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## G = C2×C4⋊Dic9order 288 = 25·32

### Direct product of C2 and C4⋊Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×C4⋊Dic9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — C2×C4⋊Dic9
 Lower central C9 — C18 — C2×C4⋊Dic9
 Upper central C1 — C23 — 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, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 432 in 138 conjugacy classes, 92 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×4], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×8], C23, C9, Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×6], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C18 [×3], C18 [×4], C2×Dic3 [×8], C2×C12 [×6], C22×C6, C2×C4⋊C4, Dic9 [×4], C36 [×4], C2×C18, C2×C18 [×6], C4⋊Dic3 [×4], C22×Dic3 [×2], C22×C12, C2×Dic9 [×4], C2×Dic9 [×4], C2×C36 [×6], C22×C18, C2×C4⋊Dic3, C4⋊Dic9 [×4], C22×Dic9 [×2], C22×C36, C2×C4⋊Dic9
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D9, Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C2×C4⋊C4, Dic9 [×4], D18 [×3], C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, Dic18 [×2], D36 [×2], C2×Dic9 [×6], C22×D9, C2×C4⋊Dic3, C4⋊Dic9 [×4], C2×Dic18, C2×D36, C22×Dic9, C2×C4⋊Dic9

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

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 2 2 2 2 2 18 ··· 18 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - - + + + - + - + + - + image C1 C2 C2 C2 C4 S3 D4 Q8 Dic3 D6 D6 D9 Dic6 D12 Dic9 D18 D18 Dic18 D36 kernel C2×C4⋊Dic9 C4⋊Dic9 C22×Dic9 C22×C36 C2×C36 C22×C12 C2×C18 C2×C18 C2×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 2 1 8 1 2 2 4 2 1 3 4 4 12 6 3 12 12

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 36 0 0 0 0 0 36
,
 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 32 27 0 0 0 10 5
,
 1 0 0 0 0 0 26 20 0 0 0 17 6 0 0 0 0 0 31 11 0 0 0 26 20
,
 36 0 0 0 0 0 30 7 0 0 0 14 7 0 0 0 0 0 8 9 0 0 0 1 29

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[36,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,32,10,0,0,0,27,5],[1,0,0,0,0,0,26,17,0,0,0,20,6,0,0,0,0,0,31,26,0,0,0,11,20],[36,0,0,0,0,0,30,14,0,0,0,7,7,0,0,0,0,0,8,1,0,0,0,9,29] >;

C2×C4⋊Dic9 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes {\rm Dic}_9
% in TeX

G:=Group("C2xC4:Dic9");
// GroupNames label

G:=SmallGroup(288,135);
// by ID

G=gap.SmallGroup(288,135);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,100,6725,292,9414]);
// 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,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽