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G = C22×C6×C12order 288 = 25·32

Abelian group of type [2,2,6,12]

Aliases: C22×C6×C12, SmallGroup(288,1018)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6×C12
 Chief series C1 — C2 — C6 — C3×C6 — C3×C12 — C6×C12 — C2×C6×C12 — C22×C6×C12
 Lower central C1 — C22×C6×C12
 Upper central C1 — C22×C6×C12

Generators and relations for C22×C6×C12
G = < a,b,c,d | a2=b2=c6=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 708, all normal (8 characteristic)
C1, C2, C2 [×14], C3 [×4], C4 [×8], C22 [×35], C6 [×60], C2×C4 [×28], C23 [×15], C32, C12 [×32], C2×C6 [×140], C22×C4 [×14], C24, C3×C6, C3×C6 [×14], C2×C12 [×112], C22×C6 [×60], C23×C4, C3×C12 [×8], C62 [×35], C22×C12 [×56], C23×C6 [×4], C6×C12 [×28], C2×C62 [×15], C23×C12 [×4], C2×C6×C12 [×14], C22×C62, C22×C6×C12
Quotients: C1, C2 [×15], C3 [×4], C4 [×8], C22 [×35], C6 [×60], C2×C4 [×28], C23 [×15], C32, C12 [×32], C2×C6 [×140], C22×C4 [×14], C24, C3×C6 [×15], C2×C12 [×112], C22×C6 [×60], C23×C4, C3×C12 [×8], C62 [×35], C22×C12 [×56], C23×C6 [×4], C6×C12 [×28], C2×C62 [×15], C23×C12 [×4], C2×C6×C12 [×14], C22×C62, C22×C6×C12

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

288 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C4 C6 C6 C12 kernel C22×C6×C12 C2×C6×C12 C22×C62 C23×C12 C2×C62 C22×C12 C23×C6 C22×C6 # reps 1 14 1 8 16 112 8 128

Matrix representation of C22×C6×C12 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 12 0 0 0 0 1
,
 4 0 0 0 0 3 0 0 0 0 9 0 0 0 0 4
,
 9 0 0 0 0 4 0 0 0 0 2 0 0 0 0 9
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,1],[4,0,0,0,0,3,0,0,0,0,9,0,0,0,0,4],[9,0,0,0,0,4,0,0,0,0,2,0,0,0,0,9] >;

C22×C6×C12 in GAP, Magma, Sage, TeX

C_2^2\times C_6\times C_{12}
% in TeX

G:=Group("C2^2xC6xC12");
// GroupNames label

G:=SmallGroup(288,1018);
// by ID

G=gap.SmallGroup(288,1018);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,1008]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^6=d^12=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

׿
×
𝔽