Copied to
clipboard

## 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, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C9, Dic3, C12, C2×C6, C22×C4, C22×C4, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, C22×C6, C22×Q8, Dic9, C36, C2×C18, C2×Dic6, C22×Dic3, C22×C12, Dic18, C2×Dic9, C2×C36, C22×C18, C22×Dic6, C2×Dic18, C22×Dic9, C22×C36, C22×Dic18
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, D9, Dic6, C22×S3, C22×Q8, D18, C2×Dic6, S3×C23, Dic18, C22×D9, C22×Dic6, C2×Dic18, C23×D9, C22×Dic18

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

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

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

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

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