direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C4×Dic9, C18⋊C42, C23.34D18, (C2×C36)⋊7C4, C36⋊8(C2×C4), C9⋊2(C2×C42), (C2×C12).413D6, (C2×C4).101D18, C6.11(C4×Dic3), C22.15(C4×D9), (C22×C4).12D9, (C2×C18).40C23, (C22×C12).33S3, (C22×C36).13C2, C18.17(C22×C4), C12.55(C2×Dic3), (C2×C12).23Dic3, (C22×C6).134D6, C2.2(C22×Dic9), (C2×C36).111C22, (C22×Dic9).7C2, C6.23(C22×Dic3), C22.13(C2×Dic9), C22.19(C22×D9), (C22×C18).32C22, (C2×Dic9).48C22, C2.3(C2×C4×D9), C3.(C2×C4×Dic3), C6.56(S3×C2×C4), (C2×C6).42(C4×S3), (C2×C18).16(C2×C4), (C2×C6).38(C2×Dic3), (C2×C6).197(C22×S3), SmallGroup(288,132)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C2×C4×Dic9 |
Generators and relations for C2×C4×Dic9
G = < a,b,c,d | a2=b4=c18=1, d2=c9, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 432 in 162 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C9, Dic3, C12, C2×C6, C2×C6, C42, C22×C4, C22×C4, C18, C18, C2×Dic3, C2×C12, C22×C6, C2×C42, Dic9, C36, C2×C18, C2×C18, C4×Dic3, C22×Dic3, C22×C12, C2×Dic9, C2×C36, C22×C18, C2×C4×Dic3, C4×Dic9, C22×Dic9, C22×C36, C2×C4×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, D9, C4×S3, C2×Dic3, C22×S3, C2×C42, Dic9, D18, C4×Dic3, S3×C2×C4, C22×Dic3, C4×D9, C2×Dic9, C22×D9, C2×C4×Dic3, C4×Dic9, C2×C4×D9, C22×Dic9, C2×C4×Dic9
►Regular action on 288 points
Generators in S288
(1 68)(2 69)(3 70)(4 71)(5 72)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 109)(53 110)(54 111)(91 192)(92 193)(93 194)(94 195)(95 196)(96 197)(97 198)(98 181)(99 182)(100 183)(101 184)(102 185)(103 186)(104 187)(105 188)(106 189)(107 190)(108 191)(127 205)(128 206)(129 207)(130 208)(131 209)(132 210)(133 211)(134 212)(135 213)(136 214)(137 215)(138 216)(139 199)(140 200)(141 201)(142 202)(143 203)(144 204)(145 265)(146 266)(147 267)(148 268)(149 269)(150 270)(151 253)(152 254)(153 255)(154 256)(155 257)(156 258)(157 259)(158 260)(159 261)(160 262)(161 263)(162 264)(163 249)(164 250)(165 251)(166 252)(167 235)(168 236)(169 237)(170 238)(171 239)(172 240)(173 241)(174 242)(175 243)(176 244)(177 245)(178 246)(179 247)(180 248)(217 271)(218 272)(219 273)(220 274)(221 275)(222 276)(223 277)(224 278)(225 279)(226 280)(227 281)(228 282)(229 283)(230 284)(231 285)(232 286)(233 287)(234 288)
(1 171 147 227)(2 172 148 228)(3 173 149 229)(4 174 150 230)(5 175 151 231)(6 176 152 232)(7 177 153 233)(8 178 154 234)(9 179 155 217)(10 180 156 218)(11 163 157 219)(12 164 158 220)(13 165 159 221)(14 166 160 222)(15 167 161 223)(16 168 162 224)(17 169 145 225)(18 170 146 226)(19 199 107 54)(20 200 108 37)(21 201 91 38)(22 202 92 39)(23 203 93 40)(24 204 94 41)(25 205 95 42)(26 206 96 43)(27 207 97 44)(28 208 98 45)(29 209 99 46)(30 210 100 47)(31 211 101 48)(32 212 102 49)(33 213 103 50)(34 214 104 51)(35 215 105 52)(36 216 106 53)(55 244 254 286)(56 245 255 287)(57 246 256 288)(58 247 257 271)(59 248 258 272)(60 249 259 273)(61 250 260 274)(62 251 261 275)(63 252 262 276)(64 235 263 277)(65 236 264 278)(66 237 265 279)(67 238 266 280)(68 239 267 281)(69 240 268 282)(70 241 269 283)(71 242 270 284)(72 243 253 285)(73 127 196 117)(74 128 197 118)(75 129 198 119)(76 130 181 120)(77 131 182 121)(78 132 183 122)(79 133 184 123)(80 134 185 124)(81 135 186 125)(82 136 187 126)(83 137 188 109)(84 138 189 110)(85 139 190 111)(86 140 191 112)(87 141 192 113)(88 142 193 114)(89 143 194 115)(90 144 195 116)
(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)
(1 185 10 194)(2 184 11 193)(3 183 12 192)(4 182 13 191)(5 181 14 190)(6 198 15 189)(7 197 16 188)(8 196 17 187)(9 195 18 186)(19 253 28 262)(20 270 29 261)(21 269 30 260)(22 268 31 259)(23 267 32 258)(24 266 33 257)(25 265 34 256)(26 264 35 255)(27 263 36 254)(37 242 46 251)(38 241 47 250)(39 240 48 249)(40 239 49 248)(41 238 50 247)(42 237 51 246)(43 236 52 245)(44 235 53 244)(45 252 54 243)(55 97 64 106)(56 96 65 105)(57 95 66 104)(58 94 67 103)(59 93 68 102)(60 92 69 101)(61 91 70 100)(62 108 71 99)(63 107 72 98)(73 145 82 154)(74 162 83 153)(75 161 84 152)(76 160 85 151)(77 159 86 150)(78 158 87 149)(79 157 88 148)(80 156 89 147)(81 155 90 146)(109 177 118 168)(110 176 119 167)(111 175 120 166)(112 174 121 165)(113 173 122 164)(114 172 123 163)(115 171 124 180)(116 170 125 179)(117 169 126 178)(127 225 136 234)(128 224 137 233)(129 223 138 232)(130 222 139 231)(131 221 140 230)(132 220 141 229)(133 219 142 228)(134 218 143 227)(135 217 144 226)(199 285 208 276)(200 284 209 275)(201 283 210 274)(202 282 211 273)(203 281 212 272)(204 280 213 271)(205 279 214 288)(206 278 215 287)(207 277 216 286)
G:=sub<Sym(288)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,109)(53,110)(54,111)(91,192)(92,193)(93,194)(94,195)(95,196)(96,197)(97,198)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,215)(138,216)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,265)(146,266)(147,267)(148,268)(149,269)(150,270)(151,253)(152,254)(153,255)(154,256)(155,257)(156,258)(157,259)(158,260)(159,261)(160,262)(161,263)(162,264)(163,249)(164,250)(165,251)(166,252)(167,235)(168,236)(169,237)(170,238)(171,239)(172,240)(173,241)(174,242)(175,243)(176,244)(177,245)(178,246)(179,247)(180,248)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,281)(228,282)(229,283)(230,284)(231,285)(232,286)(233,287)(234,288), (1,171,147,227)(2,172,148,228)(3,173,149,229)(4,174,150,230)(5,175,151,231)(6,176,152,232)(7,177,153,233)(8,178,154,234)(9,179,155,217)(10,180,156,218)(11,163,157,219)(12,164,158,220)(13,165,159,221)(14,166,160,222)(15,167,161,223)(16,168,162,224)(17,169,145,225)(18,170,146,226)(19,199,107,54)(20,200,108,37)(21,201,91,38)(22,202,92,39)(23,203,93,40)(24,204,94,41)(25,205,95,42)(26,206,96,43)(27,207,97,44)(28,208,98,45)(29,209,99,46)(30,210,100,47)(31,211,101,48)(32,212,102,49)(33,213,103,50)(34,214,104,51)(35,215,105,52)(36,216,106,53)(55,244,254,286)(56,245,255,287)(57,246,256,288)(58,247,257,271)(59,248,258,272)(60,249,259,273)(61,250,260,274)(62,251,261,275)(63,252,262,276)(64,235,263,277)(65,236,264,278)(66,237,265,279)(67,238,266,280)(68,239,267,281)(69,240,268,282)(70,241,269,283)(71,242,270,284)(72,243,253,285)(73,127,196,117)(74,128,197,118)(75,129,198,119)(76,130,181,120)(77,131,182,121)(78,132,183,122)(79,133,184,123)(80,134,185,124)(81,135,186,125)(82,136,187,126)(83,137,188,109)(84,138,189,110)(85,139,190,111)(86,140,191,112)(87,141,192,113)(88,142,193,114)(89,143,194,115)(90,144,195,116), (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), (1,185,10,194)(2,184,11,193)(3,183,12,192)(4,182,13,191)(5,181,14,190)(6,198,15,189)(7,197,16,188)(8,196,17,187)(9,195,18,186)(19,253,28,262)(20,270,29,261)(21,269,30,260)(22,268,31,259)(23,267,32,258)(24,266,33,257)(25,265,34,256)(26,264,35,255)(27,263,36,254)(37,242,46,251)(38,241,47,250)(39,240,48,249)(40,239,49,248)(41,238,50,247)(42,237,51,246)(43,236,52,245)(44,235,53,244)(45,252,54,243)(55,97,64,106)(56,96,65,105)(57,95,66,104)(58,94,67,103)(59,93,68,102)(60,92,69,101)(61,91,70,100)(62,108,71,99)(63,107,72,98)(73,145,82,154)(74,162,83,153)(75,161,84,152)(76,160,85,151)(77,159,86,150)(78,158,87,149)(79,157,88,148)(80,156,89,147)(81,155,90,146)(109,177,118,168)(110,176,119,167)(111,175,120,166)(112,174,121,165)(113,173,122,164)(114,172,123,163)(115,171,124,180)(116,170,125,179)(117,169,126,178)(127,225,136,234)(128,224,137,233)(129,223,138,232)(130,222,139,231)(131,221,140,230)(132,220,141,229)(133,219,142,228)(134,218,143,227)(135,217,144,226)(199,285,208,276)(200,284,209,275)(201,283,210,274)(202,282,211,273)(203,281,212,272)(204,280,213,271)(205,279,214,288)(206,278,215,287)(207,277,216,286)>;
G:=Group( (1,68)(2,69)(3,70)(4,71)(5,72)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,109)(53,110)(54,111)(91,192)(92,193)(93,194)(94,195)(95,196)(96,197)(97,198)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(127,205)(128,206)(129,207)(130,208)(131,209)(132,210)(133,211)(134,212)(135,213)(136,214)(137,215)(138,216)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)(145,265)(146,266)(147,267)(148,268)(149,269)(150,270)(151,253)(152,254)(153,255)(154,256)(155,257)(156,258)(157,259)(158,260)(159,261)(160,262)(161,263)(162,264)(163,249)(164,250)(165,251)(166,252)(167,235)(168,236)(169,237)(170,238)(171,239)(172,240)(173,241)(174,242)(175,243)(176,244)(177,245)(178,246)(179,247)(180,248)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,281)(228,282)(229,283)(230,284)(231,285)(232,286)(233,287)(234,288), (1,171,147,227)(2,172,148,228)(3,173,149,229)(4,174,150,230)(5,175,151,231)(6,176,152,232)(7,177,153,233)(8,178,154,234)(9,179,155,217)(10,180,156,218)(11,163,157,219)(12,164,158,220)(13,165,159,221)(14,166,160,222)(15,167,161,223)(16,168,162,224)(17,169,145,225)(18,170,146,226)(19,199,107,54)(20,200,108,37)(21,201,91,38)(22,202,92,39)(23,203,93,40)(24,204,94,41)(25,205,95,42)(26,206,96,43)(27,207,97,44)(28,208,98,45)(29,209,99,46)(30,210,100,47)(31,211,101,48)(32,212,102,49)(33,213,103,50)(34,214,104,51)(35,215,105,52)(36,216,106,53)(55,244,254,286)(56,245,255,287)(57,246,256,288)(58,247,257,271)(59,248,258,272)(60,249,259,273)(61,250,260,274)(62,251,261,275)(63,252,262,276)(64,235,263,277)(65,236,264,278)(66,237,265,279)(67,238,266,280)(68,239,267,281)(69,240,268,282)(70,241,269,283)(71,242,270,284)(72,243,253,285)(73,127,196,117)(74,128,197,118)(75,129,198,119)(76,130,181,120)(77,131,182,121)(78,132,183,122)(79,133,184,123)(80,134,185,124)(81,135,186,125)(82,136,187,126)(83,137,188,109)(84,138,189,110)(85,139,190,111)(86,140,191,112)(87,141,192,113)(88,142,193,114)(89,143,194,115)(90,144,195,116), (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), (1,185,10,194)(2,184,11,193)(3,183,12,192)(4,182,13,191)(5,181,14,190)(6,198,15,189)(7,197,16,188)(8,196,17,187)(9,195,18,186)(19,253,28,262)(20,270,29,261)(21,269,30,260)(22,268,31,259)(23,267,32,258)(24,266,33,257)(25,265,34,256)(26,264,35,255)(27,263,36,254)(37,242,46,251)(38,241,47,250)(39,240,48,249)(40,239,49,248)(41,238,50,247)(42,237,51,246)(43,236,52,245)(44,235,53,244)(45,252,54,243)(55,97,64,106)(56,96,65,105)(57,95,66,104)(58,94,67,103)(59,93,68,102)(60,92,69,101)(61,91,70,100)(62,108,71,99)(63,107,72,98)(73,145,82,154)(74,162,83,153)(75,161,84,152)(76,160,85,151)(77,159,86,150)(78,158,87,149)(79,157,88,148)(80,156,89,147)(81,155,90,146)(109,177,118,168)(110,176,119,167)(111,175,120,166)(112,174,121,165)(113,173,122,164)(114,172,123,163)(115,171,124,180)(116,170,125,179)(117,169,126,178)(127,225,136,234)(128,224,137,233)(129,223,138,232)(130,222,139,231)(131,221,140,230)(132,220,141,229)(133,219,142,228)(134,218,143,227)(135,217,144,226)(199,285,208,276)(200,284,209,275)(201,283,210,274)(202,282,211,273)(203,281,212,272)(204,280,213,271)(205,279,214,288)(206,278,215,287)(207,277,216,286) );
G=PermutationGroup([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,109),(53,110),(54,111),(91,192),(92,193),(93,194),(94,195),(95,196),(96,197),(97,198),(98,181),(99,182),(100,183),(101,184),(102,185),(103,186),(104,187),(105,188),(106,189),(107,190),(108,191),(127,205),(128,206),(129,207),(130,208),(131,209),(132,210),(133,211),(134,212),(135,213),(136,214),(137,215),(138,216),(139,199),(140,200),(141,201),(142,202),(143,203),(144,204),(145,265),(146,266),(147,267),(148,268),(149,269),(150,270),(151,253),(152,254),(153,255),(154,256),(155,257),(156,258),(157,259),(158,260),(159,261),(160,262),(161,263),(162,264),(163,249),(164,250),(165,251),(166,252),(167,235),(168,236),(169,237),(170,238),(171,239),(172,240),(173,241),(174,242),(175,243),(176,244),(177,245),(178,246),(179,247),(180,248),(217,271),(218,272),(219,273),(220,274),(221,275),(222,276),(223,277),(224,278),(225,279),(226,280),(227,281),(228,282),(229,283),(230,284),(231,285),(232,286),(233,287),(234,288)], [(1,171,147,227),(2,172,148,228),(3,173,149,229),(4,174,150,230),(5,175,151,231),(6,176,152,232),(7,177,153,233),(8,178,154,234),(9,179,155,217),(10,180,156,218),(11,163,157,219),(12,164,158,220),(13,165,159,221),(14,166,160,222),(15,167,161,223),(16,168,162,224),(17,169,145,225),(18,170,146,226),(19,199,107,54),(20,200,108,37),(21,201,91,38),(22,202,92,39),(23,203,93,40),(24,204,94,41),(25,205,95,42),(26,206,96,43),(27,207,97,44),(28,208,98,45),(29,209,99,46),(30,210,100,47),(31,211,101,48),(32,212,102,49),(33,213,103,50),(34,214,104,51),(35,215,105,52),(36,216,106,53),(55,244,254,286),(56,245,255,287),(57,246,256,288),(58,247,257,271),(59,248,258,272),(60,249,259,273),(61,250,260,274),(62,251,261,275),(63,252,262,276),(64,235,263,277),(65,236,264,278),(66,237,265,279),(67,238,266,280),(68,239,267,281),(69,240,268,282),(70,241,269,283),(71,242,270,284),(72,243,253,285),(73,127,196,117),(74,128,197,118),(75,129,198,119),(76,130,181,120),(77,131,182,121),(78,132,183,122),(79,133,184,123),(80,134,185,124),(81,135,186,125),(82,136,187,126),(83,137,188,109),(84,138,189,110),(85,139,190,111),(86,140,191,112),(87,141,192,113),(88,142,193,114),(89,143,194,115),(90,144,195,116)], [(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)], [(1,185,10,194),(2,184,11,193),(3,183,12,192),(4,182,13,191),(5,181,14,190),(6,198,15,189),(7,197,16,188),(8,196,17,187),(9,195,18,186),(19,253,28,262),(20,270,29,261),(21,269,30,260),(22,268,31,259),(23,267,32,258),(24,266,33,257),(25,265,34,256),(26,264,35,255),(27,263,36,254),(37,242,46,251),(38,241,47,250),(39,240,48,249),(40,239,49,248),(41,238,50,247),(42,237,51,246),(43,236,52,245),(44,235,53,244),(45,252,54,243),(55,97,64,106),(56,96,65,105),(57,95,66,104),(58,94,67,103),(59,93,68,102),(60,92,69,101),(61,91,70,100),(62,108,71,99),(63,107,72,98),(73,145,82,154),(74,162,83,153),(75,161,84,152),(76,160,85,151),(77,159,86,150),(78,158,87,149),(79,157,88,148),(80,156,89,147),(81,155,90,146),(109,177,118,168),(110,176,119,167),(111,175,120,166),(112,174,121,165),(113,173,122,164),(114,172,123,163),(115,171,124,180),(116,170,125,179),(117,169,126,178),(127,225,136,234),(128,224,137,233),(129,223,138,232),(130,222,139,231),(131,221,140,230),(132,220,141,229),(133,219,142,228),(134,218,143,227),(135,217,144,226),(199,285,208,276),(200,284,209,275),(201,283,210,274),(202,282,211,273),(203,281,212,272),(204,280,213,271),(205,279,214,288),(206,278,215,287),(207,277,216,286)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6G | 9A | 9B | 9C | 12A | ··· | 12H | 18A | ··· | 18U | 36A | ··· | 36X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D6 | D9 | C4×S3 | Dic9 | D18 | D18 | C4×D9 |
| kernel | C2×C4×Dic9 | C4×Dic9 | C22×Dic9 | C22×C36 | C2×Dic9 | C2×C36 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 4 | 2 | 1 | 16 | 8 | 1 | 4 | 2 | 1 | 3 | 8 | 12 | 6 | 3 | 24 |
Matrix representation of C2×C4×Dic9 ►in GL5(𝔽37)
| 36 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 36 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 36 | 36 | 0 | 0 |
| 0 | 0 | 0 | 17 | 31 |
| 0 | 0 | 0 | 6 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 6 | 26 | 0 | 0 |
| 0 | 20 | 31 | 0 | 0 |
| 0 | 0 | 0 | 36 | 29 |
| 0 | 0 | 0 | 28 | 1 |
G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[36,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,0,36,0,0,0,1,36,0,0,0,0,0,17,6,0,0,0,31,11],[1,0,0,0,0,0,6,20,0,0,0,26,31,0,0,0,0,0,36,28,0,0,0,29,1] >;
C2×C4×Dic9 in GAP, Magma, Sage, TeX
C_2\times C_4\times {\rm Dic}_9 % in TeX
G:=Group("C2xC4xDic9"); // GroupNames label
G:=SmallGroup(288,132);
// by ID
G=gap.SmallGroup(288,132);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,100,6725,292,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^18=1,d^2=c^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations