)(84 229 127)(85 230 128)(86 231 129)(87 232 130)(88 233 131)(89 234 132)(90 235 133)(91 236 134)(92 237 135)(93 238 136)(94 239 137)(95 240 138)(96 241 139)(97 242 140)(98 243 141)(99 244 142)(100 245 143)(101 246 144)(102 247 109)(103 248 110)(104 249 111)(105 250 112)(106 251 113)(107 252 114)(108 217 115)(145 269 311)(146 270 312)(147 271 313)(148 272 314)(149 273 315)(150 274 316)(151 275 317)(152 276 318)(153 277 319)(154 278 320)(155 279 321)(156 280 322)(157 281 323)(158 282 324)(159 283 289)(160 284 290)(161 285 291)(162 286 292)(163 287 293)(164 288 294)(165 253 295)(166 254 296)(167 255 297)(168 256 298)(169 257 299)(170 258 300)(171 259 301)(172 260 302)(173 261 303)(174 262 304)(175 263 305)(176 264 306)(177 265 307)(178 266 308)(179 267 309)(180 268 310)
(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)(289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324)

G:=sub<Sym(324)| (1,314,236)(2,315,237)(3,316,238)(4,317,239)(5,318,240)(6,319,241)(7,320,242)(8,321,243)(9,322,244)(10,323,245)(11,324,246)(12,289,247)(13,290,248)(14,291,249)(15,292,250)(16,293,251)(17,294,252)(18,295,217)(19,296,218)(20,297,219)(21,298,220)(22,299,221)(23,300,222)(24,301,223)(25,302,224)(26,303,225)(27,304,226)(28,305,227)(29,306,228)(30,307,229)(31,308,230)(32,309,231)(33,310,232)(34,311,233)(35,312,234)(36,313,235)(37,285,104)(38,286,105)(39,287,106)(40,288,107)(41,253,108)(42,254,73)(43,255,74)(44,256,75)(45,257,76)(46,258,77)(47,259,78)(48,260,79)(49,261,80)(50,262,81)(51,263,82)(52,264,83)(53,265,84)(54,266,85)(55,267,86)(56,268,87)(57,269,88)(58,270,89)(59,271,90)(60,272,91)(61,273,92)(62,274,93)(63,275,94)(64,276,95)(65,277,96)(66,278,97)(67,279,98)(68,280,99)(69,281,100)(70,282,101)(71,283,102)(72,284,103)(109,181,159)(110,182,160)(111,183,161)(112,184,162)(113,185,163)(114,186,164)(115,187,165)(116,188,166)(117,189,167)(118,190,168)(119,191,169)(120,192,170)(121,193,171)(122,194,172)(123,195,173)(124,196,174)(125,197,175)(126,198,176)(127,199,177)(128,200,178)(129,201,179)(130,202,180)(131,203,145)(132,204,146)(133,205,147)(134,206,148)(135,207,149)(136,208,150)(137,209,151)(138,210,152)(139,211,153)(140,212,154)(141,213,155)(142,214,156)(143,215,157)(144,216,158), (1,206,60)(2,207,61)(3,208,62)(4,209,63)(5,210,64)(6,211,65)(7,212,66)(8,213,67)(9,214,68)(10,215,69)(11,216,70)(12,181,71)(13,182,72)(14,183,37)(15,184,38)(16,185,39)(17,186,40)(18,187,41)(19,188,42)(20,189,43)(21,190,44)(22,191,45)(23,192,46)(24,193,47)(25,194,48)(26,195,49)(27,196,50)(28,197,51)(29,198,52)(30,199,53)(31,200,54)(32,201,55)(33,202,56)(34,203,57)(35,204,58)(36,205,59)(73,218,116)(74,219,117)(75,220,118)(76,221,119)(77,222,120)(78,223,121)(79,224,122)(80,225,123)(81,226,124)(82,227,125)(83,228,126)(84,229,127)(85,230,128)(86,231,129)(87,232,130)(88,233,131)(89,234,132)(90,235,133)(91,236,134)(92,237,135)(93,238,136)(94,239,137)(95,240,138)(96,241,139)(97,242,140)(98,243,141)(99,244,142)(100,245,143)(101,246,144)(102,247,109)(103,248,110)(104,249,111)(105,250,112)(106,251,113)(107,252,114)(108,217,115)(145,269,311)(146,270,312)(147,271,313)(148,272,314)(149,273,315)(150,274,316)(151,275,317)(152,276,318)(153,277,319)(154,278,320)(155,279,321)(156,280,322)(157,281,323)(158,282,324)(159,283,289)(160,284,290)(161,285,291)(162,286,292)(163,287,293)(164,288,294)(165,253,295)(166,254,296)(167,255,297)(168,256,298)(169,257,299)(170,258,300)(171,259,301)(172,260,302)(173,261,303)(174,262,304)(175,263,305)(176,264,306)(177,265,307)(178,266,308)(179,267,309)(180,268,310), (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)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)>;

G:=Group( (1,314,236)(2,315,237)(3,316,238)(4,317,239)(5,318,240)(6,319,241)(7,320,242)(8,321,243)(9,322,244)(10,323,245)(11,324,246)(12,289,247)(13,290,248)(14,291,249)(15,292,250)(16,293,251)(17,294,252)(18,295,217)(19,296,218)(20,297,219)(21,298,220)(22,299,221)(23,300,222)(24,301,223)(25,302,224)(26,303,225)(27,304,226)(28,305,227)(29,306,228)(30,307,229)(31,308,230)(32,309,231)(33,310,232)(34,311,233)(35,312,234)(36,313,235)(37,285,104)(38,286,105)(39,287,106)(40,288,107)(41,253,108)(42,254,73)(43,255,74)(44,256,75)(45,257,76)(46,258,77)(47,259,78)(48,260,79)(49,261,80)(50,262,81)(51,263,82)(52,264,83)(53,265,84)(54,266,85)(55,267,86)(56,268,87)(57,269,88)(58,270,89)(59,271,90)(60,272,91)(61,273,92)(62,274,93)(63,275,94)(64,276,95)(65,277,96)(66,278,97)(67,279,98)(68,280,99)(69,281,100)(70,282,101)(71,283,102)(72,284,103)(109,181,159)(110,182,160)(111,183,161)(112,184,162)(113,185,163)(114,186,164)(115,187,165)(116,188,166)(117,189,167)(118,190,168)(119,191,169)(120,192,170)(121,193,171)(122,194,172)(123,195,173)(124,196,174)(125,197,175)(126,198,176)(127,199,177)(128,200,178)(129,201,179)(130,202,180)(131,203,145)(132,204,146)(133,205,147)(134,206,148)(135,207,149)(136,208,150)(137,209,151)(138,210,152)(139,211,153)(140,212,154)(141,213,155)(142,214,156)(143,215,157)(144,216,158), (1,206,60)(2,207,61)(3,208,62)(4,209,63)(5,210,64)(6,211,65)(7,212,66)(8,213,67)(9,214,68)(10,215,69)(11,216,70)(12,181,71)(13,182,72)(14,183,37)(15,184,38)(16,185,39)(17,186,40)(18,187,41)(19,188,42)(20,189,43)(21,190,44)(22,191,45)(23,192,46)(24,193,47)(25,194,48)(26,195,49)(27,196,50)(28,197,51)(29,198,52)(30,199,53)(31,200,54)(32,201,55)(33,202,56)(34,203,57)(35,204,58)(36,205,59)(73,218,116)(74,219,117)(75,220,118)(76,221,119)(77,222,120)(78,223,121)(79,224,122)(80,225,123)(81,226,124)(82,227,125)(83,228,126)(84,229,127)(85,230,128)(86,231,129)(87,232,130)(88,233,131)(89,234,132)(90,235,133)(91,236,134)(92,237,135)(93,238,136)(94,239,137)(95,240,138)(96,241,139)(97,242,140)(98,243,141)(99,244,142)(100,245,143)(101,246,144)(102,247,109)(103,248,110)(104,249,111)(105,250,112)(106,251,113)(107,252,114)(108,217,115)(145,269,311)(146,270,312)(147,271,313)(148,272,314)(149,273,315)(150,274,316)(151,275,317)(152,276,318)(153,277,319)(154,278,320)(155,279,321)(156,280,322)(157,281,323)(158,282,324)(159,283,289)(160,284,290)(161,285,291)(162,286,292)(163,287,293)(164,288,294)(165,253,295)(166,254,296)(167,255,297)(168,256,298)(169,257,299)(170,258,300)(171,259,301)(172,260,302)(173,261,303)(174,262,304)(175,263,305)(176,264,306)(177,265,307)(178,266,308)(179,267,309)(180,268,310), (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)(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324) );

G=PermutationGroup([[(1,314,236),(2,315,237),(3,316,238),(4,317,239),(5,318,240),(6,319,241),(7,320,242),(8,321,243),(9,322,244),(10,323,245),(11,324,246),(12,289,247),(13,290,248),(14,291,249),(15,292,250),(16,293,251),(17,294,252),(18,295,217),(19,296,218),(20,297,219),(21,298,220),(22,299,221),(23,300,222),(24,301,223),(25,302,224),(26,303,225),(27,304,226),(28,305,227),(29,306,228),(30,307,229),(31,308,230),(32,309,231),(33,310,232),(34,311,233),(35,312,234),(36,313,235),(37,285,104),(38,286,105),(39,287,106),(40,288,107),(41,253,108),(42,254,73),(43,255,74),(44,256,75),(45,257,76),(46,258,77),(47,259,78),(48,260,79),(49,261,80),(50,262,81),(51,263,82),(52,264,83),(53,265,84),(54,266,85),(55,267,86),(56,268,87),(57,269,88),(58,270,89),(59,271,90),(60,272,91),(61,273,92),(62,274,93),(63,275,94),(64,276,95),(65,277,96),(66,278,97),(67,279,98),(68,280,99),(69,281,100),(70,282,101),(71,283,102),(72,284,103),(109,181,159),(110,182,160),(111,183,161),(112,184,162),(113,185,163),(114,186,164),(115,187,165),(116,188,166),(117,189,167),(118,190,168),(119,191,169),(120,192,170),(121,193,171),(122,194,172),(123,195,173),(124,196,174),(125,197,175),(126,198,176),(127,199,177),(128,200,178),(129,201,179),(130,202,180),(131,203,145),(132,204,146),(133,205,147),(134,206,148),(135,207,149),(136,208,150),(137,209,151),(138,210,152),(139,211,153),(140,212,154),(141,213,155),(142,214,156),(143,215,157),(144,216,158)], [(1,206,60),(2,207,61),(3,208,62),(4,209,63),(5,210,64),(6,211,65),(7,212,66),(8,213,67),(9,214,68),(10,215,69),(11,216,70),(12,181,71),(13,182,72),(14,183,37),(15,184,38),(16,185,39),(17,186,40),(18,187,41),(19,188,42),(20,189,43),(21,190,44),(22,191,45),(23,192,46),(24,193,47),(25,194,48),(26,195,49),(27,196,50),(28,197,51),(29,198,52),(30,199,53),(31,200,54),(32,201,55),(33,202,56),(34,203,57),(35,204,58),(36,205,59),(73,218,116),(74,219,117),(75,220,118),(76,221,119),(77,222,120),(78,223,121),(79,224,122),(80,225,123),(81,226,124),(82,227,125),(83,228,126),(84,229,127),(85,230,128),(86,231,129),(87,232,130),(88,233,131),(89,234,132),(90,235,133),(91,236,134),(92,237,135),(93,238,136),(94,239,137),(95,240,138),(96,241,139),(97,242,140),(98,243,141),(99,244,142),(100,245,143),(101,246,144),(102,247,109),(103,248,110),(104,249,111),(105,250,112),(106,251,113),(107,252,114),(108,217,115),(145,269,311),(146,270,312),(147,271,313),(148,272,314),(149,273,315),(150,274,316),(151,275,317),(152,276,318),(153,277,319),(154,278,320),(155,279,321),(156,280,322),(157,281,323),(158,282,324),(159,283,289),(160,284,290),(161,285,291),(162,286,292),(163,287,293),(164,288,294),(165,253,295),(166,254,296),(167,255,297),(168,256,298),(169,257,299),(170,258,300),(171,259,301),(172,260,302),(173,261,303),(174,262,304),(175,263,305),(176,264,306),(177,265,307),(178,266,308),(179,267,309),(180,268,310)], [(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),(289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324)]])

324 conjugacy classes

 class 1 2 3A ··· 3Z 4A 4B 6A ··· 6Z 9A ··· 9BB 12A ··· 12AZ 18A ··· 18BB 36A ··· 36DD order 1 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

324 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C4 C6 C6 C9 C12 C12 C18 C36 kernel C32×C36 C32×C18 C3×C36 C32×C12 C32×C9 C3×C18 C32×C6 C3×C12 C3×C9 C33 C3×C6 C32 # reps 1 1 24 2 2 24 2 54 48 4 54 108

Matrix representation of C32×C36 in GL3(𝔽37) generated by

 26 0 0 0 10 0 0 0 1
,
 10 0 0 0 10 0 0 0 26
,
 22 0 0 0 2 0 0 0 15
G:=sub<GL(3,GF(37))| [26,0,0,0,10,0,0,0,1],[10,0,0,0,10,0,0,0,26],[22,0,0,0,2,0,0,0,15] >;

C32×C36 in GAP, Magma, Sage, TeX

C_3^2\times C_{36}
% in TeX

G:=Group("C3^2xC36");
// GroupNames label

G:=SmallGroup(324,105);
// by ID

G=gap.SmallGroup(324,105);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-3,324,655]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^36=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

׿
×
𝔽