direct product, metabelian, supersoluble, monomial
Aliases: C2×C12⋊Dic3, C62.94D4, C62.19Q8, C62.246C23, (C6×C12)⋊12C4, C6⋊2(C4⋊Dic3), C12⋊7(C2×Dic3), (C2×C12)⋊5Dic3, (C2×C6).42D12, C6.62(C2×D12), (C2×C12).388D6, C6.46(C2×Dic6), (C2×C6).21Dic6, C62.113(C2×C4), (C22×C12).24S3, (C22×C6).155D6, (C6×C12).304C22, C6.34(C22×Dic3), (C2×C62).107C22, C22.15(C12⋊S3), C22.5(C32⋊4Q8), (C2×C6×C12).9C2, (C3×C6)⋊9(C4⋊C4), C32⋊15(C2×C4⋊C4), C4⋊2(C2×C3⋊Dic3), C3⋊3(C2×C4⋊Dic3), (C3×C12)⋊22(C2×C4), (C3×C6).60(C2×Q8), C2.2(C2×C12⋊S3), (C2×C4)⋊3(C3⋊Dic3), (C3×C6).202(C2×D4), C23.36(C2×C3⋊S3), C2.3(C2×C32⋊4Q8), (C22×C4).8(C3⋊S3), (C2×C6).54(C2×Dic3), C2.4(C22×C3⋊Dic3), (C2×C6).263(C22×S3), (C3×C6).122(C22×C4), C22.21(C22×C3⋊S3), C22.14(C2×C3⋊Dic3), (C22×C3⋊Dic3).12C2, (C2×C3⋊Dic3).162C22, (C2×C4).85(C2×C3⋊S3), SmallGroup(288,782)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C12⋊Dic3 |
Generators and relations for C2×C12⋊Dic3
G = < a,b,c,d | a2=b12=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 756 in 276 conjugacy classes, 173 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C2×C6, C4⋊C4, C22×C4, C22×C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C2×C4⋊C4, C3⋊Dic3, C3×C12, C62, C62, C4⋊Dic3, C22×Dic3, C22×C12, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C2×C62, C2×C4⋊Dic3, C12⋊Dic3, C22×C3⋊Dic3, C2×C6×C12, C2×C12⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C3⋊S3, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C3⋊Dic3, C2×C3⋊S3, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C32⋊4Q8, C12⋊S3, C2×C3⋊Dic3, C22×C3⋊S3, C2×C4⋊Dic3, C12⋊Dic3, C2×C32⋊4Q8, C2×C12⋊S3, C22×C3⋊Dic3, C2×C12⋊Dic3
►Regular action on 288 points
Generators in S288
(1 272)(2 273)(3 274)(4 275)(5 276)(6 265)(7 266)(8 267)(9 268)(10 269)(11 270)(12 271)(13 195)(14 196)(15 197)(16 198)(17 199)(18 200)(19 201)(20 202)(21 203)(22 204)(23 193)(24 194)(25 209)(26 210)(27 211)(28 212)(29 213)(30 214)(31 215)(32 216)(33 205)(34 206)(35 207)(36 208)(37 117)(38 118)(39 119)(40 120)(41 109)(42 110)(43 111)(44 112)(45 113)(46 114)(47 115)(48 116)(49 252)(50 241)(51 242)(52 243)(53 244)(54 245)(55 246)(56 247)(57 248)(58 249)(59 250)(60 251)(61 261)(62 262)(63 263)(64 264)(65 253)(66 254)(67 255)(68 256)(69 257)(70 258)(71 259)(72 260)(73 97)(74 98)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(85 192)(86 181)(87 182)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(121 140)(122 141)(123 142)(124 143)(125 144)(126 133)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(145 236)(146 237)(147 238)(148 239)(149 240)(150 229)(151 230)(152 231)(153 232)(154 233)(155 234)(156 235)(157 171)(158 172)(159 173)(160 174)(161 175)(162 176)(163 177)(164 178)(165 179)(166 180)(167 169)(168 170)(217 280)(218 281)(219 282)(220 283)(221 284)(222 285)(223 286)(224 287)(225 288)(226 277)(227 278)(228 279)
(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)
(1 119 244 147 138 157)(2 120 245 148 139 158)(3 109 246 149 140 159)(4 110 247 150 141 160)(5 111 248 151 142 161)(6 112 249 152 143 162)(7 113 250 153 144 163)(8 114 251 154 133 164)(9 115 252 155 134 165)(10 116 241 156 135 166)(11 117 242 145 136 167)(12 118 243 146 137 168)(13 190 81 31 220 66)(14 191 82 32 221 67)(15 192 83 33 222 68)(16 181 84 34 223 69)(17 182 73 35 224 70)(18 183 74 36 225 71)(19 184 75 25 226 72)(20 185 76 26 227 61)(21 186 77 27 228 62)(22 187 78 28 217 63)(23 188 79 29 218 64)(24 189 80 30 219 65)(37 51 236 129 169 270)(38 52 237 130 170 271)(39 53 238 131 171 272)(40 54 239 132 172 273)(41 55 240 121 173 274)(42 56 229 122 174 275)(43 57 230 123 175 276)(44 58 231 124 176 265)(45 59 232 125 177 266)(46 60 233 126 178 267)(47 49 234 127 179 268)(48 50 235 128 180 269)(85 107 205 285 256 197)(86 108 206 286 257 198)(87 97 207 287 258 199)(88 98 208 288 259 200)(89 99 209 277 260 201)(90 100 210 278 261 202)(91 101 211 279 262 203)(92 102 212 280 263 204)(93 103 213 281 264 193)(94 104 214 282 253 194)(95 105 215 283 254 195)(96 106 216 284 255 196)
(1 67 147 82)(2 66 148 81)(3 65 149 80)(4 64 150 79)(5 63 151 78)(6 62 152 77)(7 61 153 76)(8 72 154 75)(9 71 155 74)(10 70 156 73)(11 69 145 84)(12 68 146 83)(13 245 31 158)(14 244 32 157)(15 243 33 168)(16 242 34 167)(17 241 35 166)(18 252 36 165)(19 251 25 164)(20 250 26 163)(21 249 27 162)(22 248 28 161)(23 247 29 160)(24 246 30 159)(37 286 129 86)(38 285 130 85)(39 284 131 96)(40 283 132 95)(41 282 121 94)(42 281 122 93)(43 280 123 92)(44 279 124 91)(45 278 125 90)(46 277 126 89)(47 288 127 88)(48 287 128 87)(49 208 179 200)(50 207 180 199)(51 206 169 198)(52 205 170 197)(53 216 171 196)(54 215 172 195)(55 214 173 194)(56 213 174 193)(57 212 175 204)(58 211 176 203)(59 210 177 202)(60 209 178 201)(97 269 258 235)(98 268 259 234)(99 267 260 233)(100 266 261 232)(101 265 262 231)(102 276 263 230)(103 275 264 229)(104 274 253 240)(105 273 254 239)(106 272 255 238)(107 271 256 237)(108 270 257 236)(109 219 140 189)(110 218 141 188)(111 217 142 187)(112 228 143 186)(113 227 144 185)(114 226 133 184)(115 225 134 183)(116 224 135 182)(117 223 136 181)(118 222 137 192)(119 221 138 191)(120 220 139 190)
G:=sub<Sym(288)| (1,272)(2,273)(3,274)(4,275)(5,276)(6,265)(7,266)(8,267)(9,268)(10,269)(11,270)(12,271)(13,195)(14,196)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,193)(24,194)(25,209)(26,210)(27,211)(28,212)(29,213)(30,214)(31,215)(32,216)(33,205)(34,206)(35,207)(36,208)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,252)(50,241)(51,242)(52,243)(53,244)(54,245)(55,246)(56,247)(57,248)(58,249)(59,250)(60,251)(61,261)(62,262)(63,263)(64,264)(65,253)(66,254)(67,255)(68,256)(69,257)(70,258)(71,259)(72,260)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,192)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(145,236)(146,237)(147,238)(148,239)(149,240)(150,229)(151,230)(152,231)(153,232)(154,233)(155,234)(156,235)(157,171)(158,172)(159,173)(160,174)(161,175)(162,176)(163,177)(164,178)(165,179)(166,180)(167,169)(168,170)(217,280)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)(224,287)(225,288)(226,277)(227,278)(228,279), (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), (1,119,244,147,138,157)(2,120,245,148,139,158)(3,109,246,149,140,159)(4,110,247,150,141,160)(5,111,248,151,142,161)(6,112,249,152,143,162)(7,113,250,153,144,163)(8,114,251,154,133,164)(9,115,252,155,134,165)(10,116,241,156,135,166)(11,117,242,145,136,167)(12,118,243,146,137,168)(13,190,81,31,220,66)(14,191,82,32,221,67)(15,192,83,33,222,68)(16,181,84,34,223,69)(17,182,73,35,224,70)(18,183,74,36,225,71)(19,184,75,25,226,72)(20,185,76,26,227,61)(21,186,77,27,228,62)(22,187,78,28,217,63)(23,188,79,29,218,64)(24,189,80,30,219,65)(37,51,236,129,169,270)(38,52,237,130,170,271)(39,53,238,131,171,272)(40,54,239,132,172,273)(41,55,240,121,173,274)(42,56,229,122,174,275)(43,57,230,123,175,276)(44,58,231,124,176,265)(45,59,232,125,177,266)(46,60,233,126,178,267)(47,49,234,127,179,268)(48,50,235,128,180,269)(85,107,205,285,256,197)(86,108,206,286,257,198)(87,97,207,287,258,199)(88,98,208,288,259,200)(89,99,209,277,260,201)(90,100,210,278,261,202)(91,101,211,279,262,203)(92,102,212,280,263,204)(93,103,213,281,264,193)(94,104,214,282,253,194)(95,105,215,283,254,195)(96,106,216,284,255,196), (1,67,147,82)(2,66,148,81)(3,65,149,80)(4,64,150,79)(5,63,151,78)(6,62,152,77)(7,61,153,76)(8,72,154,75)(9,71,155,74)(10,70,156,73)(11,69,145,84)(12,68,146,83)(13,245,31,158)(14,244,32,157)(15,243,33,168)(16,242,34,167)(17,241,35,166)(18,252,36,165)(19,251,25,164)(20,250,26,163)(21,249,27,162)(22,248,28,161)(23,247,29,160)(24,246,30,159)(37,286,129,86)(38,285,130,85)(39,284,131,96)(40,283,132,95)(41,282,121,94)(42,281,122,93)(43,280,123,92)(44,279,124,91)(45,278,125,90)(46,277,126,89)(47,288,127,88)(48,287,128,87)(49,208,179,200)(50,207,180,199)(51,206,169,198)(52,205,170,197)(53,216,171,196)(54,215,172,195)(55,214,173,194)(56,213,174,193)(57,212,175,204)(58,211,176,203)(59,210,177,202)(60,209,178,201)(97,269,258,235)(98,268,259,234)(99,267,260,233)(100,266,261,232)(101,265,262,231)(102,276,263,230)(103,275,264,229)(104,274,253,240)(105,273,254,239)(106,272,255,238)(107,271,256,237)(108,270,257,236)(109,219,140,189)(110,218,141,188)(111,217,142,187)(112,228,143,186)(113,227,144,185)(114,226,133,184)(115,225,134,183)(116,224,135,182)(117,223,136,181)(118,222,137,192)(119,221,138,191)(120,220,139,190)>;
G:=Group( (1,272)(2,273)(3,274)(4,275)(5,276)(6,265)(7,266)(8,267)(9,268)(10,269)(11,270)(12,271)(13,195)(14,196)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,193)(24,194)(25,209)(26,210)(27,211)(28,212)(29,213)(30,214)(31,215)(32,216)(33,205)(34,206)(35,207)(36,208)(37,117)(38,118)(39,119)(40,120)(41,109)(42,110)(43,111)(44,112)(45,113)(46,114)(47,115)(48,116)(49,252)(50,241)(51,242)(52,243)(53,244)(54,245)(55,246)(56,247)(57,248)(58,249)(59,250)(60,251)(61,261)(62,262)(63,263)(64,264)(65,253)(66,254)(67,255)(68,256)(69,257)(70,258)(71,259)(72,260)(73,97)(74,98)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(85,192)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(121,140)(122,141)(123,142)(124,143)(125,144)(126,133)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(145,236)(146,237)(147,238)(148,239)(149,240)(150,229)(151,230)(152,231)(153,232)(154,233)(155,234)(156,235)(157,171)(158,172)(159,173)(160,174)(161,175)(162,176)(163,177)(164,178)(165,179)(166,180)(167,169)(168,170)(217,280)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)(224,287)(225,288)(226,277)(227,278)(228,279), (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), (1,119,244,147,138,157)(2,120,245,148,139,158)(3,109,246,149,140,159)(4,110,247,150,141,160)(5,111,248,151,142,161)(6,112,249,152,143,162)(7,113,250,153,144,163)(8,114,251,154,133,164)(9,115,252,155,134,165)(10,116,241,156,135,166)(11,117,242,145,136,167)(12,118,243,146,137,168)(13,190,81,31,220,66)(14,191,82,32,221,67)(15,192,83,33,222,68)(16,181,84,34,223,69)(17,182,73,35,224,70)(18,183,74,36,225,71)(19,184,75,25,226,72)(20,185,76,26,227,61)(21,186,77,27,228,62)(22,187,78,28,217,63)(23,188,79,29,218,64)(24,189,80,30,219,65)(37,51,236,129,169,270)(38,52,237,130,170,271)(39,53,238,131,171,272)(40,54,239,132,172,273)(41,55,240,121,173,274)(42,56,229,122,174,275)(43,57,230,123,175,276)(44,58,231,124,176,265)(45,59,232,125,177,266)(46,60,233,126,178,267)(47,49,234,127,179,268)(48,50,235,128,180,269)(85,107,205,285,256,197)(86,108,206,286,257,198)(87,97,207,287,258,199)(88,98,208,288,259,200)(89,99,209,277,260,201)(90,100,210,278,261,202)(91,101,211,279,262,203)(92,102,212,280,263,204)(93,103,213,281,264,193)(94,104,214,282,253,194)(95,105,215,283,254,195)(96,106,216,284,255,196), (1,67,147,82)(2,66,148,81)(3,65,149,80)(4,64,150,79)(5,63,151,78)(6,62,152,77)(7,61,153,76)(8,72,154,75)(9,71,155,74)(10,70,156,73)(11,69,145,84)(12,68,146,83)(13,245,31,158)(14,244,32,157)(15,243,33,168)(16,242,34,167)(17,241,35,166)(18,252,36,165)(19,251,25,164)(20,250,26,163)(21,249,27,162)(22,248,28,161)(23,247,29,160)(24,246,30,159)(37,286,129,86)(38,285,130,85)(39,284,131,96)(40,283,132,95)(41,282,121,94)(42,281,122,93)(43,280,123,92)(44,279,124,91)(45,278,125,90)(46,277,126,89)(47,288,127,88)(48,287,128,87)(49,208,179,200)(50,207,180,199)(51,206,169,198)(52,205,170,197)(53,216,171,196)(54,215,172,195)(55,214,173,194)(56,213,174,193)(57,212,175,204)(58,211,176,203)(59,210,177,202)(60,209,178,201)(97,269,258,235)(98,268,259,234)(99,267,260,233)(100,266,261,232)(101,265,262,231)(102,276,263,230)(103,275,264,229)(104,274,253,240)(105,273,254,239)(106,272,255,238)(107,271,256,237)(108,270,257,236)(109,219,140,189)(110,218,141,188)(111,217,142,187)(112,228,143,186)(113,227,144,185)(114,226,133,184)(115,225,134,183)(116,224,135,182)(117,223,136,181)(118,222,137,192)(119,221,138,191)(120,220,139,190) );
G=PermutationGroup([[(1,272),(2,273),(3,274),(4,275),(5,276),(6,265),(7,266),(8,267),(9,268),(10,269),(11,270),(12,271),(13,195),(14,196),(15,197),(16,198),(17,199),(18,200),(19,201),(20,202),(21,203),(22,204),(23,193),(24,194),(25,209),(26,210),(27,211),(28,212),(29,213),(30,214),(31,215),(32,216),(33,205),(34,206),(35,207),(36,208),(37,117),(38,118),(39,119),(40,120),(41,109),(42,110),(43,111),(44,112),(45,113),(46,114),(47,115),(48,116),(49,252),(50,241),(51,242),(52,243),(53,244),(54,245),(55,246),(56,247),(57,248),(58,249),(59,250),(60,251),(61,261),(62,262),(63,263),(64,264),(65,253),(66,254),(67,255),(68,256),(69,257),(70,258),(71,259),(72,260),(73,97),(74,98),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(85,192),(86,181),(87,182),(88,183),(89,184),(90,185),(91,186),(92,187),(93,188),(94,189),(95,190),(96,191),(121,140),(122,141),(123,142),(124,143),(125,144),(126,133),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(145,236),(146,237),(147,238),(148,239),(149,240),(150,229),(151,230),(152,231),(153,232),(154,233),(155,234),(156,235),(157,171),(158,172),(159,173),(160,174),(161,175),(162,176),(163,177),(164,178),(165,179),(166,180),(167,169),(168,170),(217,280),(218,281),(219,282),(220,283),(221,284),(222,285),(223,286),(224,287),(225,288),(226,277),(227,278),(228,279)], [(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)], [(1,119,244,147,138,157),(2,120,245,148,139,158),(3,109,246,149,140,159),(4,110,247,150,141,160),(5,111,248,151,142,161),(6,112,249,152,143,162),(7,113,250,153,144,163),(8,114,251,154,133,164),(9,115,252,155,134,165),(10,116,241,156,135,166),(11,117,242,145,136,167),(12,118,243,146,137,168),(13,190,81,31,220,66),(14,191,82,32,221,67),(15,192,83,33,222,68),(16,181,84,34,223,69),(17,182,73,35,224,70),(18,183,74,36,225,71),(19,184,75,25,226,72),(20,185,76,26,227,61),(21,186,77,27,228,62),(22,187,78,28,217,63),(23,188,79,29,218,64),(24,189,80,30,219,65),(37,51,236,129,169,270),(38,52,237,130,170,271),(39,53,238,131,171,272),(40,54,239,132,172,273),(41,55,240,121,173,274),(42,56,229,122,174,275),(43,57,230,123,175,276),(44,58,231,124,176,265),(45,59,232,125,177,266),(46,60,233,126,178,267),(47,49,234,127,179,268),(48,50,235,128,180,269),(85,107,205,285,256,197),(86,108,206,286,257,198),(87,97,207,287,258,199),(88,98,208,288,259,200),(89,99,209,277,260,201),(90,100,210,278,261,202),(91,101,211,279,262,203),(92,102,212,280,263,204),(93,103,213,281,264,193),(94,104,214,282,253,194),(95,105,215,283,254,195),(96,106,216,284,255,196)], [(1,67,147,82),(2,66,148,81),(3,65,149,80),(4,64,150,79),(5,63,151,78),(6,62,152,77),(7,61,153,76),(8,72,154,75),(9,71,155,74),(10,70,156,73),(11,69,145,84),(12,68,146,83),(13,245,31,158),(14,244,32,157),(15,243,33,168),(16,242,34,167),(17,241,35,166),(18,252,36,165),(19,251,25,164),(20,250,26,163),(21,249,27,162),(22,248,28,161),(23,247,29,160),(24,246,30,159),(37,286,129,86),(38,285,130,85),(39,284,131,96),(40,283,132,95),(41,282,121,94),(42,281,122,93),(43,280,123,92),(44,279,124,91),(45,278,125,90),(46,277,126,89),(47,288,127,88),(48,287,128,87),(49,208,179,200),(50,207,180,199),(51,206,169,198),(52,205,170,197),(53,216,171,196),(54,215,172,195),(55,214,173,194),(56,213,174,193),(57,212,175,204),(58,211,176,203),(59,210,177,202),(60,209,178,201),(97,269,258,235),(98,268,259,234),(99,267,260,233),(100,266,261,232),(101,265,262,231),(102,276,263,230),(103,275,264,229),(104,274,253,240),(105,273,254,239),(106,272,255,238),(107,271,256,237),(108,270,257,236),(109,219,140,189),(110,218,141,188),(111,217,142,187),(112,228,143,186),(113,227,144,185),(114,226,133,184),(115,225,134,183),(116,224,135,182),(117,223,136,181),(118,222,137,192),(119,221,138,191),(120,220,139,190)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6AB | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 18 | ··· | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | - | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | Dic3 | D6 | D6 | Dic6 | D12 |
| kernel | C2×C12⋊Dic3 | C12⋊Dic3 | C22×C3⋊Dic3 | C2×C6×C12 | C6×C12 | C22×C12 | C62 | C62 | C2×C12 | C2×C12 | C22×C6 | C2×C6 | C2×C6 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 2 | 2 | 16 | 8 | 4 | 16 | 16 |
Matrix representation of C2×C12⋊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 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 0 | 10 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 2 | 3 | 0 | 0 | 0 | 0 |
| 12 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 0 | 0 | 8 | 6 |
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],[9,9,0,0,0,0,0,3,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,6,10,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,4,12,0,0,0,0,0,10],[2,12,0,0,0,0,3,11,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,7,8,0,0,0,0,10,6] >;
C2×C12⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times C_{12}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xC12:Dic3"); // GroupNames label
G:=SmallGroup(288,782);
// by ID
G=gap.SmallGroup(288,782);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,100,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations