direct product, abelian, monomial
Aliases: C22×C6×C12, SmallGroup(288,1018)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C6×C12 |
| C1 — C22×C6×C12 |
| C1 — C22×C6×C12 |
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
Generators and relations
G = < a,b,c,d | a2=b2=c6=d12=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >
►Regular action on 288 points
Generators in S288
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 25)(12 26)(13 230)(14 231)(15 232)(16 233)(17 234)(18 235)(19 236)(20 237)(21 238)(22 239)(23 240)(24 229)(37 252)(38 241)(39 242)(40 243)(41 244)(42 245)(43 246)(44 247)(45 248)(46 249)(47 250)(48 251)(49 105)(50 106)(51 107)(52 108)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 171)(62 172)(63 173)(64 174)(65 175)(66 176)(67 177)(68 178)(69 179)(70 180)(71 169)(72 170)(73 228)(74 217)(75 218)(76 219)(77 220)(78 221)(79 222)(80 223)(81 224)(82 225)(83 226)(84 227)(85 196)(86 197)(87 198)(88 199)(89 200)(90 201)(91 202)(92 203)(93 204)(94 193)(95 194)(96 195)(109 156)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 155)(121 259)(122 260)(123 261)(124 262)(125 263)(126 264)(127 253)(128 254)(129 255)(130 256)(131 257)(132 258)(133 286)(134 287)(135 288)(136 277)(137 278)(138 279)(139 280)(140 281)(141 282)(142 283)(143 284)(144 285)(157 192)(158 181)(159 182)(160 183)(161 184)(162 185)(163 186)(164 187)(165 188)(166 189)(167 190)(168 191)(205 265)(206 266)(207 267)(208 268)(209 269)(210 270)(211 271)(212 272)(213 273)(214 274)(215 275)(216 276)
(1 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 177)(26 178)(27 179)(28 180)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 208)(38 209)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 205)(47 206)(48 207)(49 140)(50 141)(51 142)(52 143)(53 144)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(85 254)(86 255)(87 256)(88 257)(89 258)(90 259)(91 260)(92 261)(93 262)(94 263)(95 264)(96 253)(97 285)(98 286)(99 287)(100 288)(101 277)(102 278)(103 279)(104 280)(105 281)(106 282)(107 283)(108 284)(109 184)(110 185)(111 186)(112 187)(113 188)(114 189)(115 190)(116 191)(117 192)(118 181)(119 182)(120 183)(121 201)(122 202)(123 203)(124 204)(125 193)(126 194)(127 195)(128 196)(129 197)(130 198)(131 199)(132 200)(145 162)(146 163)(147 164)(148 165)(149 166)(150 167)(151 168)(152 157)(153 158)(154 159)(155 160)(156 161)(217 231)(218 232)(219 233)(220 234)(221 235)(222 236)(223 237)(224 238)(225 239)(226 240)(227 229)(228 230)(241 269)(242 270)(243 271)(244 272)(245 273)(246 274)(247 275)(248 276)(249 265)(250 266)(251 267)(252 268)
(1 188 105 258 267 223)(2 189 106 259 268 224)(3 190 107 260 269 225)(4 191 108 261 270 226)(5 192 97 262 271 227)(6 181 98 263 272 228)(7 182 99 264 273 217)(8 183 100 253 274 218)(9 184 101 254 275 219)(10 185 102 255 276 220)(11 186 103 256 265 221)(12 187 104 257 266 222)(13 172 153 133 193 41)(14 173 154 134 194 42)(15 174 155 135 195 43)(16 175 156 136 196 44)(17 176 145 137 197 45)(18 177 146 138 198 46)(19 178 147 139 199 47)(20 179 148 140 200 48)(21 180 149 141 201 37)(22 169 150 142 202 38)(23 170 151 143 203 39)(24 171 152 144 204 40)(25 163 59 130 205 78)(26 164 60 131 206 79)(27 165 49 132 207 80)(28 166 50 121 208 81)(29 167 51 122 209 82)(30 168 52 123 210 83)(31 157 53 124 211 84)(32 158 54 125 212 73)(33 159 55 126 213 74)(34 160 56 127 214 75)(35 161 57 128 215 76)(36 162 58 129 216 77)(61 117 285 93 243 229)(62 118 286 94 244 230)(63 119 287 95 245 231)(64 120 288 96 246 232)(65 109 277 85 247 233)(66 110 278 86 248 234)(67 111 279 87 249 235)(68 112 280 88 250 236)(69 113 281 89 251 237)(70 114 282 90 252 238)(71 115 283 91 241 239)(72 116 284 92 242 240)
(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)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,230)(14,231)(15,232)(16,233)(17,234)(18,235)(19,236)(20,237)(21,238)(22,239)(23,240)(24,229)(37,252)(38,241)(39,242)(40,243)(41,244)(42,245)(43,246)(44,247)(45,248)(46,249)(47,250)(48,251)(49,105)(50,106)(51,107)(52,108)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,169)(72,170)(73,228)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,193)(95,194)(96,195)(109,156)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,286)(134,287)(135,288)(136,277)(137,278)(138,279)(139,280)(140,281)(141,282)(142,283)(143,284)(144,285)(157,192)(158,181)(159,182)(160,183)(161,184)(162,185)(163,186)(164,187)(165,188)(166,189)(167,190)(168,191)(205,265)(206,266)(207,267)(208,268)(209,269)(210,270)(211,271)(212,272)(213,273)(214,274)(215,275)(216,276), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,177)(26,178)(27,179)(28,180)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,205)(47,206)(48,207)(49,140)(50,141)(51,142)(52,143)(53,144)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(85,254)(86,255)(87,256)(88,257)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,253)(97,285)(98,286)(99,287)(100,288)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,181)(119,182)(120,183)(121,201)(122,202)(123,203)(124,204)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,168)(152,157)(153,158)(154,159)(155,160)(156,161)(217,231)(218,232)(219,233)(220,234)(221,235)(222,236)(223,237)(224,238)(225,239)(226,240)(227,229)(228,230)(241,269)(242,270)(243,271)(244,272)(245,273)(246,274)(247,275)(248,276)(249,265)(250,266)(251,267)(252,268), (1,188,105,258,267,223)(2,189,106,259,268,224)(3,190,107,260,269,225)(4,191,108,261,270,226)(5,192,97,262,271,227)(6,181,98,263,272,228)(7,182,99,264,273,217)(8,183,100,253,274,218)(9,184,101,254,275,219)(10,185,102,255,276,220)(11,186,103,256,265,221)(12,187,104,257,266,222)(13,172,153,133,193,41)(14,173,154,134,194,42)(15,174,155,135,195,43)(16,175,156,136,196,44)(17,176,145,137,197,45)(18,177,146,138,198,46)(19,178,147,139,199,47)(20,179,148,140,200,48)(21,180,149,141,201,37)(22,169,150,142,202,38)(23,170,151,143,203,39)(24,171,152,144,204,40)(25,163,59,130,205,78)(26,164,60,131,206,79)(27,165,49,132,207,80)(28,166,50,121,208,81)(29,167,51,122,209,82)(30,168,52,123,210,83)(31,157,53,124,211,84)(32,158,54,125,212,73)(33,159,55,126,213,74)(34,160,56,127,214,75)(35,161,57,128,215,76)(36,162,58,129,216,77)(61,117,285,93,243,229)(62,118,286,94,244,230)(63,119,287,95,245,231)(64,120,288,96,246,232)(65,109,277,85,247,233)(66,110,278,86,248,234)(67,111,279,87,249,235)(68,112,280,88,250,236)(69,113,281,89,251,237)(70,114,282,90,252,238)(71,115,283,91,241,239)(72,116,284,92,242,240), (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( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,25)(12,26)(13,230)(14,231)(15,232)(16,233)(17,234)(18,235)(19,236)(20,237)(21,238)(22,239)(23,240)(24,229)(37,252)(38,241)(39,242)(40,243)(41,244)(42,245)(43,246)(44,247)(45,248)(46,249)(47,250)(48,251)(49,105)(50,106)(51,107)(52,108)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,171)(62,172)(63,173)(64,174)(65,175)(66,176)(67,177)(68,178)(69,179)(70,180)(71,169)(72,170)(73,228)(74,217)(75,218)(76,219)(77,220)(78,221)(79,222)(80,223)(81,224)(82,225)(83,226)(84,227)(85,196)(86,197)(87,198)(88,199)(89,200)(90,201)(91,202)(92,203)(93,204)(94,193)(95,194)(96,195)(109,156)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,286)(134,287)(135,288)(136,277)(137,278)(138,279)(139,280)(140,281)(141,282)(142,283)(143,284)(144,285)(157,192)(158,181)(159,182)(160,183)(161,184)(162,185)(163,186)(164,187)(165,188)(166,189)(167,190)(168,191)(205,265)(206,266)(207,267)(208,268)(209,269)(210,270)(211,271)(212,272)(213,273)(214,274)(215,275)(216,276), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,177)(26,178)(27,179)(28,180)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,205)(47,206)(48,207)(49,140)(50,141)(51,142)(52,143)(53,144)(54,133)(55,134)(56,135)(57,136)(58,137)(59,138)(60,139)(85,254)(86,255)(87,256)(88,257)(89,258)(90,259)(91,260)(92,261)(93,262)(94,263)(95,264)(96,253)(97,285)(98,286)(99,287)(100,288)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,184)(110,185)(111,186)(112,187)(113,188)(114,189)(115,190)(116,191)(117,192)(118,181)(119,182)(120,183)(121,201)(122,202)(123,203)(124,204)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)(145,162)(146,163)(147,164)(148,165)(149,166)(150,167)(151,168)(152,157)(153,158)(154,159)(155,160)(156,161)(217,231)(218,232)(219,233)(220,234)(221,235)(222,236)(223,237)(224,238)(225,239)(226,240)(227,229)(228,230)(241,269)(242,270)(243,271)(244,272)(245,273)(246,274)(247,275)(248,276)(249,265)(250,266)(251,267)(252,268), (1,188,105,258,267,223)(2,189,106,259,268,224)(3,190,107,260,269,225)(4,191,108,261,270,226)(5,192,97,262,271,227)(6,181,98,263,272,228)(7,182,99,264,273,217)(8,183,100,253,274,218)(9,184,101,254,275,219)(10,185,102,255,276,220)(11,186,103,256,265,221)(12,187,104,257,266,222)(13,172,153,133,193,41)(14,173,154,134,194,42)(15,174,155,135,195,43)(16,175,156,136,196,44)(17,176,145,137,197,45)(18,177,146,138,198,46)(19,178,147,139,199,47)(20,179,148,140,200,48)(21,180,149,141,201,37)(22,169,150,142,202,38)(23,170,151,143,203,39)(24,171,152,144,204,40)(25,163,59,130,205,78)(26,164,60,131,206,79)(27,165,49,132,207,80)(28,166,50,121,208,81)(29,167,51,122,209,82)(30,168,52,123,210,83)(31,157,53,124,211,84)(32,158,54,125,212,73)(33,159,55,126,213,74)(34,160,56,127,214,75)(35,161,57,128,215,76)(36,162,58,129,216,77)(61,117,285,93,243,229)(62,118,286,94,244,230)(63,119,287,95,245,231)(64,120,288,96,246,232)(65,109,277,85,247,233)(66,110,278,86,248,234)(67,111,279,87,249,235)(68,112,280,88,250,236)(69,113,281,89,251,237)(70,114,282,90,252,238)(71,115,283,91,241,239)(72,116,284,92,242,240), (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([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,25),(12,26),(13,230),(14,231),(15,232),(16,233),(17,234),(18,235),(19,236),(20,237),(21,238),(22,239),(23,240),(24,229),(37,252),(38,241),(39,242),(40,243),(41,244),(42,245),(43,246),(44,247),(45,248),(46,249),(47,250),(48,251),(49,105),(50,106),(51,107),(52,108),(53,97),(54,98),(55,99),(56,100),(57,101),(58,102),(59,103),(60,104),(61,171),(62,172),(63,173),(64,174),(65,175),(66,176),(67,177),(68,178),(69,179),(70,180),(71,169),(72,170),(73,228),(74,217),(75,218),(76,219),(77,220),(78,221),(79,222),(80,223),(81,224),(82,225),(83,226),(84,227),(85,196),(86,197),(87,198),(88,199),(89,200),(90,201),(91,202),(92,203),(93,204),(94,193),(95,194),(96,195),(109,156),(110,145),(111,146),(112,147),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,155),(121,259),(122,260),(123,261),(124,262),(125,263),(126,264),(127,253),(128,254),(129,255),(130,256),(131,257),(132,258),(133,286),(134,287),(135,288),(136,277),(137,278),(138,279),(139,280),(140,281),(141,282),(142,283),(143,284),(144,285),(157,192),(158,181),(159,182),(160,183),(161,184),(162,185),(163,186),(164,187),(165,188),(166,189),(167,190),(168,191),(205,265),(206,266),(207,267),(208,268),(209,269),(210,270),(211,271),(212,272),(213,273),(214,274),(215,275),(216,276)], [(1,69),(2,70),(3,71),(4,72),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,177),(26,178),(27,179),(28,180),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,208),(38,209),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,205),(47,206),(48,207),(49,140),(50,141),(51,142),(52,143),(53,144),(54,133),(55,134),(56,135),(57,136),(58,137),(59,138),(60,139),(85,254),(86,255),(87,256),(88,257),(89,258),(90,259),(91,260),(92,261),(93,262),(94,263),(95,264),(96,253),(97,285),(98,286),(99,287),(100,288),(101,277),(102,278),(103,279),(104,280),(105,281),(106,282),(107,283),(108,284),(109,184),(110,185),(111,186),(112,187),(113,188),(114,189),(115,190),(116,191),(117,192),(118,181),(119,182),(120,183),(121,201),(122,202),(123,203),(124,204),(125,193),(126,194),(127,195),(128,196),(129,197),(130,198),(131,199),(132,200),(145,162),(146,163),(147,164),(148,165),(149,166),(150,167),(151,168),(152,157),(153,158),(154,159),(155,160),(156,161),(217,231),(218,232),(219,233),(220,234),(221,235),(222,236),(223,237),(224,238),(225,239),(226,240),(227,229),(228,230),(241,269),(242,270),(243,271),(244,272),(245,273),(246,274),(247,275),(248,276),(249,265),(250,266),(251,267),(252,268)], [(1,188,105,258,267,223),(2,189,106,259,268,224),(3,190,107,260,269,225),(4,191,108,261,270,226),(5,192,97,262,271,227),(6,181,98,263,272,228),(7,182,99,264,273,217),(8,183,100,253,274,218),(9,184,101,254,275,219),(10,185,102,255,276,220),(11,186,103,256,265,221),(12,187,104,257,266,222),(13,172,153,133,193,41),(14,173,154,134,194,42),(15,174,155,135,195,43),(16,175,156,136,196,44),(17,176,145,137,197,45),(18,177,146,138,198,46),(19,178,147,139,199,47),(20,179,148,140,200,48),(21,180,149,141,201,37),(22,169,150,142,202,38),(23,170,151,143,203,39),(24,171,152,144,204,40),(25,163,59,130,205,78),(26,164,60,131,206,79),(27,165,49,132,207,80),(28,166,50,121,208,81),(29,167,51,122,209,82),(30,168,52,123,210,83),(31,157,53,124,211,84),(32,158,54,125,212,73),(33,159,55,126,213,74),(34,160,56,127,214,75),(35,161,57,128,215,76),(36,162,58,129,216,77),(61,117,285,93,243,229),(62,118,286,94,244,230),(63,119,287,95,245,231),(64,120,288,96,246,232),(65,109,277,85,247,233),(66,110,278,86,248,234),(67,111,279,87,249,235),(68,112,280,88,250,236),(69,113,281,89,251,237),(70,114,282,90,252,238),(71,115,283,91,241,239),(72,116,284,92,242,240)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀