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G = C22×Dic18order 288 = 25·32

Direct product of C22 and Dic18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C22×Dic18
 Chief series C1 — C3 — C9 — C18 — Dic9 — C2×Dic9 — C22×Dic9 — C22×Dic18
 Lower central C9 — C18 — C22×Dic18
 Upper central C1 — C23 — C22×C4

Generators and relations for C22×Dic18
G = < a,b,c,d | a2=b2=c36=1, d2=c18, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 744 in 234 conjugacy classes, 132 normal (13 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C9, Dic3 [×8], C12 [×4], C2×C6 [×7], C22×C4, C22×C4 [×2], C2×Q8 [×12], C18, C18 [×6], Dic6 [×16], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C22×Q8, Dic9 [×8], C36 [×4], C2×C18 [×7], C2×Dic6 [×12], C22×Dic3 [×2], C22×C12, Dic18 [×16], C2×Dic9 [×12], C2×C36 [×6], C22×C18, C22×Dic6, C2×Dic18 [×12], C22×Dic9 [×2], C22×C36, C22×Dic18
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C24, D9, Dic6 [×4], C22×S3 [×7], C22×Q8, D18 [×7], C2×Dic6 [×6], S3×C23, Dic18 [×4], C22×D9 [×7], C22×Dic6, C2×Dic18 [×6], C23×D9, C22×Dic18

Smallest permutation representation of C22×Dic18
Regular action on 288 points
Generators in S288
(1 288)(2 253)(3 254)(4 255)(5 256)(6 257)(7 258)(8 259)(9 260)(10 261)(11 262)(12 263)(13 264)(14 265)(15 266)(16 267)(17 268)(18 269)(19 270)(20 271)(21 272)(22 273)(23 274)(24 275)(25 276)(26 277)(27 278)(28 279)(29 280)(30 281)(31 282)(32 283)(33 284)(34 285)(35 286)(36 287)(37 213)(38 214)(39 215)(40 216)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 197)(58 198)(59 199)(60 200)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 211)(72 212)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(145 241)(146 242)(147 243)(148 244)(149 245)(150 246)(151 247)(152 248)(153 249)(154 250)(155 251)(156 252)(157 217)(158 218)(159 219)(160 220)(161 221)(162 222)(163 223)(164 224)(165 225)(166 226)(167 227)(168 228)(169 229)(170 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)
(1 180)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 169)(27 170)(28 171)(29 172)(30 173)(31 174)(32 175)(33 176)(34 177)(35 178)(36 179)(37 87)(38 88)(39 89)(40 90)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(109 206)(110 207)(111 208)(112 209)(113 210)(114 211)(115 212)(116 213)(117 214)(118 215)(119 216)(120 181)(121 182)(122 183)(123 184)(124 185)(125 186)(126 187)(127 188)(128 189)(129 190)(130 191)(131 192)(132 193)(133 194)(134 195)(135 196)(136 197)(137 198)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 205)(217 265)(218 266)(219 267)(220 268)(221 269)(222 270)(223 271)(224 272)(225 273)(226 274)(227 275)(228 276)(229 277)(230 278)(231 279)(232 280)(233 281)(234 282)(235 283)(236 284)(237 285)(238 286)(239 287)(240 288)(241 253)(242 254)(243 255)(244 256)(245 257)(246 258)(247 259)(248 260)(249 261)(250 262)(251 263)(252 264)
(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 42 19 60)(2 41 20 59)(3 40 21 58)(4 39 22 57)(5 38 23 56)(6 37 24 55)(7 72 25 54)(8 71 26 53)(9 70 27 52)(10 69 28 51)(11 68 29 50)(12 67 30 49)(13 66 31 48)(14 65 32 47)(15 64 33 46)(16 63 34 45)(17 62 35 44)(18 61 36 43)(73 145 91 163)(74 180 92 162)(75 179 93 161)(76 178 94 160)(77 177 95 159)(78 176 96 158)(79 175 97 157)(80 174 98 156)(81 173 99 155)(82 172 100 154)(83 171 101 153)(84 170 102 152)(85 169 103 151)(86 168 104 150)(87 167 105 149)(88 166 106 148)(89 165 107 147)(90 164 108 146)(109 234 127 252)(110 233 128 251)(111 232 129 250)(112 231 130 249)(113 230 131 248)(114 229 132 247)(115 228 133 246)(116 227 134 245)(117 226 135 244)(118 225 136 243)(119 224 137 242)(120 223 138 241)(121 222 139 240)(122 221 140 239)(123 220 141 238)(124 219 142 237)(125 218 143 236)(126 217 144 235)(181 271 199 253)(182 270 200 288)(183 269 201 287)(184 268 202 286)(185 267 203 285)(186 266 204 284)(187 265 205 283)(188 264 206 282)(189 263 207 281)(190 262 208 280)(191 261 209 279)(192 260 210 278)(193 259 211 277)(194 258 212 276)(195 257 213 275)(196 256 214 274)(197 255 215 273)(198 254 216 272)

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

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

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

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 2 2 2 2 2 2 2 2 2 type + + + + + - + + + - + + - image C1 C2 C2 C2 S3 Q8 D6 D6 D9 Dic6 D18 D18 Dic18 kernel C22×Dic18 C2×Dic18 C22×Dic9 C22×C36 C22×C12 C2×C18 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 1 4 6 1 3 8 18 3 24

Matrix representation of C22×Dic18 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 36 0 0 0 0 1 1 0 0 0 0 0 0 6 11 0 0 0 0 26 17 0 0 0 0 0 0 1 2 0 0 0 0 36 36
,
 33 25 0 0 0 0 29 4 0 0 0 0 0 0 26 17 0 0 0 0 6 11 0 0 0 0 0 0 26 31 0 0 0 0 8 11

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,36,1,0,0,0,0,0,0,6,26,0,0,0,0,11,17,0,0,0,0,0,0,1,36,0,0,0,0,2,36],[33,29,0,0,0,0,25,4,0,0,0,0,0,0,26,6,0,0,0,0,17,11,0,0,0,0,0,0,26,8,0,0,0,0,31,11] >;

C22×Dic18 in GAP, Magma, Sage, TeX

C_2^2\times {\rm Dic}_{18}
% in TeX

G:=Group("C2^2xDic18");
// GroupNames label

G:=SmallGroup(288,352);
// by ID

G=gap.SmallGroup(288,352);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,675,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^36=1,d^2=c^18,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