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## G = C2×C4×Dic9order 288 = 25·32

### Direct product of C2×C4 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C2×C4×Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — C2×C4×Dic9
 Lower central C9 — C2×C4×Dic9
 Upper central C1 — C22×C4

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 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×6], C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], C23, C9, Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C42 [×4], C22×C4, C22×C4 [×2], C18, C18 [×6], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C2×C42, Dic9 [×8], C36 [×4], C2×C18, C2×C18 [×6], C4×Dic3 [×4], C22×Dic3 [×2], C22×C12, C2×Dic9 [×12], C2×C36 [×6], C22×C18, C2×C4×Dic3, C4×Dic9 [×4], C22×Dic9 [×2], C22×C36, C2×C4×Dic9
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], C23, Dic3 [×4], D6 [×3], C42 [×4], C22×C4 [×3], D9, C4×S3 [×4], C2×Dic3 [×6], C22×S3, C2×C42, Dic9 [×4], D18 [×3], C4×Dic3 [×4], S3×C2×C4 [×2], C22×Dic3, C4×D9 [×4], C2×Dic9 [×6], C22×D9, C2×C4×Dic3, C4×Dic9 [×4], C2×C4×D9 [×2], C22×Dic9, C2×C4×Dic9

Smallest permutation representation of C2×C4×Dic9
Regular action on 288 points
Generators in S288
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 197)(13 198)(14 181)(15 182)(16 183)(17 184)(18 185)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 55)(26 56)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 91)(47 92)(48 93)(49 94)(50 95)(51 96)(52 97)(53 98)(54 99)(73 246)(74 247)(75 248)(76 249)(77 250)(78 251)(79 252)(80 235)(81 236)(82 237)(83 238)(84 239)(85 240)(86 241)(87 242)(88 243)(89 244)(90 245)(109 199)(110 200)(111 201)(112 202)(113 203)(114 204)(115 205)(116 206)(117 207)(118 208)(119 209)(120 210)(121 211)(122 212)(123 213)(124 214)(125 215)(126 216)(127 219)(128 220)(129 221)(130 222)(131 223)(132 224)(133 225)(134 226)(135 227)(136 228)(137 229)(138 230)(139 231)(140 232)(141 233)(142 234)(143 217)(144 218)(145 267)(146 268)(147 269)(148 270)(149 253)(150 254)(151 255)(152 256)(153 257)(154 258)(155 259)(156 260)(157 261)(158 262)(159 263)(160 264)(161 265)(162 266)(163 276)(164 277)(165 278)(166 279)(167 280)(168 281)(169 282)(170 283)(171 284)(172 285)(173 286)(174 287)(175 288)(176 271)(177 272)(178 273)(179 274)(180 275)
(1 112 72 157)(2 113 55 158)(3 114 56 159)(4 115 57 160)(5 116 58 161)(6 117 59 162)(7 118 60 145)(8 119 61 146)(9 120 62 147)(10 121 63 148)(11 122 64 149)(12 123 65 150)(13 124 66 151)(14 125 67 152)(15 126 68 153)(16 109 69 154)(17 110 70 155)(18 111 71 156)(19 256 181 215)(20 257 182 216)(21 258 183 199)(22 259 184 200)(23 260 185 201)(24 261 186 202)(25 262 187 203)(26 263 188 204)(27 264 189 205)(28 265 190 206)(29 266 191 207)(30 267 192 208)(31 268 193 209)(32 269 194 210)(33 270 195 211)(34 253 196 212)(35 254 197 213)(36 255 198 214)(37 239 171 135)(38 240 172 136)(39 241 173 137)(40 242 174 138)(41 243 175 139)(42 244 176 140)(43 245 177 141)(44 246 178 142)(45 247 179 143)(46 248 180 144)(47 249 163 127)(48 250 164 128)(49 251 165 129)(50 252 166 130)(51 235 167 131)(52 236 168 132)(53 237 169 133)(54 238 170 134)(73 273 234 107)(74 274 217 108)(75 275 218 91)(76 276 219 92)(77 277 220 93)(78 278 221 94)(79 279 222 95)(80 280 223 96)(81 281 224 97)(82 282 225 98)(83 283 226 99)(84 284 227 100)(85 285 228 101)(86 286 229 102)(87 287 230 103)(88 288 231 104)(89 271 232 105)(90 272 233 106)
(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)
(1 130 10 139)(2 129 11 138)(3 128 12 137)(4 127 13 136)(5 144 14 135)(6 143 15 134)(7 142 16 133)(8 141 17 132)(9 140 18 131)(19 84 28 75)(20 83 29 74)(21 82 30 73)(22 81 31 90)(23 80 32 89)(24 79 33 88)(25 78 34 87)(26 77 35 86)(27 76 36 85)(37 116 46 125)(38 115 47 124)(39 114 48 123)(40 113 49 122)(41 112 50 121)(42 111 51 120)(43 110 52 119)(44 109 53 118)(45 126 54 117)(55 251 64 242)(56 250 65 241)(57 249 66 240)(58 248 67 239)(59 247 68 238)(60 246 69 237)(61 245 70 236)(62 244 71 235)(63 243 72 252)(91 215 100 206)(92 214 101 205)(93 213 102 204)(94 212 103 203)(95 211 104 202)(96 210 105 201)(97 209 106 200)(98 208 107 199)(99 207 108 216)(145 178 154 169)(146 177 155 168)(147 176 156 167)(148 175 157 166)(149 174 158 165)(150 173 159 164)(151 172 160 163)(152 171 161 180)(153 170 162 179)(181 227 190 218)(182 226 191 217)(183 225 192 234)(184 224 193 233)(185 223 194 232)(186 222 195 231)(187 221 196 230)(188 220 197 229)(189 219 198 228)(253 287 262 278)(254 286 263 277)(255 285 264 276)(256 284 265 275)(257 283 266 274)(258 282 267 273)(259 281 268 272)(260 280 269 271)(261 279 270 288)

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

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

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

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

׿
×
𝔽