direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×Dic9, C24.3D9, C18.14C24, C23.41D18, C9⋊2(C23×C4), (C22×C18)⋊5C4, C18⋊2(C22×C4), C2.2(C23×D9), C3.(C23×Dic3), (C23×C6).10S3, C6.51(S3×C23), (C23×C18).3C2, (C2×C18).69C23, (C22×C6).150D6, C6.30(C22×Dic3), (C22×C6).16Dic3, C22.33(C22×D9), (C22×C18).47C22, (C2×C18)⋊9(C2×C4), (C2×C6).50(C2×Dic3), (C2×C6).227(C22×S3), SmallGroup(288,365)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C23×Dic9 |
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, C3, C4, C22, C6, C6, C2×C4, C23, C9, Dic3, C2×C6, C22×C4, C24, C18, C18, C2×Dic3, C22×C6, C23×C4, Dic9, C2×C18, C22×Dic3, C23×C6, C2×Dic9, C22×C18, C23×Dic3, C22×Dic9, C23×C18, C23×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C24, D9, C2×Dic3, C22×S3, C23×C4, Dic9, D18, C22×Dic3, S3×C23, C2×Dic9, C22×D9, C23×Dic3, C22×Dic9, C23×D9, C23×Dic9
►Regular action on 288 points
Generators in S288
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 91)(34 92)(35 93)(36 94)(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)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 127)(71 128)(72 129)(145 225)(146 226)(147 227)(148 228)(149 229)(150 230)(151 231)(152 232)(153 233)(154 234)(155 217)(156 218)(157 219)(158 220)(159 221)(160 222)(161 223)(162 224)(163 251)(164 252)(165 235)(166 236)(167 237)(168 238)(169 239)(170 240)(171 241)(172 242)(173 243)(174 244)(175 245)(176 246)(177 247)(178 248)(179 249)(180 250)(181 265)(182 266)(183 267)(184 268)(185 269)(186 270)(187 253)(188 254)(189 255)(190 256)(191 257)(192 258)(193 259)(194 260)(195 261)(196 262)(197 263)(198 264)(199 280)(200 281)(201 282)(202 283)(203 284)(204 285)(205 286)(206 287)(207 288)(208 271)(209 272)(210 273)(211 274)(212 275)(213 276)(214 277)(215 278)(216 279)
(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 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 55)(73 126)(74 109)(75 110)(76 111)(77 112)(78 113)(79 114)(80 115)(81 116)(82 117)(83 118)(84 119)(85 120)(86 121)(87 122)(88 123)(89 124)(90 125)(91 127)(92 128)(93 129)(94 130)(95 131)(96 132)(97 133)(98 134)(99 135)(100 136)(101 137)(102 138)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)(145 187)(146 188)(147 189)(148 190)(149 191)(150 192)(151 193)(152 194)(153 195)(154 196)(155 197)(156 198)(157 181)(158 182)(159 183)(160 184)(161 185)(162 186)(163 203)(164 204)(165 205)(166 206)(167 207)(168 208)(169 209)(170 210)(171 211)(172 212)(173 213)(174 214)(175 215)(176 216)(177 199)(178 200)(179 201)(180 202)(217 263)(218 264)(219 265)(220 266)(221 267)(222 268)(223 269)(224 270)(225 253)(226 254)(227 255)(228 256)(229 257)(230 258)(231 259)(232 260)(233 261)(234 262)(235 286)(236 287)(237 288)(238 271)(239 272)(240 273)(241 274)(242 275)(243 276)(244 277)(245 278)(246 279)(247 280)(248 281)(249 282)(250 283)(251 284)(252 285)
(1 33)(2 34)(3 35)(4 36)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(37 70)(38 71)(39 72)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)(85 97)(86 98)(87 99)(88 100)(89 101)(90 102)(109 140)(110 141)(111 142)(112 143)(113 144)(114 127)(115 128)(116 129)(117 130)(118 131)(119 132)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 163)(155 164)(156 165)(157 166)(158 167)(159 168)(160 169)(161 170)(162 171)(181 206)(182 207)(183 208)(184 209)(185 210)(186 211)(187 212)(188 213)(189 214)(190 215)(191 216)(192 199)(193 200)(194 201)(195 202)(196 203)(197 204)(198 205)(217 252)(218 235)(219 236)(220 237)(221 238)(222 239)(223 240)(224 241)(225 242)(226 243)(227 244)(228 245)(229 246)(230 247)(231 248)(232 249)(233 250)(234 251)(253 275)(254 276)(255 277)(256 278)(257 279)(258 280)(259 281)(260 282)(261 283)(262 284)(263 285)(264 286)(265 287)(266 288)(267 271)(268 272)(269 273)(270 274)
(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)
(1 220 10 229)(2 219 11 228)(3 218 12 227)(4 217 13 226)(5 234 14 225)(6 233 15 224)(7 232 16 223)(8 231 17 222)(9 230 18 221)(19 251 28 242)(20 250 29 241)(21 249 30 240)(22 248 31 239)(23 247 32 238)(24 246 33 237)(25 245 34 236)(26 244 35 235)(27 243 36 252)(37 266 46 257)(38 265 47 256)(39 264 48 255)(40 263 49 254)(41 262 50 253)(42 261 51 270)(43 260 52 269)(44 259 53 268)(45 258 54 267)(55 285 64 276)(56 284 65 275)(57 283 66 274)(58 282 67 273)(59 281 68 272)(60 280 69 271)(61 279 70 288)(62 278 71 287)(63 277 72 286)(73 146 82 155)(74 145 83 154)(75 162 84 153)(76 161 85 152)(77 160 86 151)(78 159 87 150)(79 158 88 149)(80 157 89 148)(81 156 90 147)(91 167 100 176)(92 166 101 175)(93 165 102 174)(94 164 103 173)(95 163 104 172)(96 180 105 171)(97 179 106 170)(98 178 107 169)(99 177 108 168)(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)(127 207 136 216)(128 206 137 215)(129 205 138 214)(130 204 139 213)(131 203 140 212)(132 202 141 211)(133 201 142 210)(134 200 143 209)(135 199 144 208)
G:=sub<Sym(288)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(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)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,127)(71,128)(72,129)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,251)(164,252)(165,235)(166,236)(167,237)(168,238)(169,239)(170,240)(171,241)(172,242)(173,243)(174,244)(175,245)(176,246)(177,247)(178,248)(179,249)(180,250)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,253)(188,254)(189,255)(190,256)(191,257)(192,258)(193,259)(194,260)(195,261)(196,262)(197,263)(198,264)(199,280)(200,281)(201,282)(202,283)(203,284)(204,285)(205,286)(206,287)(207,288)(208,271)(209,272)(210,273)(211,274)(212,275)(213,276)(214,277)(215,278)(216,279), (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)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,55)(73,126)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,193)(152,194)(153,195)(154,196)(155,197)(156,198)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,199)(178,200)(179,201)(180,202)(217,263)(218,264)(219,265)(220,266)(221,267)(222,268)(223,269)(224,270)(225,253)(226,254)(227,255)(228,256)(229,257)(230,258)(231,259)(232,260)(233,261)(234,262)(235,286)(236,287)(237,288)(238,271)(239,272)(240,273)(241,274)(242,275)(243,276)(244,277)(245,278)(246,279)(247,280)(248,281)(249,282)(250,283)(251,284)(252,285), (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,70)(38,71)(39,72)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(109,140)(110,141)(111,142)(112,143)(113,144)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,199)(193,200)(194,201)(195,202)(196,203)(197,204)(198,205)(217,252)(218,235)(219,236)(220,237)(221,238)(222,239)(223,240)(224,241)(225,242)(226,243)(227,244)(228,245)(229,246)(230,247)(231,248)(232,249)(233,250)(234,251)(253,275)(254,276)(255,277)(256,278)(257,279)(258,280)(259,281)(260,282)(261,283)(262,284)(263,285)(264,286)(265,287)(266,288)(267,271)(268,272)(269,273)(270,274), (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), (1,220,10,229)(2,219,11,228)(3,218,12,227)(4,217,13,226)(5,234,14,225)(6,233,15,224)(7,232,16,223)(8,231,17,222)(9,230,18,221)(19,251,28,242)(20,250,29,241)(21,249,30,240)(22,248,31,239)(23,247,32,238)(24,246,33,237)(25,245,34,236)(26,244,35,235)(27,243,36,252)(37,266,46,257)(38,265,47,256)(39,264,48,255)(40,263,49,254)(41,262,50,253)(42,261,51,270)(43,260,52,269)(44,259,53,268)(45,258,54,267)(55,285,64,276)(56,284,65,275)(57,283,66,274)(58,282,67,273)(59,281,68,272)(60,280,69,271)(61,279,70,288)(62,278,71,287)(63,277,72,286)(73,146,82,155)(74,145,83,154)(75,162,84,153)(76,161,85,152)(77,160,86,151)(78,159,87,150)(79,158,88,149)(80,157,89,148)(81,156,90,147)(91,167,100,176)(92,166,101,175)(93,165,102,174)(94,164,103,173)(95,163,104,172)(96,180,105,171)(97,179,106,170)(98,178,107,169)(99,177,108,168)(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)(127,207,136,216)(128,206,137,215)(129,205,138,214)(130,204,139,213)(131,203,140,212)(132,202,141,211)(133,201,142,210)(134,200,143,209)(135,199,144,208)>;
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,85)(8,86)(9,87)(10,88)(11,89)(12,90)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(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)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,141)(67,142)(68,143)(69,144)(70,127)(71,128)(72,129)(145,225)(146,226)(147,227)(148,228)(149,229)(150,230)(151,231)(152,232)(153,233)(154,234)(155,217)(156,218)(157,219)(158,220)(159,221)(160,222)(161,223)(162,224)(163,251)(164,252)(165,235)(166,236)(167,237)(168,238)(169,239)(170,240)(171,241)(172,242)(173,243)(174,244)(175,245)(176,246)(177,247)(178,248)(179,249)(180,250)(181,265)(182,266)(183,267)(184,268)(185,269)(186,270)(187,253)(188,254)(189,255)(190,256)(191,257)(192,258)(193,259)(194,260)(195,261)(196,262)(197,263)(198,264)(199,280)(200,281)(201,282)(202,283)(203,284)(204,285)(205,286)(206,287)(207,288)(208,271)(209,272)(210,273)(211,274)(212,275)(213,276)(214,277)(215,278)(216,279), (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)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,55)(73,126)(74,109)(75,110)(76,111)(77,112)(78,113)(79,114)(80,115)(81,116)(82,117)(83,118)(84,119)(85,120)(86,121)(87,122)(88,123)(89,124)(90,125)(91,127)(92,128)(93,129)(94,130)(95,131)(96,132)(97,133)(98,134)(99,135)(100,136)(101,137)(102,138)(103,139)(104,140)(105,141)(106,142)(107,143)(108,144)(145,187)(146,188)(147,189)(148,190)(149,191)(150,192)(151,193)(152,194)(153,195)(154,196)(155,197)(156,198)(157,181)(158,182)(159,183)(160,184)(161,185)(162,186)(163,203)(164,204)(165,205)(166,206)(167,207)(168,208)(169,209)(170,210)(171,211)(172,212)(173,213)(174,214)(175,215)(176,216)(177,199)(178,200)(179,201)(180,202)(217,263)(218,264)(219,265)(220,266)(221,267)(222,268)(223,269)(224,270)(225,253)(226,254)(227,255)(228,256)(229,257)(230,258)(231,259)(232,260)(233,261)(234,262)(235,286)(236,287)(237,288)(238,271)(239,272)(240,273)(241,274)(242,275)(243,276)(244,277)(245,278)(246,279)(247,280)(248,281)(249,282)(250,283)(251,284)(252,285), (1,33)(2,34)(3,35)(4,36)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(37,70)(38,71)(39,72)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)(85,97)(86,98)(87,99)(88,100)(89,101)(90,102)(109,140)(110,141)(111,142)(112,143)(113,144)(114,127)(115,128)(116,129)(117,130)(118,131)(119,132)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,163)(155,164)(156,165)(157,166)(158,167)(159,168)(160,169)(161,170)(162,171)(181,206)(182,207)(183,208)(184,209)(185,210)(186,211)(187,212)(188,213)(189,214)(190,215)(191,216)(192,199)(193,200)(194,201)(195,202)(196,203)(197,204)(198,205)(217,252)(218,235)(219,236)(220,237)(221,238)(222,239)(223,240)(224,241)(225,242)(226,243)(227,244)(228,245)(229,246)(230,247)(231,248)(232,249)(233,250)(234,251)(253,275)(254,276)(255,277)(256,278)(257,279)(258,280)(259,281)(260,282)(261,283)(262,284)(263,285)(264,286)(265,287)(266,288)(267,271)(268,272)(269,273)(270,274), (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), (1,220,10,229)(2,219,11,228)(3,218,12,227)(4,217,13,226)(5,234,14,225)(6,233,15,224)(7,232,16,223)(8,231,17,222)(9,230,18,221)(19,251,28,242)(20,250,29,241)(21,249,30,240)(22,248,31,239)(23,247,32,238)(24,246,33,237)(25,245,34,236)(26,244,35,235)(27,243,36,252)(37,266,46,257)(38,265,47,256)(39,264,48,255)(40,263,49,254)(41,262,50,253)(42,261,51,270)(43,260,52,269)(44,259,53,268)(45,258,54,267)(55,285,64,276)(56,284,65,275)(57,283,66,274)(58,282,67,273)(59,281,68,272)(60,280,69,271)(61,279,70,288)(62,278,71,287)(63,277,72,286)(73,146,82,155)(74,145,83,154)(75,162,84,153)(76,161,85,152)(77,160,86,151)(78,159,87,150)(79,158,88,149)(80,157,89,148)(81,156,90,147)(91,167,100,176)(92,166,101,175)(93,165,102,174)(94,164,103,173)(95,163,104,172)(96,180,105,171)(97,179,106,170)(98,178,107,169)(99,177,108,168)(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)(127,207,136,216)(128,206,137,215)(129,205,138,214)(130,204,139,213)(131,203,140,212)(132,202,141,211)(133,201,142,210)(134,200,143,209)(135,199,144,208) );
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,85),(8,86),(9,87),(10,88),(11,89),(12,90),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,91),(34,92),(35,93),(36,94),(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),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,141),(67,142),(68,143),(69,144),(70,127),(71,128),(72,129),(145,225),(146,226),(147,227),(148,228),(149,229),(150,230),(151,231),(152,232),(153,233),(154,234),(155,217),(156,218),(157,219),(158,220),(159,221),(160,222),(161,223),(162,224),(163,251),(164,252),(165,235),(166,236),(167,237),(168,238),(169,239),(170,240),(171,241),(172,242),(173,243),(174,244),(175,245),(176,246),(177,247),(178,248),(179,249),(180,250),(181,265),(182,266),(183,267),(184,268),(185,269),(186,270),(187,253),(188,254),(189,255),(190,256),(191,257),(192,258),(193,259),(194,260),(195,261),(196,262),(197,263),(198,264),(199,280),(200,281),(201,282),(202,283),(203,284),(204,285),(205,286),(206,287),(207,288),(208,271),(209,272),(210,273),(211,274),(212,275),(213,276),(214,277),(215,278),(216,279)], [(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),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,55),(73,126),(74,109),(75,110),(76,111),(77,112),(78,113),(79,114),(80,115),(81,116),(82,117),(83,118),(84,119),(85,120),(86,121),(87,122),(88,123),(89,124),(90,125),(91,127),(92,128),(93,129),(94,130),(95,131),(96,132),(97,133),(98,134),(99,135),(100,136),(101,137),(102,138),(103,139),(104,140),(105,141),(106,142),(107,143),(108,144),(145,187),(146,188),(147,189),(148,190),(149,191),(150,192),(151,193),(152,194),(153,195),(154,196),(155,197),(156,198),(157,181),(158,182),(159,183),(160,184),(161,185),(162,186),(163,203),(164,204),(165,205),(166,206),(167,207),(168,208),(169,209),(170,210),(171,211),(172,212),(173,213),(174,214),(175,215),(176,216),(177,199),(178,200),(179,201),(180,202),(217,263),(218,264),(219,265),(220,266),(221,267),(222,268),(223,269),(224,270),(225,253),(226,254),(227,255),(228,256),(229,257),(230,258),(231,259),(232,260),(233,261),(234,262),(235,286),(236,287),(237,288),(238,271),(239,272),(240,273),(241,274),(242,275),(243,276),(244,277),(245,278),(246,279),(247,280),(248,281),(249,282),(250,283),(251,284),(252,285)], [(1,33),(2,34),(3,35),(4,36),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(37,70),(38,71),(39,72),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96),(85,97),(86,98),(87,99),(88,100),(89,101),(90,102),(109,140),(110,141),(111,142),(112,143),(113,144),(114,127),(115,128),(116,129),(117,130),(118,131),(119,132),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,163),(155,164),(156,165),(157,166),(158,167),(159,168),(160,169),(161,170),(162,171),(181,206),(182,207),(183,208),(184,209),(185,210),(186,211),(187,212),(188,213),(189,214),(190,215),(191,216),(192,199),(193,200),(194,201),(195,202),(196,203),(197,204),(198,205),(217,252),(218,235),(219,236),(220,237),(221,238),(222,239),(223,240),(224,241),(225,242),(226,243),(227,244),(228,245),(229,246),(230,247),(231,248),(232,249),(233,250),(234,251),(253,275),(254,276),(255,277),(256,278),(257,279),(258,280),(259,281),(260,282),(261,283),(262,284),(263,285),(264,286),(265,287),(266,288),(267,271),(268,272),(269,273),(270,274)], [(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)], [(1,220,10,229),(2,219,11,228),(3,218,12,227),(4,217,13,226),(5,234,14,225),(6,233,15,224),(7,232,16,223),(8,231,17,222),(9,230,18,221),(19,251,28,242),(20,250,29,241),(21,249,30,240),(22,248,31,239),(23,247,32,238),(24,246,33,237),(25,245,34,236),(26,244,35,235),(27,243,36,252),(37,266,46,257),(38,265,47,256),(39,264,48,255),(40,263,49,254),(41,262,50,253),(42,261,51,270),(43,260,52,269),(44,259,53,268),(45,258,54,267),(55,285,64,276),(56,284,65,275),(57,283,66,274),(58,282,67,273),(59,281,68,272),(60,280,69,271),(61,279,70,288),(62,278,71,287),(63,277,72,286),(73,146,82,155),(74,145,83,154),(75,162,84,153),(76,161,85,152),(77,160,86,151),(78,159,87,150),(79,158,88,149),(80,157,89,148),(81,156,90,147),(91,167,100,176),(92,166,101,175),(93,165,102,174),(94,164,103,173),(95,163,104,172),(96,180,105,171),(97,179,106,170),(98,178,107,169),(99,177,108,168),(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),(127,207,136,216),(128,206,137,215),(129,205,138,214),(130,204,139,213),(131,203,140,212),(132,202,141,211),(133,201,142,210),(134,200,143,209),(135,199,144,208)]])
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