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## G = C24×C18order 288 = 25·32

### Abelian group of type [2,2,2,2,18]

Aliases: C24×C18, SmallGroup(288,840)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C24×C18
G = < a,b,c,d,e | a2=b2=c2=d2=e18=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Subgroups: 1122, all normal (6 characteristic)
C1, C2 [×31], C3, C22 [×155], C6 [×31], C23 [×155], C9, C2×C6 [×155], C24 [×31], C18 [×31], C22×C6 [×155], C25, C2×C18 [×155], C23×C6 [×31], C22×C18 [×155], C24×C6, C23×C18 [×31], C24×C18
Quotients: C1, C2 [×31], C3, C22 [×155], C6 [×31], C23 [×155], C9, C2×C6 [×155], C24 [×31], C18 [×31], C22×C6 [×155], C25, C2×C18 [×155], C23×C6 [×31], C22×C18 [×155], C24×C6, C23×C18 [×31], C24×C18

Smallest permutation representation of C24×C18
Regular action on 288 points
Generators in S288
(1 158)(2 159)(3 160)(4 161)(5 162)(6 145)(7 146)(8 147)(9 148)(10 149)(11 150)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 166)(20 167)(21 168)(22 169)(23 170)(24 171)(25 172)(26 173)(27 174)(28 175)(29 176)(30 177)(31 178)(32 179)(33 180)(34 163)(35 164)(36 165)(37 196)(38 197)(39 198)(40 181)(41 182)(42 183)(43 184)(44 185)(45 186)(46 187)(47 188)(48 189)(49 190)(50 191)(51 192)(52 193)(53 194)(54 195)(55 199)(56 200)(57 201)(58 202)(59 203)(60 204)(61 205)(62 206)(63 207)(64 208)(65 209)(66 210)(67 211)(68 212)(69 213)(70 214)(71 215)(72 216)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 217)(90 218)(91 240)(92 241)(93 242)(94 243)(95 244)(96 245)(97 246)(98 247)(99 248)(100 249)(101 250)(102 251)(103 252)(104 235)(105 236)(106 237)(107 238)(108 239)(109 258)(110 259)(111 260)(112 261)(113 262)(114 263)(115 264)(116 265)(117 266)(118 267)(119 268)(120 269)(121 270)(122 253)(123 254)(124 255)(125 256)(126 257)(127 274)(128 275)(129 276)(130 277)(131 278)(132 279)(133 280)(134 281)(135 282)(136 283)(137 284)(138 285)(139 286)(140 287)(141 288)(142 271)(143 272)(144 273)
(1 89)(2 90)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 91)(30 92)(31 93)(32 94)(33 95)(34 96)(35 97)(36 98)(37 119)(38 120)(39 121)(40 122)(41 123)(42 124)(43 125)(44 126)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 132)(56 133)(57 134)(58 135)(59 136)(60 137)(61 138)(62 139)(63 140)(64 141)(65 142)(66 143)(67 144)(68 127)(69 128)(70 129)(71 130)(72 131)(145 222)(146 223)(147 224)(148 225)(149 226)(150 227)(151 228)(152 229)(153 230)(154 231)(155 232)(156 233)(157 234)(158 217)(159 218)(160 219)(161 220)(162 221)(163 245)(164 246)(165 247)(166 248)(167 249)(168 250)(169 251)(170 252)(171 235)(172 236)(173 237)(174 238)(175 239)(176 240)(177 241)(178 242)(179 243)(180 244)(181 253)(182 254)(183 255)(184 256)(185 257)(186 258)(187 259)(188 260)(189 261)(190 262)(191 263)(192 264)(193 265)(194 266)(195 267)(196 268)(197 269)(198 270)(199 279)(200 280)(201 281)(202 282)(203 283)(204 284)(205 285)(206 286)(207 287)(208 288)(209 271)(210 272)(211 273)(212 274)(213 275)(214 276)(215 277)(216 278)
(1 53)(2 54)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 68)(20 69)(21 70)(22 71)(23 72)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(73 119)(74 120)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 109)(82 110)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 137)(92 138)(93 139)(94 140)(95 141)(96 142)(97 143)(98 144)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(145 181)(146 182)(147 183)(148 184)(149 185)(150 186)(151 187)(152 188)(153 189)(154 190)(155 191)(156 192)(157 193)(158 194)(159 195)(160 196)(161 197)(162 198)(163 209)(164 210)(165 211)(166 212)(167 213)(168 214)(169 215)(170 216)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(217 266)(218 267)(219 268)(220 269)(221 270)(222 253)(223 254)(224 255)(225 256)(226 257)(227 258)(228 259)(229 260)(230 261)(231 262)(232 263)(233 264)(234 265)(235 279)(236 280)(237 281)(238 282)(239 283)(240 284)(241 285)(242 286)(243 287)(244 288)(245 271)(246 272)(247 273)(248 274)(249 275)(250 276)(251 277)(252 278)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(73 96)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 91)(87 92)(88 93)(89 94)(90 95)(109 132)(110 133)(111 134)(112 135)(113 136)(114 137)(115 138)(116 139)(117 140)(118 141)(119 142)(120 143)(121 144)(122 127)(123 128)(124 129)(125 130)(126 131)(145 166)(146 167)(147 168)(148 169)(149 170)(150 171)(151 172)(152 173)(153 174)(154 175)(155 176)(156 177)(157 178)(158 179)(159 180)(160 163)(161 164)(162 165)(181 212)(182 213)(183 214)(184 215)(185 216)(186 199)(187 200)(188 201)(189 202)(190 203)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(197 210)(198 211)(217 243)(218 244)(219 245)(220 246)(221 247)(222 248)(223 249)(224 250)(225 251)(226 252)(227 235)(228 236)(229 237)(230 238)(231 239)(232 240)(233 241)(234 242)(253 274)(254 275)(255 276)(256 277)(257 278)(258 279)(259 280)(260 281)(261 282)(262 283)(263 284)(264 285)(265 286)(266 287)(267 288)(268 271)(269 272)(270 273)
(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)

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

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

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

288 conjugacy classes

 class 1 2A ··· 2AE 3A 3B 6A ··· 6BJ 9A ··· 9F 18A ··· 18GD order 1 2 ··· 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

288 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 kernel C24×C18 C23×C18 C24×C6 C23×C6 C25 C24 # reps 1 31 2 62 6 186

Matrix representation of C24×C18 in GL5(𝔽19)

 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18
,
 18 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 1
,
 15 0 0 0 0 0 9 0 0 0 0 0 15 0 0 0 0 0 5 0 0 0 0 0 14

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18],[18,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,1],[15,0,0,0,0,0,9,0,0,0,0,0,15,0,0,0,0,0,5,0,0,0,0,0,14] >;

C24×C18 in GAP, Magma, Sage, TeX

C_2^4\times C_{18}
% in TeX

G:=Group("C2^4xC18");
// GroupNames label

G:=SmallGroup(288,840);
// by ID

G=gap.SmallGroup(288,840);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,166]);
// Polycyclic

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

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