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## G = C23×Dic9order 288 = 25·32

### Direct product of C23 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C23×Dic9
 Chief series C1 — C3 — C9 — C18 — Dic9 — C2×Dic9 — C22×Dic9 — C23×Dic9
 Lower central C9 — C23×Dic9
 Upper central C1 — C24

Generators and relations for C23×Dic9
G = < a,b,c,d,e | a2=b2=c2=d18=1, e2=d9, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 864 in 354 conjugacy classes, 252 normal (10 characteristic)
C1, C2, C2 [×14], C3, C4 [×8], C22 [×35], C6, C6 [×14], C2×C4 [×28], C23 [×15], C9, Dic3 [×8], C2×C6 [×35], C22×C4 [×14], C24, C18, C18 [×14], C2×Dic3 [×28], C22×C6 [×15], C23×C4, Dic9 [×8], C2×C18 [×35], C22×Dic3 [×14], C23×C6, C2×Dic9 [×28], C22×C18 [×15], C23×Dic3, C22×Dic9 [×14], C23×C18, C23×Dic9
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, D9, C2×Dic3 [×28], C22×S3 [×7], C23×C4, Dic9 [×8], D18 [×7], C22×Dic3 [×14], S3×C23, C2×Dic9 [×28], C22×D9 [×7], C23×Dic3, C22×Dic9 [×14], C23×D9, C23×Dic9

Smallest permutation representation of C23×Dic9
Regular action on 288 points
Generators in S288
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 81)(17 82)(18 83)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 109)(51 110)(52 111)(53 112)(54 113)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 127)(64 128)(65 129)(66 130)(67 131)(68 132)(69 133)(70 134)(71 135)(72 136)(145 219)(146 220)(147 221)(148 222)(149 223)(150 224)(151 225)(152 226)(153 227)(154 228)(155 229)(156 230)(157 231)(158 232)(159 233)(160 234)(161 217)(162 218)(163 239)(164 240)(165 241)(166 242)(167 243)(168 244)(169 245)(170 246)(171 247)(172 248)(173 249)(174 250)(175 251)(176 252)(177 235)(178 236)(179 237)(180 238)(181 260)(182 261)(183 262)(184 263)(185 264)(186 265)(187 266)(188 267)(189 268)(190 269)(191 270)(192 253)(193 254)(194 255)(195 256)(196 257)(197 258)(198 259)(199 278)(200 279)(201 280)(202 281)(203 282)(204 283)(205 284)(206 285)(207 286)(208 287)(209 288)(210 271)(211 272)(212 273)(213 274)(214 275)(215 276)(216 277)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 127)(104 128)(105 129)(106 130)(107 131)(108 132)(145 194)(146 195)(147 196)(148 197)(149 198)(150 181)(151 182)(152 183)(153 184)(154 185)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 199)(164 200)(165 201)(166 202)(167 203)(168 204)(169 205)(170 206)(171 207)(172 208)(173 209)(174 210)(175 211)(176 212)(177 213)(178 214)(179 215)(180 216)(217 253)(218 254)(219 255)(220 256)(221 257)(222 258)(223 259)(224 260)(225 261)(226 262)(227 263)(228 264)(229 265)(230 266)(231 267)(232 268)(233 269)(234 270)(235 274)(236 275)(237 276)(238 277)(239 278)(240 279)(241 280)(242 281)(243 282)(244 283)(245 284)(246 285)(247 286)(248 287)(249 288)(250 271)(251 272)(252 273)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(37 69)(38 70)(39 71)(40 72)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 91)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(109 128)(110 129)(111 130)(112 131)(113 132)(114 133)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 127)(145 170)(146 171)(147 172)(148 173)(149 174)(150 175)(151 176)(152 177)(153 178)(154 179)(155 180)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(162 169)(181 211)(182 212)(183 213)(184 214)(185 215)(186 216)(187 199)(188 200)(189 201)(190 202)(191 203)(192 204)(193 205)(194 206)(195 207)(196 208)(197 209)(198 210)(217 244)(218 245)(219 246)(220 247)(221 248)(222 249)(223 250)(224 251)(225 252)(226 235)(227 236)(228 237)(229 238)(230 239)(231 240)(232 241)(233 242)(234 243)(253 283)(254 284)(255 285)(256 286)(257 287)(258 288)(259 271)(260 272)(261 273)(262 274)(263 275)(264 276)(265 277)(266 278)(267 279)(268 280)(269 281)(270 282)
(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)
(1 225 10 234)(2 224 11 233)(3 223 12 232)(4 222 13 231)(5 221 14 230)(6 220 15 229)(7 219 16 228)(8 218 17 227)(9 217 18 226)(19 242 28 251)(20 241 29 250)(21 240 30 249)(22 239 31 248)(23 238 32 247)(24 237 33 246)(25 236 34 245)(26 235 35 244)(27 252 36 243)(37 261 46 270)(38 260 47 269)(39 259 48 268)(40 258 49 267)(41 257 50 266)(42 256 51 265)(43 255 52 264)(44 254 53 263)(45 253 54 262)(55 287 64 278)(56 286 65 277)(57 285 66 276)(58 284 67 275)(59 283 68 274)(60 282 69 273)(61 281 70 272)(62 280 71 271)(63 279 72 288)(73 162 82 153)(74 161 83 152)(75 160 84 151)(76 159 85 150)(77 158 86 149)(78 157 87 148)(79 156 88 147)(80 155 89 146)(81 154 90 145)(91 176 100 167)(92 175 101 166)(93 174 102 165)(94 173 103 164)(95 172 104 163)(96 171 105 180)(97 170 106 179)(98 169 107 178)(99 168 108 177)(109 187 118 196)(110 186 119 195)(111 185 120 194)(112 184 121 193)(113 183 122 192)(114 182 123 191)(115 181 124 190)(116 198 125 189)(117 197 126 188)(127 200 136 209)(128 199 137 208)(129 216 138 207)(130 215 139 206)(131 214 140 205)(132 213 141 204)(133 212 142 203)(134 211 143 202)(135 210 144 201)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2O 3 4A ··· 4P 6A ··· 6O 9A 9B 9C 18A ··· 18AS order 1 2 ··· 2 3 4 ··· 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 ··· 1 2 9 ··· 9 2 ··· 2 2 2 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + - + + - + image C1 C2 C2 C4 S3 Dic3 D6 D9 Dic9 D18 kernel C23×Dic9 C22×Dic9 C23×C18 C22×C18 C23×C6 C22×C6 C22×C6 C24 C23 C23 # reps 1 14 1 16 1 8 7 3 24 21

Matrix representation of C23×Dic9 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 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 36 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 1 0 0 0 0 0 0 36 1 0 0 0 0 36 0 0 0 0 0 0 0 34 0 0 0 0 0 32 12
,
 31 0 0 0 0 0 0 36 0 0 0 0 0 0 27 5 0 0 0 0 32 10 0 0 0 0 0 0 4 25 0 0 0 0 29 33

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,34,32,0,0,0,0,0,12],[31,0,0,0,0,0,0,36,0,0,0,0,0,0,27,32,0,0,0,0,5,10,0,0,0,0,0,0,4,29,0,0,0,0,25,33] >;

C23×Dic9 in GAP, Magma, Sage, TeX

C_2^3\times {\rm Dic}_9
% in TeX

G:=Group("C2^3xDic9");
// GroupNames label

G:=SmallGroup(288,365);
// by ID

G=gap.SmallGroup(288,365);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^18=1,e^2=d^9,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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