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G = C32×C36order 324 = 22·34

Abelian group of type [3,3,36]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C36, SmallGroup(324,105)

Series: Derived Chief Lower central Upper central

C1 — C32×C36
C1C3C6C3×C6C32×C6C32×C18 — C32×C36
C1 — C32×C36
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 [×12], C4, C6, C6 [×12], C9 [×9], C32 [×13], C12, C12 [×12], C18 [×9], C3×C6 [×13], C3×C9 [×12], C33, C36 [×9], C3×C12 [×13], C3×C18 [×12], C32×C6, C32×C9, C3×C36 [×12], C32×C12, C32×C18, C32×C36
Quotients: C1, C2, C3 [×13], C4, C6 [×13], C9 [×9], C32 [×13], C12 [×13], C18 [×9], C3×C6 [×13], C3×C9 [×12], C33, C36 [×9], C3×C12 [×13], C3×C18 [×12], C32×C6, C32×C9, C3×C36 [×12], C32×C12, C32×C18, C32×C36

Smallest permutation representation of C32×C36
Regular action on 324 points
Generators in S324
(1 284 230)(2 285 231)(3 286 232)(4 287 233)(5 288 234)(6 253 235)(7 254 236)(8 255 237)(9 256 238)(10 257 239)(11 258 240)(12 259 241)(13 260 242)(14 261 243)(15 262 244)(16 263 245)(17 264 246)(18 265 247)(19 266 248)(20 267 249)(21 268 250)(22 269 251)(23 270 252)(24 271 217)(25 272 218)(26 273 219)(27 274 220)(28 275 221)(29 276 222)(30 277 223)(31 278 224)(32 279 225)(33 280 226)(34 281 227)(35 282 228)(36 283 229)(37 185 173)(38 186 174)(39 187 175)(40 188 176)(41 189 177)(42 190 178)(43 191 179)(44 192 180)(45 193 145)(46 194 146)(47 195 147)(48 196 148)(49 197 149)(50 198 150)(51 199 151)(52 200 152)(53 201 153)(54 202 154)(55 203 155)(56 204 156)(57 205 157)(58 206 158)(59 207 159)(60 208 160)(61 209 161)(62 210 162)(63 211 163)(64 212 164)(65 213 165)(66 214 166)(67 215 167)(68 216 168)(69 181 169)(70 182 170)(71 183 171)(72 184 172)(73 135 305)(74 136 306)(75 137 307)(76 138 308)(77 139 309)(78 140 310)(79 141 311)(80 142 312)(81 143 313)(82 144 314)(83 109 315)(84 110 316)(85 111 317)(86 112 318)(87 113 319)(88 114 320)(89 115 321)(90 116 322)(91 117 323)(92 118 324)(93 119 289)(94 120 290)(95 121 291)(96 122 292)(97 123 293)(98 124 294)(99 125 295)(100 126 296)(101 127 297)(102 128 298)(103 129 299)(104 130 300)(105 131 301)(106 132 302)(107 133 303)(108 134 304)
(1 82 215)(2 83 216)(3 84 181)(4 85 182)(5 86 183)(6 87 184)(7 88 185)(8 89 186)(9 90 187)(10 91 188)(11 92 189)(12 93 190)(13 94 191)(14 95 192)(15 96 193)(16 97 194)(17 98 195)(18 99 196)(19 100 197)(20 101 198)(21 102 199)(22 103 200)(23 104 201)(24 105 202)(25 106 203)(26 107 204)(27 108 205)(28 73 206)(29 74 207)(30 75 208)(31 76 209)(32 77 210)(33 78 211)(34 79 212)(35 80 213)(36 81 214)(37 236 320)(38 237 321)(39 238 322)(40 239 323)(41 240 324)(42 241 289)(43 242 290)(44 243 291)(45 244 292)(46 245 293)(47 246 294)(48 247 295)(49 248 296)(50 249 297)(51 250 298)(52 251 299)(53 252 300)(54 217 301)(55 218 302)(56 219 303)(57 220 304)(58 221 305)(59 222 306)(60 223 307)(61 224 308)(62 225 309)(63 226 310)(64 227 311)(65 228 312)(66 229 313)(67 230 314)(68 231 315)(69 232 316)(70 233 317)(71 234 318)(72 235 319)(109 168 285)(110 169 286)(111 170 287)(112 171 288)(113 172 253)(114 173 254)(115 174 255)(116 175 256)(117 176 257)(118 177 258)(119 178 259)(120 179 260)(121 180 261)(122 145 262)(123 146 263)(124 147 264)(125 148 265)(126 149 266)(127 150 267)(128 151 268)(129 152 269)(130 153 270)(131 154 271)(132 155 272)(133 156 273)(134 157 274)(135 158 275)(136 159 276)(137 160 277)(138 161 278)(139 162 279)(140 163 280)(141 164 281)(142 165 282)(143 166 283)(144 167 284)
(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)| (1,284,230)(2,285,231)(3,286,232)(4,287,233)(5,288,234)(6,253,235)(7,254,236)(8,255,237)(9,256,238)(10,257,239)(11,258,240)(12,259,241)(13,260,242)(14,261,243)(15,262,244)(16,263,245)(17,264,246)(18,265,247)(19,266,248)(20,267,249)(21,268,250)(22,269,251)(23,270,252)(24,271,217)(25,272,218)(26,273,219)(27,274,220)(28,275,221)(29,276,222)(30,277,223)(31,278,224)(32,279,225)(33,280,226)(34,281,227)(35,282,228)(36,283,229)(37,185,173)(38,186,174)(39,187,175)(40,188,176)(41,189,177)(42,190,178)(43,191,179)(44,192,180)(45,193,145)(46,194,146)(47,195,147)(48,196,148)(49,197,149)(50,198,150)(51,199,151)(52,200,152)(53,201,153)(54,202,154)(55,203,155)(56,204,156)(57,205,157)(58,206,158)(59,207,159)(60,208,160)(61,209,161)(62,210,162)(63,211,163)(64,212,164)(65,213,165)(66,214,166)(67,215,167)(68,216,168)(69,181,169)(70,182,170)(71,183,171)(72,184,172)(73,135,305)(74,136,306)(75,137,307)(76,138,308)(77,139,309)(78,140,310)(79,141,311)(80,142,312)(81,143,313)(82,144,314)(83,109,315)(84,110,316)(85,111,317)(86,112,318)(87,113,319)(88,114,320)(89,115,321)(90,116,322)(91,117,323)(92,118,324)(93,119,289)(94,120,290)(95,121,291)(96,122,292)(97,123,293)(98,124,294)(99,125,295)(100,126,296)(101,127,297)(102,128,298)(103,129,299)(104,130,300)(105,131,301)(106,132,302)(107,133,303)(108,134,304), (1,82,215)(2,83,216)(3,84,181)(4,85,182)(5,86,183)(6,87,184)(7,88,185)(8,89,186)(9,90,187)(10,91,188)(11,92,189)(12,93,190)(13,94,191)(14,95,192)(15,96,193)(16,97,194)(17,98,195)(18,99,196)(19,100,197)(20,101,198)(21,102,199)(22,103,200)(23,104,201)(24,105,202)(25,106,203)(26,107,204)(27,108,205)(28,73,206)(29,74,207)(30,75,208)(31,76,209)(32,77,210)(33,78,211)(34,79,212)(35,80,213)(36,81,214)(37,236,320)(38,237,321)(39,238,322)(40,239,323)(41,240,324)(42,241,289)(43,242,290)(44,243,291)(45,244,292)(46,245,293)(47,246,294)(48,247,295)(49,248,296)(50,249,297)(51,250,298)(52,251,299)(53,252,300)(54,217,301)(55,218,302)(56,219,303)(57,220,304)(58,221,305)(59,222,306)(60,223,307)(61,224,308)(62,225,309)(63,226,310)(64,227,311)(65,228,312)(66,229,313)(67,230,314)(68,231,315)(69,232,316)(70,233,317)(71,234,318)(72,235,319)(109,168,285)(110,169,286)(111,170,287)(112,171,288)(113,172,253)(114,173,254)(115,174,255)(116,175,256)(117,176,257)(118,177,258)(119,178,259)(120,179,260)(121,180,261)(122,145,262)(123,146,263)(124,147,264)(125,148,265)(126,149,266)(127,150,267)(128,151,268)(129,152,269)(130,153,270)(131,154,271)(132,155,272)(133,156,273)(134,157,274)(135,158,275)(136,159,276)(137,160,277)(138,161,278)(139,162,279)(140,163,280)(141,164,281)(142,165,282)(143,166,283)(144,167,284), (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( (1,284,230)(2,285,231)(3,286,232)(4,287,233)(5,288,234)(6,253,235)(7,254,236)(8,255,237)(9,256,238)(10,257,239)(11,258,240)(12,259,241)(13,260,242)(14,261,243)(15,262,244)(16,263,245)(17,264,246)(18,265,247)(19,266,248)(20,267,249)(21,268,250)(22,269,251)(23,270,252)(24,271,217)(25,272,218)(26,273,219)(27,274,220)(28,275,221)(29,276,222)(30,277,223)(31,278,224)(32,279,225)(33,280,226)(34,281,227)(35,282,228)(36,283,229)(37,185,173)(38,186,174)(39,187,175)(40,188,176)(41,189,177)(42,190,178)(43,191,179)(44,192,180)(45,193,145)(46,194,146)(47,195,147)(48,196,148)(49,197,149)(50,198,150)(51,199,151)(52,200,152)(53,201,153)(54,202,154)(55,203,155)(56,204,156)(57,205,157)(58,206,158)(59,207,159)(60,208,160)(61,209,161)(62,210,162)(63,211,163)(64,212,164)(65,213,165)(66,214,166)(67,215,167)(68,216,168)(69,181,169)(70,182,170)(71,183,171)(72,184,172)(73,135,305)(74,136,306)(75,137,307)(76,138,308)(77,139,309)(78,140,310)(79,141,311)(80,142,312)(81,143,313)(82,144,314)(83,109,315)(84,110,316)(85,111,317)(86,112,318)(87,113,319)(88,114,320)(89,115,321)(90,116,322)(91,117,323)(92,118,324)(93,119,289)(94,120,290)(95,121,291)(96,122,292)(97,123,293)(98,124,294)(99,125,295)(100,126,296)(101,127,297)(102,128,298)(103,129,299)(104,130,300)(105,131,301)(106,132,302)(107,133,303)(108,134,304), (1,82,215)(2,83,216)(3,84,181)(4,85,182)(5,86,183)(6,87,184)(7,88,185)(8,89,186)(9,90,187)(10,91,188)(11,92,189)(12,93,190)(13,94,191)(14,95,192)(15,96,193)(16,97,194)(17,98,195)(18,99,196)(19,100,197)(20,101,198)(21,102,199)(22,103,200)(23,104,201)(24,105,202)(25,106,203)(26,107,204)(27,108,205)(28,73,206)(29,74,207)(30,75,208)(31,76,209)(32,77,210)(33,78,211)(34,79,212)(35,80,213)(36,81,214)(37,236,320)(38,237,321)(39,238,322)(40,239,323)(41,240,324)(42,241,289)(43,242,290)(44,243,291)(45,244,292)(46,245,293)(47,246,294)(48,247,295)(49,248,296)(50,249,297)(51,250,298)(52,251,299)(53,252,300)(54,217,301)(55,218,302)(56,219,303)(57,220,304)(58,221,305)(59,222,306)(60,223,307)(61,224,308)(62,225,309)(63,226,310)(64,227,311)(65,228,312)(66,229,313)(67,230,314)(68,231,315)(69,232,316)(70,233,317)(71,234,318)(72,235,319)(109,168,285)(110,169,286)(111,170,287)(112,171,288)(113,172,253)(114,173,254)(115,174,255)(116,175,256)(117,176,257)(118,177,258)(119,178,259)(120,179,260)(121,180,261)(122,145,262)(123,146,263)(124,147,264)(125,148,265)(126,149,266)(127,150,267)(128,151,268)(129,152,269)(130,153,270)(131,154,271)(132,155,272)(133,156,273)(134,157,274)(135,158,275)(136,159,276)(137,160,277)(138,161,278)(139,162,279)(140,163,280)(141,164,281)(142,165,282)(143,166,283)(144,167,284), (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([(1,284,230),(2,285,231),(3,286,232),(4,287,233),(5,288,234),(6,253,235),(7,254,236),(8,255,237),(9,256,238),(10,257,239),(11,258,240),(12,259,241),(13,260,242),(14,261,243),(15,262,244),(16,263,245),(17,264,246),(18,265,247),(19,266,248),(20,267,249),(21,268,250),(22,269,251),(23,270,252),(24,271,217),(25,272,218),(26,273,219),(27,274,220),(28,275,221),(29,276,222),(30,277,223),(31,278,224),(32,279,225),(33,280,226),(34,281,227),(35,282,228),(36,283,229),(37,185,173),(38,186,174),(39,187,175),(40,188,176),(41,189,177),(42,190,178),(43,191,179),(44,192,180),(45,193,145),(46,194,146),(47,195,147),(48,196,148),(49,197,149),(50,198,150),(51,199,151),(52,200,152),(53,201,153),(54,202,154),(55,203,155),(56,204,156),(57,205,157),(58,206,158),(59,207,159),(60,208,160),(61,209,161),(62,210,162),(63,211,163),(64,212,164),(65,213,165),(66,214,166),(67,215,167),(68,216,168),(69,181,169),(70,182,170),(71,183,171),(72,184,172),(73,135,305),(74,136,306),(75,137,307),(76,138,308),(77,139,309),(78,140,310),(79,141,311),(80,142,312),(81,143,313),(82,144,314),(83,109,315),(84,110,316),(85,111,317),(86,112,318),(87,113,319),(88,114,320),(89,115,321),(90,116,322),(91,117,323),(92,118,324),(93,119,289),(94,120,290),(95,121,291),(96,122,292),(97,123,293),(98,124,294),(99,125,295),(100,126,296),(101,127,297),(102,128,298),(103,129,299),(104,130,300),(105,131,301),(106,132,302),(107,133,303),(108,134,304)], [(1,82,215),(2,83,216),(3,84,181),(4,85,182),(5,86,183),(6,87,184),(7,88,185),(8,89,186),(9,90,187),(10,91,188),(11,92,189),(12,93,190),(13,94,191),(14,95,192),(15,96,193),(16,97,194),(17,98,195),(18,99,196),(19,100,197),(20,101,198),(21,102,199),(22,103,200),(23,104,201),(24,105,202),(25,106,203),(26,107,204),(27,108,205),(28,73,206),(29,74,207),(30,75,208),(31,76,209),(32,77,210),(33,78,211),(34,79,212),(35,80,213),(36,81,214),(37,236,320),(38,237,321),(39,238,322),(40,239,323),(41,240,324),(42,241,289),(43,242,290),(44,243,291),(45,244,292),(46,245,293),(47,246,294),(48,247,295),(49,248,296),(50,249,297),(51,250,298),(52,251,299),(53,252,300),(54,217,301),(55,218,302),(56,219,303),(57,220,304),(58,221,305),(59,222,306),(60,223,307),(61,224,308),(62,225,309),(63,226,310),(64,227,311),(65,228,312),(66,229,313),(67,230,314),(68,231,315),(69,232,316),(70,233,317),(71,234,318),(72,235,319),(109,168,285),(110,169,286),(111,170,287),(112,171,288),(113,172,253),(114,173,254),(115,174,255),(116,175,256),(117,176,257),(118,177,258),(119,178,259),(120,179,260),(121,180,261),(122,145,262),(123,146,263),(124,147,264),(125,148,265),(126,149,266),(127,150,267),(128,151,268),(129,152,269),(130,153,270),(131,154,271),(132,155,272),(133,156,273),(134,157,274),(135,158,275),(136,159,276),(137,160,277),(138,161,278),(139,162,279),(140,163,280),(141,164,281),(142,165,282),(143,166,283),(144,167,284)], [(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···3Z4A4B6A···6Z9A···9BB12A···12AZ18A···18BB36A···36DD
order123···3446···69···912···1218···1836···36
size111···1111···11···11···11···11···1

324 irreducible representations

dim111111111111
type++
imageC1C2C3C3C4C6C6C9C12C12C18C36
kernelC32×C36C32×C18C3×C36C32×C12C32×C9C3×C18C32×C6C3×C12C3×C9C33C3×C6C32
# reps1124222425448454108

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

2600
0100
001
,
1000
0100
0026
,
2200
020
0015
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