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## G = C23×C36order 288 = 25·32

### Abelian group of type [2,2,2,36]

Aliases: C23×C36, SmallGroup(288,367)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23×C36
 Chief series C1 — C3 — C6 — C18 — C36 — C2×C36 — C22×C36 — C23×C36
 Lower central C1 — C23×C36
 Upper central C1 — C23×C36

Generators and relations for C23×C36
G = < a,b,c,d | a2=b2=c2=d36=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 354, all normal (12 characteristic)
C1, C2, C2 [×14], C3, C4 [×8], C22 [×35], C6, C6 [×14], C2×C4 [×28], C23 [×15], C9, C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C18, C18 [×14], C2×C12 [×28], C22×C6 [×15], C23×C4, C36 [×8], C2×C18 [×35], C22×C12 [×14], C23×C6, C2×C36 [×28], C22×C18 [×15], C23×C12, C22×C36 [×14], C23×C18, C23×C36
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C9, C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C18 [×15], C2×C12 [×28], C22×C6 [×15], C23×C4, C36 [×8], C2×C18 [×35], C22×C12 [×14], C23×C6, C2×C36 [×28], C22×C18 [×15], C23×C12, C22×C36 [×14], C23×C18, C23×C36

Smallest permutation representation of C23×C36
Regular action on 288 points
Generators in S288
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(21 158)(22 159)(23 160)(24 161)(25 162)(26 163)(27 164)(28 165)(29 166)(30 167)(31 168)(32 169)(33 170)(34 171)(35 172)(36 173)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 283)(74 284)(75 285)(76 286)(77 287)(78 288)(79 253)(80 254)(81 255)(82 256)(83 257)(84 258)(85 259)(86 260)(87 261)(88 262)(89 263)(90 264)(91 265)(92 266)(93 267)(94 268)(95 269)(96 270)(97 271)(98 272)(99 273)(100 274)(101 275)(102 276)(103 277)(104 278)(105 279)(106 280)(107 281)(108 282)(181 220)(182 221)(183 222)(184 223)(185 224)(186 225)(187 226)(188 227)(189 228)(190 229)(191 230)(192 231)(193 232)(194 233)(195 234)(196 235)(197 236)(198 237)(199 238)(200 239)(201 240)(202 241)(203 242)(204 243)(205 244)(206 245)(207 246)(208 247)(209 248)(210 249)(211 250)(212 251)(213 252)(214 217)(215 218)(216 219)
(1 262)(2 263)(3 264)(4 265)(5 266)(6 267)(7 268)(8 269)(9 270)(10 271)(11 272)(12 273)(13 274)(14 275)(15 276)(16 277)(17 278)(18 279)(19 280)(20 281)(21 282)(22 283)(23 284)(24 285)(25 286)(26 287)(27 288)(28 253)(29 254)(30 255)(31 256)(32 257)(33 258)(34 259)(35 260)(36 261)(37 215)(38 216)(39 181)(40 182)(41 183)(42 184)(43 185)(44 186)(45 187)(46 188)(47 189)(48 190)(49 191)(50 192)(51 193)(52 194)(53 195)(54 196)(55 197)(56 198)(57 199)(58 200)(59 201)(60 202)(61 203)(62 204)(63 205)(64 206)(65 207)(66 208)(67 209)(68 210)(69 211)(70 212)(71 213)(72 214)(73 159)(74 160)(75 161)(76 162)(77 163)(78 164)(79 165)(80 166)(81 167)(82 168)(83 169)(84 170)(85 171)(86 172)(87 173)(88 174)(89 175)(90 176)(91 177)(92 178)(93 179)(94 180)(95 145)(96 146)(97 147)(98 148)(99 149)(100 150)(101 151)(102 152)(103 153)(104 154)(105 155)(106 156)(107 157)(108 158)(109 232)(110 233)(111 234)(112 235)(113 236)(114 237)(115 238)(116 239)(117 240)(118 241)(119 242)(120 243)(121 244)(122 245)(123 246)(124 247)(125 248)(126 249)(127 250)(128 251)(129 252)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 224)(138 225)(139 226)(140 227)(141 228)(142 229)(143 230)(144 231)
(1 191)(2 192)(3 193)(4 194)(5 195)(6 196)(7 197)(8 198)(9 199)(10 200)(11 201)(12 202)(13 203)(14 204)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 181)(28 182)(29 183)(30 184)(31 185)(32 186)(33 187)(34 188)(35 189)(36 190)(37 286)(38 287)(39 288)(40 253)(41 254)(42 255)(43 256)(44 257)(45 258)(46 259)(47 260)(48 261)(49 262)(50 263)(51 264)(52 265)(53 266)(54 267)(55 268)(56 269)(57 270)(58 271)(59 272)(60 273)(61 274)(62 275)(63 276)(64 277)(65 278)(66 279)(67 280)(68 281)(69 282)(70 283)(71 284)(72 285)(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)(145 237)(146 238)(147 239)(148 240)(149 241)(150 242)(151 243)(152 244)(153 245)(154 246)(155 247)(156 248)(157 249)(158 250)(159 251)(160 252)(161 217)(162 218)(163 219)(164 220)(165 221)(166 222)(167 223)(168 224)(169 225)(170 226)(171 227)(172 228)(173 229)(174 230)(175 231)(176 232)(177 233)(178 234)(179 235)(180 236)
(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)| (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,158)(22,159)(23,160)(24,161)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,169)(33,170)(34,171)(35,172)(36,173)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,283)(74,284)(75,285)(76,286)(77,287)(78,288)(79,253)(80,254)(81,255)(82,256)(83,257)(84,258)(85,259)(86,260)(87,261)(88,262)(89,263)(90,264)(91,265)(92,266)(93,267)(94,268)(95,269)(96,270)(97,271)(98,272)(99,273)(100,274)(101,275)(102,276)(103,277)(104,278)(105,279)(106,280)(107,281)(108,282)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(196,235)(197,236)(198,237)(199,238)(200,239)(201,240)(202,241)(203,242)(204,243)(205,244)(206,245)(207,246)(208,247)(209,248)(210,249)(211,250)(212,251)(213,252)(214,217)(215,218)(216,219), (1,262)(2,263)(3,264)(4,265)(5,266)(6,267)(7,268)(8,269)(9,270)(10,271)(11,272)(12,273)(13,274)(14,275)(15,276)(16,277)(17,278)(18,279)(19,280)(20,281)(21,282)(22,283)(23,284)(24,285)(25,286)(26,287)(27,288)(28,253)(29,254)(30,255)(31,256)(32,257)(33,258)(34,259)(35,260)(36,261)(37,215)(38,216)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,191)(50,192)(51,193)(52,194)(53,195)(54,196)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,211)(70,212)(71,213)(72,214)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)(120,243)(121,244)(122,245)(123,246)(124,247)(125,248)(126,249)(127,250)(128,251)(129,252)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,181)(28,182)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,286)(38,287)(39,288)(40,253)(41,254)(42,255)(43,256)(44,257)(45,258)(46,259)(47,260)(48,261)(49,262)(50,263)(51,264)(52,265)(53,266)(54,267)(55,268)(56,269)(57,270)(58,271)(59,272)(60,273)(61,274)(62,275)(63,276)(64,277)(65,278)(66,279)(67,280)(68,281)(69,282)(70,283)(71,284)(72,285)(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)(145,237)(146,238)(147,239)(148,240)(149,241)(150,242)(151,243)(152,244)(153,245)(154,246)(155,247)(156,248)(157,249)(158,250)(159,251)(160,252)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224)(169,225)(170,226)(171,227)(172,228)(173,229)(174,230)(175,231)(176,232)(177,233)(178,234)(179,235)(180,236), (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( (1,174)(2,175)(3,176)(4,177)(5,178)(6,179)(7,180)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,157)(21,158)(22,159)(23,160)(24,161)(25,162)(26,163)(27,164)(28,165)(29,166)(30,167)(31,168)(32,169)(33,170)(34,171)(35,172)(36,173)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,283)(74,284)(75,285)(76,286)(77,287)(78,288)(79,253)(80,254)(81,255)(82,256)(83,257)(84,258)(85,259)(86,260)(87,261)(88,262)(89,263)(90,264)(91,265)(92,266)(93,267)(94,268)(95,269)(96,270)(97,271)(98,272)(99,273)(100,274)(101,275)(102,276)(103,277)(104,278)(105,279)(106,280)(107,281)(108,282)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(196,235)(197,236)(198,237)(199,238)(200,239)(201,240)(202,241)(203,242)(204,243)(205,244)(206,245)(207,246)(208,247)(209,248)(210,249)(211,250)(212,251)(213,252)(214,217)(215,218)(216,219), (1,262)(2,263)(3,264)(4,265)(5,266)(6,267)(7,268)(8,269)(9,270)(10,271)(11,272)(12,273)(13,274)(14,275)(15,276)(16,277)(17,278)(18,279)(19,280)(20,281)(21,282)(22,283)(23,284)(24,285)(25,286)(26,287)(27,288)(28,253)(29,254)(30,255)(31,256)(32,257)(33,258)(34,259)(35,260)(36,261)(37,215)(38,216)(39,181)(40,182)(41,183)(42,184)(43,185)(44,186)(45,187)(46,188)(47,189)(48,190)(49,191)(50,192)(51,193)(52,194)(53,195)(54,196)(55,197)(56,198)(57,199)(58,200)(59,201)(60,202)(61,203)(62,204)(63,205)(64,206)(65,207)(66,208)(67,209)(68,210)(69,211)(70,212)(71,213)(72,214)(73,159)(74,160)(75,161)(76,162)(77,163)(78,164)(79,165)(80,166)(81,167)(82,168)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,175)(90,176)(91,177)(92,178)(93,179)(94,180)(95,145)(96,146)(97,147)(98,148)(99,149)(100,150)(101,151)(102,152)(103,153)(104,154)(105,155)(106,156)(107,157)(108,158)(109,232)(110,233)(111,234)(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)(120,243)(121,244)(122,245)(123,246)(124,247)(125,248)(126,249)(127,250)(128,251)(129,252)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,224)(138,225)(139,226)(140,227)(141,228)(142,229)(143,230)(144,231), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,181)(28,182)(29,183)(30,184)(31,185)(32,186)(33,187)(34,188)(35,189)(36,190)(37,286)(38,287)(39,288)(40,253)(41,254)(42,255)(43,256)(44,257)(45,258)(46,259)(47,260)(48,261)(49,262)(50,263)(51,264)(52,265)(53,266)(54,267)(55,268)(56,269)(57,270)(58,271)(59,272)(60,273)(61,274)(62,275)(63,276)(64,277)(65,278)(66,279)(67,280)(68,281)(69,282)(70,283)(71,284)(72,285)(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)(145,237)(146,238)(147,239)(148,240)(149,241)(150,242)(151,243)(152,244)(153,245)(154,246)(155,247)(156,248)(157,249)(158,250)(159,251)(160,252)(161,217)(162,218)(163,219)(164,220)(165,221)(166,222)(167,223)(168,224)(169,225)(170,226)(171,227)(172,228)(173,229)(174,230)(175,231)(176,232)(177,233)(178,234)(179,235)(180,236), (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([(1,174),(2,175),(3,176),(4,177),(5,178),(6,179),(7,180),(8,145),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,157),(21,158),(22,159),(23,160),(24,161),(25,162),(26,163),(27,164),(28,165),(29,166),(30,167),(31,168),(32,169),(33,170),(34,171),(35,172),(36,173),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130),(73,283),(74,284),(75,285),(76,286),(77,287),(78,288),(79,253),(80,254),(81,255),(82,256),(83,257),(84,258),(85,259),(86,260),(87,261),(88,262),(89,263),(90,264),(91,265),(92,266),(93,267),(94,268),(95,269),(96,270),(97,271),(98,272),(99,273),(100,274),(101,275),(102,276),(103,277),(104,278),(105,279),(106,280),(107,281),(108,282),(181,220),(182,221),(183,222),(184,223),(185,224),(186,225),(187,226),(188,227),(189,228),(190,229),(191,230),(192,231),(193,232),(194,233),(195,234),(196,235),(197,236),(198,237),(199,238),(200,239),(201,240),(202,241),(203,242),(204,243),(205,244),(206,245),(207,246),(208,247),(209,248),(210,249),(211,250),(212,251),(213,252),(214,217),(215,218),(216,219)], [(1,262),(2,263),(3,264),(4,265),(5,266),(6,267),(7,268),(8,269),(9,270),(10,271),(11,272),(12,273),(13,274),(14,275),(15,276),(16,277),(17,278),(18,279),(19,280),(20,281),(21,282),(22,283),(23,284),(24,285),(25,286),(26,287),(27,288),(28,253),(29,254),(30,255),(31,256),(32,257),(33,258),(34,259),(35,260),(36,261),(37,215),(38,216),(39,181),(40,182),(41,183),(42,184),(43,185),(44,186),(45,187),(46,188),(47,189),(48,190),(49,191),(50,192),(51,193),(52,194),(53,195),(54,196),(55,197),(56,198),(57,199),(58,200),(59,201),(60,202),(61,203),(62,204),(63,205),(64,206),(65,207),(66,208),(67,209),(68,210),(69,211),(70,212),(71,213),(72,214),(73,159),(74,160),(75,161),(76,162),(77,163),(78,164),(79,165),(80,166),(81,167),(82,168),(83,169),(84,170),(85,171),(86,172),(87,173),(88,174),(89,175),(90,176),(91,177),(92,178),(93,179),(94,180),(95,145),(96,146),(97,147),(98,148),(99,149),(100,150),(101,151),(102,152),(103,153),(104,154),(105,155),(106,156),(107,157),(108,158),(109,232),(110,233),(111,234),(112,235),(113,236),(114,237),(115,238),(116,239),(117,240),(118,241),(119,242),(120,243),(121,244),(122,245),(123,246),(124,247),(125,248),(126,249),(127,250),(128,251),(129,252),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,224),(138,225),(139,226),(140,227),(141,228),(142,229),(143,230),(144,231)], [(1,191),(2,192),(3,193),(4,194),(5,195),(6,196),(7,197),(8,198),(9,199),(10,200),(11,201),(12,202),(13,203),(14,204),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,211),(22,212),(23,213),(24,214),(25,215),(26,216),(27,181),(28,182),(29,183),(30,184),(31,185),(32,186),(33,187),(34,188),(35,189),(36,190),(37,286),(38,287),(39,288),(40,253),(41,254),(42,255),(43,256),(44,257),(45,258),(46,259),(47,260),(48,261),(49,262),(50,263),(51,264),(52,265),(53,266),(54,267),(55,268),(56,269),(57,270),(58,271),(59,272),(60,273),(61,274),(62,275),(63,276),(64,277),(65,278),(66,279),(67,280),(68,281),(69,282),(70,283),(71,284),(72,285),(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),(145,237),(146,238),(147,239),(148,240),(149,241),(150,242),(151,243),(152,244),(153,245),(154,246),(155,247),(156,248),(157,249),(158,250),(159,251),(160,252),(161,217),(162,218),(163,219),(164,220),(165,221),(166,222),(167,223),(168,224),(169,225),(170,226),(171,227),(172,228),(173,229),(174,230),(175,231),(176,232),(177,233),(178,234),(179,235),(180,236)], [(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 ··· 2O 3A 3B 4A ··· 4P 6A ··· 6AD 9A ··· 9F 12A ··· 12AF 18A ··· 18CL 36A ··· 36CR 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 C23×C36 C22×C36 C23×C18 C23×C12 C22×C18 C22×C12 C23×C6 C23×C4 C22×C6 C22×C4 C24 C23 # reps 1 14 1 2 16 28 2 6 32 84 6 96

Matrix representation of C23×C36 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 36
,
 36 0 0 0 0 1 0 0 0 0 36 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 31 0 0 0 0 6 0 0 0 0 34 0 0 0 0 6
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,36],[36,0,0,0,0,1,0,0,0,0,36,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[31,0,0,0,0,6,0,0,0,0,34,0,0,0,0,6] >;

C23×C36 in GAP, Magma, Sage, TeX

C_2^3\times C_{36}
% in TeX

G:=Group("C2^3xC36");
// GroupNames label

G:=SmallGroup(288,367);
// by ID

G=gap.SmallGroup(288,367);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,336,242]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^2=d^36=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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