direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C4×C3⋊Dic3, C62.243C23, (C6×C12)⋊17C4, (C3×C6)⋊4C42, C6⋊2(C4×Dic3), C12⋊8(C2×Dic3), (C2×C12)⋊7Dic3, C32⋊8(C2×C42), (C2×C12).428D6, C62.85(C2×C4), (C22×C12).38S3, (C22×C6).152D6, (C6×C12).358C22, C6.33(C22×Dic3), (C2×C62).104C22, C6.77(S3×C2×C4), C3⋊3(C2×C4×Dic3), (C2×C6×C12).23C2, (C3×C12)⋊25(C2×C4), (C2×C6).55(C4×S3), C23.34(C2×C3⋊S3), C22.15(C4×C3⋊S3), (C2×C6).53(C2×Dic3), C2.2(C22×C3⋊Dic3), (C22×C4).12(C3⋊S3), (C2×C6).260(C22×S3), (C3×C6).108(C22×C4), C22.19(C22×C3⋊S3), C22.13(C2×C3⋊Dic3), (C22×C3⋊Dic3).17C2, (C2×C3⋊Dic3).186C22, C2.3(C2×C4×C3⋊S3), (C2×C4).101(C2×C3⋊S3), SmallGroup(288,779)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C4×C3⋊Dic3 |
| C32 — C2×C4×C3⋊Dic3 |
Generators and relations for C2×C4×C3⋊Dic3
G = < a,b,c,d,e | a2=b4=c3=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 756 in 324 conjugacy classes, 189 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C42, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C2×C42, C3⋊Dic3, C3×C12, C62, C62, C4×Dic3, C22×Dic3, C22×C12, C2×C3⋊Dic3, C6×C12, C2×C62, C2×C4×Dic3, C4×C3⋊Dic3, C22×C3⋊Dic3, C2×C6×C12, C2×C4×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C3⋊S3, C4×S3, C2×Dic3, C22×S3, C2×C42, C3⋊Dic3, C2×C3⋊S3, C4×Dic3, S3×C2×C4, C22×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C22×C3⋊S3, C2×C4×Dic3, C4×C3⋊Dic3, C2×C4×C3⋊S3, C22×C3⋊Dic3, C2×C4×C3⋊Dic3
►Regular action on 288 points
Generators in S288
(1 105)(2 106)(3 107)(4 108)(5 103)(6 104)(7 205)(8 206)(9 207)(10 208)(11 209)(12 210)(13 101)(14 102)(15 97)(16 98)(17 99)(18 100)(19 128)(20 129)(21 130)(22 131)(23 132)(24 127)(25 224)(26 225)(27 226)(28 227)(29 228)(30 223)(31 240)(32 235)(33 236)(34 237)(35 238)(36 239)(37 163)(38 164)(39 165)(40 166)(41 167)(42 168)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 75)(56 76)(57 77)(58 78)(59 73)(60 74)(61 144)(62 139)(63 140)(64 141)(65 142)(66 143)(67 134)(68 135)(69 136)(70 137)(71 138)(72 133)(79 162)(80 157)(81 158)(82 159)(83 160)(84 161)(85 151)(86 152)(87 153)(88 154)(89 155)(90 156)(91 111)(92 112)(93 113)(94 114)(95 109)(96 110)(145 249)(146 250)(147 251)(148 252)(149 247)(150 248)(169 266)(170 267)(171 268)(172 269)(173 270)(174 265)(175 221)(176 222)(177 217)(178 218)(179 219)(180 220)(181 285)(182 286)(183 287)(184 288)(185 283)(186 284)(187 271)(188 272)(189 273)(190 274)(191 275)(192 276)(193 246)(194 241)(195 242)(196 243)(197 244)(198 245)(199 259)(200 260)(201 261)(202 262)(203 263)(204 264)(211 257)(212 258)(213 253)(214 254)(215 255)(216 256)(229 282)(230 277)(231 278)(232 279)(233 280)(234 281)
(1 72 18 61)(2 67 13 62)(3 68 14 63)(4 69 15 64)(5 70 16 65)(6 71 17 66)(7 269 277 244)(8 270 278 245)(9 265 279 246)(10 266 280 241)(11 267 281 242)(12 268 282 243)(19 95 56 37)(20 96 57 38)(21 91 58 39)(22 92 59 40)(23 93 60 41)(24 94 55 42)(25 275 33 263)(26 276 34 264)(27 271 35 259)(28 272 36 260)(29 273 31 261)(30 274 32 262)(43 88 49 82)(44 89 50 83)(45 90 51 84)(46 85 52 79)(47 86 53 80)(48 87 54 81)(73 166 131 112)(74 167 132 113)(75 168 127 114)(76 163 128 109)(77 164 129 110)(78 165 130 111)(97 141 108 136)(98 142 103 137)(99 143 104 138)(100 144 105 133)(101 139 106 134)(102 140 107 135)(115 159 121 154)(116 160 122 155)(117 161 123 156)(118 162 124 151)(119 157 125 152)(120 158 126 153)(145 216 177 184)(146 211 178 185)(147 212 179 186)(148 213 180 181)(149 214 175 182)(150 215 176 183)(169 233 194 208)(170 234 195 209)(171 229 196 210)(172 230 197 205)(173 231 198 206)(174 232 193 207)(187 238 199 226)(188 239 200 227)(189 240 201 228)(190 235 202 223)(191 236 203 224)(192 237 204 225)(217 288 249 256)(218 283 250 257)(219 284 251 258)(220 285 252 253)(221 286 247 254)(222 287 248 255)
(1 44 23)(2 45 24)(3 46 19)(4 47 20)(5 48 21)(6 43 22)(7 26 284)(8 27 285)(9 28 286)(10 29 287)(11 30 288)(12 25 283)(13 51 55)(14 52 56)(15 53 57)(16 54 58)(17 49 59)(18 50 60)(31 255 280)(32 256 281)(33 257 282)(34 258 277)(35 253 278)(36 254 279)(37 63 79)(38 64 80)(39 65 81)(40 66 82)(41 61 83)(42 62 84)(67 90 94)(68 85 95)(69 86 96)(70 87 91)(71 88 92)(72 89 93)(73 99 115)(74 100 116)(75 101 117)(76 102 118)(77 97 119)(78 98 120)(103 126 130)(104 121 131)(105 122 132)(106 123 127)(107 124 128)(108 125 129)(109 135 151)(110 136 152)(111 137 153)(112 138 154)(113 133 155)(114 134 156)(139 161 168)(140 162 163)(141 157 164)(142 158 165)(143 159 166)(144 160 167)(145 170 190)(146 171 191)(147 172 192)(148 173 187)(149 174 188)(150 169 189)(175 193 200)(176 194 201)(177 195 202)(178 196 203)(179 197 204)(180 198 199)(181 206 226)(182 207 227)(183 208 228)(184 209 223)(185 210 224)(186 205 225)(211 229 236)(212 230 237)(213 231 238)(214 232 239)(215 233 240)(216 234 235)(217 242 262)(218 243 263)(219 244 264)(220 245 259)(221 246 260)(222 241 261)(247 265 272)(248 266 273)(249 267 274)(250 268 275)(251 269 276)(252 270 271)
(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)
(1 288 4 285)(2 287 5 284)(3 286 6 283)(7 24 10 21)(8 23 11 20)(9 22 12 19)(13 255 16 258)(14 254 17 257)(15 253 18 256)(25 46 28 43)(26 45 29 48)(27 44 30 47)(31 54 34 51)(32 53 35 50)(33 52 36 49)(37 246 40 243)(38 245 41 242)(39 244 42 241)(55 280 58 277)(56 279 59 282)(57 278 60 281)(61 217 64 220)(62 222 65 219)(63 221 66 218)(67 248 70 251)(68 247 71 250)(69 252 72 249)(73 229 76 232)(74 234 77 231)(75 233 78 230)(79 260 82 263)(80 259 83 262)(81 264 84 261)(85 272 88 275)(86 271 89 274)(87 276 90 273)(91 269 94 266)(92 268 95 265)(93 267 96 270)(97 213 100 216)(98 212 101 215)(99 211 102 214)(103 186 106 183)(104 185 107 182)(105 184 108 181)(109 174 112 171)(110 173 113 170)(111 172 114 169)(115 236 118 239)(116 235 119 238)(117 240 120 237)(121 224 124 227)(122 223 125 226)(123 228 126 225)(127 208 130 205)(128 207 131 210)(129 206 132 209)(133 145 136 148)(134 150 137 147)(135 149 138 146)(139 176 142 179)(140 175 143 178)(141 180 144 177)(151 188 154 191)(152 187 155 190)(153 192 156 189)(157 199 160 202)(158 204 161 201)(159 203 162 200)(163 193 166 196)(164 198 167 195)(165 197 168 194)
G:=sub<Sym(288)| (1,105)(2,106)(3,107)(4,108)(5,103)(6,104)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,101)(14,102)(15,97)(16,98)(17,99)(18,100)(19,128)(20,129)(21,130)(22,131)(23,132)(24,127)(25,224)(26,225)(27,226)(28,227)(29,228)(30,223)(31,240)(32,235)(33,236)(34,237)(35,238)(36,239)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,144)(62,139)(63,140)(64,141)(65,142)(66,143)(67,134)(68,135)(69,136)(70,137)(71,138)(72,133)(79,162)(80,157)(81,158)(82,159)(83,160)(84,161)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,111)(92,112)(93,113)(94,114)(95,109)(96,110)(145,249)(146,250)(147,251)(148,252)(149,247)(150,248)(169,266)(170,267)(171,268)(172,269)(173,270)(174,265)(175,221)(176,222)(177,217)(178,218)(179,219)(180,220)(181,285)(182,286)(183,287)(184,288)(185,283)(186,284)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,246)(194,241)(195,242)(196,243)(197,244)(198,245)(199,259)(200,260)(201,261)(202,262)(203,263)(204,264)(211,257)(212,258)(213,253)(214,254)(215,255)(216,256)(229,282)(230,277)(231,278)(232,279)(233,280)(234,281), (1,72,18,61)(2,67,13,62)(3,68,14,63)(4,69,15,64)(5,70,16,65)(6,71,17,66)(7,269,277,244)(8,270,278,245)(9,265,279,246)(10,266,280,241)(11,267,281,242)(12,268,282,243)(19,95,56,37)(20,96,57,38)(21,91,58,39)(22,92,59,40)(23,93,60,41)(24,94,55,42)(25,275,33,263)(26,276,34,264)(27,271,35,259)(28,272,36,260)(29,273,31,261)(30,274,32,262)(43,88,49,82)(44,89,50,83)(45,90,51,84)(46,85,52,79)(47,86,53,80)(48,87,54,81)(73,166,131,112)(74,167,132,113)(75,168,127,114)(76,163,128,109)(77,164,129,110)(78,165,130,111)(97,141,108,136)(98,142,103,137)(99,143,104,138)(100,144,105,133)(101,139,106,134)(102,140,107,135)(115,159,121,154)(116,160,122,155)(117,161,123,156)(118,162,124,151)(119,157,125,152)(120,158,126,153)(145,216,177,184)(146,211,178,185)(147,212,179,186)(148,213,180,181)(149,214,175,182)(150,215,176,183)(169,233,194,208)(170,234,195,209)(171,229,196,210)(172,230,197,205)(173,231,198,206)(174,232,193,207)(187,238,199,226)(188,239,200,227)(189,240,201,228)(190,235,202,223)(191,236,203,224)(192,237,204,225)(217,288,249,256)(218,283,250,257)(219,284,251,258)(220,285,252,253)(221,286,247,254)(222,287,248,255), (1,44,23)(2,45,24)(3,46,19)(4,47,20)(5,48,21)(6,43,22)(7,26,284)(8,27,285)(9,28,286)(10,29,287)(11,30,288)(12,25,283)(13,51,55)(14,52,56)(15,53,57)(16,54,58)(17,49,59)(18,50,60)(31,255,280)(32,256,281)(33,257,282)(34,258,277)(35,253,278)(36,254,279)(37,63,79)(38,64,80)(39,65,81)(40,66,82)(41,61,83)(42,62,84)(67,90,94)(68,85,95)(69,86,96)(70,87,91)(71,88,92)(72,89,93)(73,99,115)(74,100,116)(75,101,117)(76,102,118)(77,97,119)(78,98,120)(103,126,130)(104,121,131)(105,122,132)(106,123,127)(107,124,128)(108,125,129)(109,135,151)(110,136,152)(111,137,153)(112,138,154)(113,133,155)(114,134,156)(139,161,168)(140,162,163)(141,157,164)(142,158,165)(143,159,166)(144,160,167)(145,170,190)(146,171,191)(147,172,192)(148,173,187)(149,174,188)(150,169,189)(175,193,200)(176,194,201)(177,195,202)(178,196,203)(179,197,204)(180,198,199)(181,206,226)(182,207,227)(183,208,228)(184,209,223)(185,210,224)(186,205,225)(211,229,236)(212,230,237)(213,231,238)(214,232,239)(215,233,240)(216,234,235)(217,242,262)(218,243,263)(219,244,264)(220,245,259)(221,246,260)(222,241,261)(247,265,272)(248,266,273)(249,267,274)(250,268,275)(251,269,276)(252,270,271), (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), (1,288,4,285)(2,287,5,284)(3,286,6,283)(7,24,10,21)(8,23,11,20)(9,22,12,19)(13,255,16,258)(14,254,17,257)(15,253,18,256)(25,46,28,43)(26,45,29,48)(27,44,30,47)(31,54,34,51)(32,53,35,50)(33,52,36,49)(37,246,40,243)(38,245,41,242)(39,244,42,241)(55,280,58,277)(56,279,59,282)(57,278,60,281)(61,217,64,220)(62,222,65,219)(63,221,66,218)(67,248,70,251)(68,247,71,250)(69,252,72,249)(73,229,76,232)(74,234,77,231)(75,233,78,230)(79,260,82,263)(80,259,83,262)(81,264,84,261)(85,272,88,275)(86,271,89,274)(87,276,90,273)(91,269,94,266)(92,268,95,265)(93,267,96,270)(97,213,100,216)(98,212,101,215)(99,211,102,214)(103,186,106,183)(104,185,107,182)(105,184,108,181)(109,174,112,171)(110,173,113,170)(111,172,114,169)(115,236,118,239)(116,235,119,238)(117,240,120,237)(121,224,124,227)(122,223,125,226)(123,228,126,225)(127,208,130,205)(128,207,131,210)(129,206,132,209)(133,145,136,148)(134,150,137,147)(135,149,138,146)(139,176,142,179)(140,175,143,178)(141,180,144,177)(151,188,154,191)(152,187,155,190)(153,192,156,189)(157,199,160,202)(158,204,161,201)(159,203,162,200)(163,193,166,196)(164,198,167,195)(165,197,168,194)>;
G:=Group( (1,105)(2,106)(3,107)(4,108)(5,103)(6,104)(7,205)(8,206)(9,207)(10,208)(11,209)(12,210)(13,101)(14,102)(15,97)(16,98)(17,99)(18,100)(19,128)(20,129)(21,130)(22,131)(23,132)(24,127)(25,224)(26,225)(27,226)(28,227)(29,228)(30,223)(31,240)(32,235)(33,236)(34,237)(35,238)(36,239)(37,163)(38,164)(39,165)(40,166)(41,167)(42,168)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,75)(56,76)(57,77)(58,78)(59,73)(60,74)(61,144)(62,139)(63,140)(64,141)(65,142)(66,143)(67,134)(68,135)(69,136)(70,137)(71,138)(72,133)(79,162)(80,157)(81,158)(82,159)(83,160)(84,161)(85,151)(86,152)(87,153)(88,154)(89,155)(90,156)(91,111)(92,112)(93,113)(94,114)(95,109)(96,110)(145,249)(146,250)(147,251)(148,252)(149,247)(150,248)(169,266)(170,267)(171,268)(172,269)(173,270)(174,265)(175,221)(176,222)(177,217)(178,218)(179,219)(180,220)(181,285)(182,286)(183,287)(184,288)(185,283)(186,284)(187,271)(188,272)(189,273)(190,274)(191,275)(192,276)(193,246)(194,241)(195,242)(196,243)(197,244)(198,245)(199,259)(200,260)(201,261)(202,262)(203,263)(204,264)(211,257)(212,258)(213,253)(214,254)(215,255)(216,256)(229,282)(230,277)(231,278)(232,279)(233,280)(234,281), (1,72,18,61)(2,67,13,62)(3,68,14,63)(4,69,15,64)(5,70,16,65)(6,71,17,66)(7,269,277,244)(8,270,278,245)(9,265,279,246)(10,266,280,241)(11,267,281,242)(12,268,282,243)(19,95,56,37)(20,96,57,38)(21,91,58,39)(22,92,59,40)(23,93,60,41)(24,94,55,42)(25,275,33,263)(26,276,34,264)(27,271,35,259)(28,272,36,260)(29,273,31,261)(30,274,32,262)(43,88,49,82)(44,89,50,83)(45,90,51,84)(46,85,52,79)(47,86,53,80)(48,87,54,81)(73,166,131,112)(74,167,132,113)(75,168,127,114)(76,163,128,109)(77,164,129,110)(78,165,130,111)(97,141,108,136)(98,142,103,137)(99,143,104,138)(100,144,105,133)(101,139,106,134)(102,140,107,135)(115,159,121,154)(116,160,122,155)(117,161,123,156)(118,162,124,151)(119,157,125,152)(120,158,126,153)(145,216,177,184)(146,211,178,185)(147,212,179,186)(148,213,180,181)(149,214,175,182)(150,215,176,183)(169,233,194,208)(170,234,195,209)(171,229,196,210)(172,230,197,205)(173,231,198,206)(174,232,193,207)(187,238,199,226)(188,239,200,227)(189,240,201,228)(190,235,202,223)(191,236,203,224)(192,237,204,225)(217,288,249,256)(218,283,250,257)(219,284,251,258)(220,285,252,253)(221,286,247,254)(222,287,248,255), (1,44,23)(2,45,24)(3,46,19)(4,47,20)(5,48,21)(6,43,22)(7,26,284)(8,27,285)(9,28,286)(10,29,287)(11,30,288)(12,25,283)(13,51,55)(14,52,56)(15,53,57)(16,54,58)(17,49,59)(18,50,60)(31,255,280)(32,256,281)(33,257,282)(34,258,277)(35,253,278)(36,254,279)(37,63,79)(38,64,80)(39,65,81)(40,66,82)(41,61,83)(42,62,84)(67,90,94)(68,85,95)(69,86,96)(70,87,91)(71,88,92)(72,89,93)(73,99,115)(74,100,116)(75,101,117)(76,102,118)(77,97,119)(78,98,120)(103,126,130)(104,121,131)(105,122,132)(106,123,127)(107,124,128)(108,125,129)(109,135,151)(110,136,152)(111,137,153)(112,138,154)(113,133,155)(114,134,156)(139,161,168)(140,162,163)(141,157,164)(142,158,165)(143,159,166)(144,160,167)(145,170,190)(146,171,191)(147,172,192)(148,173,187)(149,174,188)(150,169,189)(175,193,200)(176,194,201)(177,195,202)(178,196,203)(179,197,204)(180,198,199)(181,206,226)(182,207,227)(183,208,228)(184,209,223)(185,210,224)(186,205,225)(211,229,236)(212,230,237)(213,231,238)(214,232,239)(215,233,240)(216,234,235)(217,242,262)(218,243,263)(219,244,264)(220,245,259)(221,246,260)(222,241,261)(247,265,272)(248,266,273)(249,267,274)(250,268,275)(251,269,276)(252,270,271), (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), (1,288,4,285)(2,287,5,284)(3,286,6,283)(7,24,10,21)(8,23,11,20)(9,22,12,19)(13,255,16,258)(14,254,17,257)(15,253,18,256)(25,46,28,43)(26,45,29,48)(27,44,30,47)(31,54,34,51)(32,53,35,50)(33,52,36,49)(37,246,40,243)(38,245,41,242)(39,244,42,241)(55,280,58,277)(56,279,59,282)(57,278,60,281)(61,217,64,220)(62,222,65,219)(63,221,66,218)(67,248,70,251)(68,247,71,250)(69,252,72,249)(73,229,76,232)(74,234,77,231)(75,233,78,230)(79,260,82,263)(80,259,83,262)(81,264,84,261)(85,272,88,275)(86,271,89,274)(87,276,90,273)(91,269,94,266)(92,268,95,265)(93,267,96,270)(97,213,100,216)(98,212,101,215)(99,211,102,214)(103,186,106,183)(104,185,107,182)(105,184,108,181)(109,174,112,171)(110,173,113,170)(111,172,114,169)(115,236,118,239)(116,235,119,238)(117,240,120,237)(121,224,124,227)(122,223,125,226)(123,228,126,225)(127,208,130,205)(128,207,131,210)(129,206,132,209)(133,145,136,148)(134,150,137,147)(135,149,138,146)(139,176,142,179)(140,175,143,178)(141,180,144,177)(151,188,154,191)(152,187,155,190)(153,192,156,189)(157,199,160,202)(158,204,161,201)(159,203,162,200)(163,193,166,196)(164,198,167,195)(165,197,168,194) );
G=PermutationGroup([[(1,105),(2,106),(3,107),(4,108),(5,103),(6,104),(7,205),(8,206),(9,207),(10,208),(11,209),(12,210),(13,101),(14,102),(15,97),(16,98),(17,99),(18,100),(19,128),(20,129),(21,130),(22,131),(23,132),(24,127),(25,224),(26,225),(27,226),(28,227),(29,228),(30,223),(31,240),(32,235),(33,236),(34,237),(35,238),(36,239),(37,163),(38,164),(39,165),(40,166),(41,167),(42,168),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,75),(56,76),(57,77),(58,78),(59,73),(60,74),(61,144),(62,139),(63,140),(64,141),(65,142),(66,143),(67,134),(68,135),(69,136),(70,137),(71,138),(72,133),(79,162),(80,157),(81,158),(82,159),(83,160),(84,161),(85,151),(86,152),(87,153),(88,154),(89,155),(90,156),(91,111),(92,112),(93,113),(94,114),(95,109),(96,110),(145,249),(146,250),(147,251),(148,252),(149,247),(150,248),(169,266),(170,267),(171,268),(172,269),(173,270),(174,265),(175,221),(176,222),(177,217),(178,218),(179,219),(180,220),(181,285),(182,286),(183,287),(184,288),(185,283),(186,284),(187,271),(188,272),(189,273),(190,274),(191,275),(192,276),(193,246),(194,241),(195,242),(196,243),(197,244),(198,245),(199,259),(200,260),(201,261),(202,262),(203,263),(204,264),(211,257),(212,258),(213,253),(214,254),(215,255),(216,256),(229,282),(230,277),(231,278),(232,279),(233,280),(234,281)], [(1,72,18,61),(2,67,13,62),(3,68,14,63),(4,69,15,64),(5,70,16,65),(6,71,17,66),(7,269,277,244),(8,270,278,245),(9,265,279,246),(10,266,280,241),(11,267,281,242),(12,268,282,243),(19,95,56,37),(20,96,57,38),(21,91,58,39),(22,92,59,40),(23,93,60,41),(24,94,55,42),(25,275,33,263),(26,276,34,264),(27,271,35,259),(28,272,36,260),(29,273,31,261),(30,274,32,262),(43,88,49,82),(44,89,50,83),(45,90,51,84),(46,85,52,79),(47,86,53,80),(48,87,54,81),(73,166,131,112),(74,167,132,113),(75,168,127,114),(76,163,128,109),(77,164,129,110),(78,165,130,111),(97,141,108,136),(98,142,103,137),(99,143,104,138),(100,144,105,133),(101,139,106,134),(102,140,107,135),(115,159,121,154),(116,160,122,155),(117,161,123,156),(118,162,124,151),(119,157,125,152),(120,158,126,153),(145,216,177,184),(146,211,178,185),(147,212,179,186),(148,213,180,181),(149,214,175,182),(150,215,176,183),(169,233,194,208),(170,234,195,209),(171,229,196,210),(172,230,197,205),(173,231,198,206),(174,232,193,207),(187,238,199,226),(188,239,200,227),(189,240,201,228),(190,235,202,223),(191,236,203,224),(192,237,204,225),(217,288,249,256),(218,283,250,257),(219,284,251,258),(220,285,252,253),(221,286,247,254),(222,287,248,255)], [(1,44,23),(2,45,24),(3,46,19),(4,47,20),(5,48,21),(6,43,22),(7,26,284),(8,27,285),(9,28,286),(10,29,287),(11,30,288),(12,25,283),(13,51,55),(14,52,56),(15,53,57),(16,54,58),(17,49,59),(18,50,60),(31,255,280),(32,256,281),(33,257,282),(34,258,277),(35,253,278),(36,254,279),(37,63,79),(38,64,80),(39,65,81),(40,66,82),(41,61,83),(42,62,84),(67,90,94),(68,85,95),(69,86,96),(70,87,91),(71,88,92),(72,89,93),(73,99,115),(74,100,116),(75,101,117),(76,102,118),(77,97,119),(78,98,120),(103,126,130),(104,121,131),(105,122,132),(106,123,127),(107,124,128),(108,125,129),(109,135,151),(110,136,152),(111,137,153),(112,138,154),(113,133,155),(114,134,156),(139,161,168),(140,162,163),(141,157,164),(142,158,165),(143,159,166),(144,160,167),(145,170,190),(146,171,191),(147,172,192),(148,173,187),(149,174,188),(150,169,189),(175,193,200),(176,194,201),(177,195,202),(178,196,203),(179,197,204),(180,198,199),(181,206,226),(182,207,227),(183,208,228),(184,209,223),(185,210,224),(186,205,225),(211,229,236),(212,230,237),(213,231,238),(214,232,239),(215,233,240),(216,234,235),(217,242,262),(218,243,263),(219,244,264),(220,245,259),(221,246,260),(222,241,261),(247,265,272),(248,266,273),(249,267,274),(250,268,275),(251,269,276),(252,270,271)], [(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)], [(1,288,4,285),(2,287,5,284),(3,286,6,283),(7,24,10,21),(8,23,11,20),(9,22,12,19),(13,255,16,258),(14,254,17,257),(15,253,18,256),(25,46,28,43),(26,45,29,48),(27,44,30,47),(31,54,34,51),(32,53,35,50),(33,52,36,49),(37,246,40,243),(38,245,41,242),(39,244,42,241),(55,280,58,277),(56,279,59,282),(57,278,60,281),(61,217,64,220),(62,222,65,219),(63,221,66,218),(67,248,70,251),(68,247,71,250),(69,252,72,249),(73,229,76,232),(74,234,77,231),(75,233,78,230),(79,260,82,263),(80,259,83,262),(81,264,84,261),(85,272,88,275),(86,271,89,274),(87,276,90,273),(91,269,94,266),(92,268,95,265),(93,267,96,270),(97,213,100,216),(98,212,101,215),(99,211,102,214),(103,186,106,183),(104,185,107,182),(105,184,108,181),(109,174,112,171),(110,173,113,170),(111,172,114,169),(115,236,118,239),(116,235,119,238),(117,240,120,237),(121,224,124,227),(122,223,125,226),(123,228,126,225),(127,208,130,205),(128,207,131,210),(129,206,132,209),(133,145,136,148),(134,150,137,147),(135,149,138,146),(139,176,142,179),(140,175,143,178),(141,180,144,177),(151,188,154,191),(152,187,155,190),(153,192,156,189),(157,199,160,202),(158,204,161,201),(159,203,162,200),(163,193,166,196),(164,198,167,195),(165,197,168,194)]])
96 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 4A | ··· | 4H | 4I | ··· | 4X | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | D6 | C4×S3 |
| kernel | C2×C4×C3⋊Dic3 | C4×C3⋊Dic3 | C22×C3⋊Dic3 | C2×C6×C12 | C2×C3⋊Dic3 | C6×C12 | C22×C12 | C2×C12 | C2×C12 | C22×C6 | C2×C6 |
| # reps | 1 | 4 | 2 | 1 | 16 | 8 | 4 | 16 | 8 | 4 | 32 |
Matrix representation of C2×C4×C3⋊Dic3 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 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 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 6 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[7,10,0,0,0,0,3,6,0,0,0,0,0,0,3,6,0,0,0,0,3,10,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;
C2×C4×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times C_4\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xC4xC3:Dic3"); // GroupNames label
G:=SmallGroup(288,779);
// by ID
G=gap.SmallGroup(288,779);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^3=d^6=1,e^2=d^3,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,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations