direct product, metabelian, supersoluble, monomial
Aliases: C2×C6.Dic6, C62.18Q8, C62.128D4, C62.244C23, C6⋊3(Dic3⋊C4), (C2×C12).359D6, C62.86(C2×C4), (C2×C6).20Dic6, C6.44(C2×Dic6), (C22×C12).17S3, (C22×C6).153D6, (C6×C12).289C22, (C2×C62).105C22, C22.4(C32⋊4Q8), C22.19(C32⋊7D4), (C2×C6×C12).6C2, C6.78(S3×C2×C4), (C3×C6)⋊8(C4⋊C4), C32⋊14(C2×C4⋊C4), (C2×C6).56(C4×S3), C3⋊4(C2×Dic3⋊C4), (C3×C6).58(C2×Q8), (C2×C3⋊Dic3)⋊11C4, C3⋊Dic3⋊15(C2×C4), (C3×C6).272(C2×D4), C6.113(C2×C3⋊D4), C23.35(C2×C3⋊S3), C22.16(C4×C3⋊S3), C2.1(C2×C32⋊7D4), (C2×C6).95(C3⋊D4), C2.2(C2×C32⋊4Q8), (C22×C4).5(C3⋊S3), (C3×C6).109(C22×C4), (C2×C6).261(C22×S3), C22.20(C22×C3⋊S3), (C22×C3⋊Dic3).11C2, (C2×C3⋊Dic3).161C22, C2.18(C2×C4×C3⋊S3), (C2×C4).67(C2×C3⋊S3), SmallGroup(288,780)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C6.Dic6 |
Generators and relations for C2×C6.Dic6
G = < a,b,c,d | a2=b6=c12=1, d2=b3c6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >
Subgroups: 756 in 276 conjugacy classes, 133 normal (17 characteristic)
C1, C2, C2, C3, 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, C2×C12, C22×C6, C2×C4⋊C4, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, Dic3⋊C4, C22×Dic3, C22×C12, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C2×C62, C2×Dic3⋊C4, C6.Dic6, C22×C3⋊Dic3, C2×C6×C12, C2×C6.Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C3⋊S3, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C2×C3⋊S3, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C32⋊7D4, C22×C3⋊S3, C2×Dic3⋊C4, C6.Dic6, C2×C32⋊4Q8, C2×C4×C3⋊S3, C2×C32⋊7D4, C2×C6.Dic6
►Regular action on 288 points
Generators in S288
(1 68)(2 69)(3 70)(4 71)(5 72)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 240)(14 229)(15 230)(16 231)(17 232)(18 233)(19 234)(20 235)(21 236)(22 237)(23 238)(24 239)(25 197)(26 198)(27 199)(28 200)(29 201)(30 202)(31 203)(32 204)(33 193)(34 194)(35 195)(36 196)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 49)(47 50)(48 51)(73 218)(74 219)(75 220)(76 221)(77 222)(78 223)(79 224)(80 225)(81 226)(82 227)(83 228)(84 217)(85 153)(86 154)(87 155)(88 156)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 206)(98 207)(99 208)(100 209)(101 210)(102 211)(103 212)(104 213)(105 214)(106 215)(107 216)(108 205)(109 266)(110 267)(111 268)(112 269)(113 270)(114 271)(115 272)(116 273)(117 274)(118 275)(119 276)(120 265)(121 259)(122 260)(123 261)(124 262)(125 263)(126 264)(127 253)(128 254)(129 255)(130 256)(131 257)(132 258)(133 166)(134 167)(135 168)(136 157)(137 158)(138 159)(139 160)(140 161)(141 162)(142 163)(143 164)(144 165)(169 244)(170 245)(171 246)(172 247)(173 248)(174 249)(175 250)(176 251)(177 252)(178 241)(179 242)(180 243)(181 282)(182 283)(183 284)(184 285)(185 286)(186 287)(187 288)(188 277)(189 278)(190 279)(191 280)(192 281)
(1 133 256 170 203 85)(2 134 257 171 204 86)(3 135 258 172 193 87)(4 136 259 173 194 88)(5 137 260 174 195 89)(6 138 261 175 196 90)(7 139 262 176 197 91)(8 140 263 177 198 92)(9 141 264 178 199 93)(10 142 253 179 200 94)(11 143 254 180 201 95)(12 144 255 169 202 96)(13 56 208 217 268 189)(14 57 209 218 269 190)(15 58 210 219 270 191)(16 59 211 220 271 192)(17 60 212 221 272 181)(18 49 213 222 273 182)(19 50 214 223 274 183)(20 51 215 224 275 184)(21 52 216 225 276 185)(22 53 205 226 265 186)(23 54 206 227 266 187)(24 55 207 228 267 188)(25 147 62 160 124 251)(26 148 63 161 125 252)(27 149 64 162 126 241)(28 150 65 163 127 242)(29 151 66 164 128 243)(30 152 67 165 129 244)(31 153 68 166 130 245)(32 154 69 167 131 246)(33 155 70 168 132 247)(34 156 71 157 121 248)(35 145 72 158 122 249)(36 146 61 159 123 250)(37 107 80 119 286 236)(38 108 81 120 287 237)(39 97 82 109 288 238)(40 98 83 110 277 239)(41 99 84 111 278 240)(42 100 73 112 279 229)(43 101 74 113 280 230)(44 102 75 114 281 231)(45 103 76 115 282 232)(46 104 77 116 283 233)(47 105 78 117 284 234)(48 106 79 118 285 235)
(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)
(1 219 176 21)(2 14 177 224)(3 217 178 19)(4 24 179 222)(5 227 180 17)(6 22 169 220)(7 225 170 15)(8 20 171 218)(9 223 172 13)(10 18 173 228)(11 221 174 23)(12 16 175 226)(25 286 166 101)(26 106 167 279)(27 284 168 99)(28 104 157 277)(29 282 158 97)(30 102 159 287)(31 280 160 107)(32 100 161 285)(33 278 162 105)(34 98 163 283)(35 288 164 103)(36 108 165 281)(37 153 113 124)(38 129 114 146)(39 151 115 122)(40 127 116 156)(41 149 117 132)(42 125 118 154)(43 147 119 130)(44 123 120 152)(45 145 109 128)(46 121 110 150)(47 155 111 126)(48 131 112 148)(49 259 267 94)(50 87 268 264)(51 257 269 92)(52 85 270 262)(53 255 271 90)(54 95 272 260)(55 253 273 88)(56 93 274 258)(57 263 275 86)(58 91 276 256)(59 261 265 96)(60 89 266 254)(61 237 244 75)(62 80 245 230)(63 235 246 73)(64 78 247 240)(65 233 248 83)(66 76 249 238)(67 231 250 81)(68 74 251 236)(69 229 252 79)(70 84 241 234)(71 239 242 77)(72 82 243 232)(133 210 197 185)(134 190 198 215)(135 208 199 183)(136 188 200 213)(137 206 201 181)(138 186 202 211)(139 216 203 191)(140 184 204 209)(141 214 193 189)(142 182 194 207)(143 212 195 187)(144 192 196 205)
G:=sub<Sym(288)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,240)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,197)(26,198)(27,199)(28,200)(29,201)(30,202)(31,203)(32,204)(33,193)(34,194)(35,195)(36,196)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,49)(47,50)(48,51)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,217)(85,153)(86,154)(87,155)(88,156)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,206)(98,207)(99,208)(100,209)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,205)(109,266)(110,267)(111,268)(112,269)(113,270)(114,271)(115,272)(116,273)(117,274)(118,275)(119,276)(120,265)(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,166)(134,167)(135,168)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(169,244)(170,245)(171,246)(172,247)(173,248)(174,249)(175,250)(176,251)(177,252)(178,241)(179,242)(180,243)(181,282)(182,283)(183,284)(184,285)(185,286)(186,287)(187,288)(188,277)(189,278)(190,279)(191,280)(192,281), (1,133,256,170,203,85)(2,134,257,171,204,86)(3,135,258,172,193,87)(4,136,259,173,194,88)(5,137,260,174,195,89)(6,138,261,175,196,90)(7,139,262,176,197,91)(8,140,263,177,198,92)(9,141,264,178,199,93)(10,142,253,179,200,94)(11,143,254,180,201,95)(12,144,255,169,202,96)(13,56,208,217,268,189)(14,57,209,218,269,190)(15,58,210,219,270,191)(16,59,211,220,271,192)(17,60,212,221,272,181)(18,49,213,222,273,182)(19,50,214,223,274,183)(20,51,215,224,275,184)(21,52,216,225,276,185)(22,53,205,226,265,186)(23,54,206,227,266,187)(24,55,207,228,267,188)(25,147,62,160,124,251)(26,148,63,161,125,252)(27,149,64,162,126,241)(28,150,65,163,127,242)(29,151,66,164,128,243)(30,152,67,165,129,244)(31,153,68,166,130,245)(32,154,69,167,131,246)(33,155,70,168,132,247)(34,156,71,157,121,248)(35,145,72,158,122,249)(36,146,61,159,123,250)(37,107,80,119,286,236)(38,108,81,120,287,237)(39,97,82,109,288,238)(40,98,83,110,277,239)(41,99,84,111,278,240)(42,100,73,112,279,229)(43,101,74,113,280,230)(44,102,75,114,281,231)(45,103,76,115,282,232)(46,104,77,116,283,233)(47,105,78,117,284,234)(48,106,79,118,285,235), (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), (1,219,176,21)(2,14,177,224)(3,217,178,19)(4,24,179,222)(5,227,180,17)(6,22,169,220)(7,225,170,15)(8,20,171,218)(9,223,172,13)(10,18,173,228)(11,221,174,23)(12,16,175,226)(25,286,166,101)(26,106,167,279)(27,284,168,99)(28,104,157,277)(29,282,158,97)(30,102,159,287)(31,280,160,107)(32,100,161,285)(33,278,162,105)(34,98,163,283)(35,288,164,103)(36,108,165,281)(37,153,113,124)(38,129,114,146)(39,151,115,122)(40,127,116,156)(41,149,117,132)(42,125,118,154)(43,147,119,130)(44,123,120,152)(45,145,109,128)(46,121,110,150)(47,155,111,126)(48,131,112,148)(49,259,267,94)(50,87,268,264)(51,257,269,92)(52,85,270,262)(53,255,271,90)(54,95,272,260)(55,253,273,88)(56,93,274,258)(57,263,275,86)(58,91,276,256)(59,261,265,96)(60,89,266,254)(61,237,244,75)(62,80,245,230)(63,235,246,73)(64,78,247,240)(65,233,248,83)(66,76,249,238)(67,231,250,81)(68,74,251,236)(69,229,252,79)(70,84,241,234)(71,239,242,77)(72,82,243,232)(133,210,197,185)(134,190,198,215)(135,208,199,183)(136,188,200,213)(137,206,201,181)(138,186,202,211)(139,216,203,191)(140,184,204,209)(141,214,193,189)(142,182,194,207)(143,212,195,187)(144,192,196,205)>;
G:=Group( (1,68)(2,69)(3,70)(4,71)(5,72)(6,61)(7,62)(8,63)(9,64)(10,65)(11,66)(12,67)(13,240)(14,229)(15,230)(16,231)(17,232)(18,233)(19,234)(20,235)(21,236)(22,237)(23,238)(24,239)(25,197)(26,198)(27,199)(28,200)(29,201)(30,202)(31,203)(32,204)(33,193)(34,194)(35,195)(36,196)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,49)(47,50)(48,51)(73,218)(74,219)(75,220)(76,221)(77,222)(78,223)(79,224)(80,225)(81,226)(82,227)(83,228)(84,217)(85,153)(86,154)(87,155)(88,156)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,206)(98,207)(99,208)(100,209)(101,210)(102,211)(103,212)(104,213)(105,214)(106,215)(107,216)(108,205)(109,266)(110,267)(111,268)(112,269)(113,270)(114,271)(115,272)(116,273)(117,274)(118,275)(119,276)(120,265)(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,253)(128,254)(129,255)(130,256)(131,257)(132,258)(133,166)(134,167)(135,168)(136,157)(137,158)(138,159)(139,160)(140,161)(141,162)(142,163)(143,164)(144,165)(169,244)(170,245)(171,246)(172,247)(173,248)(174,249)(175,250)(176,251)(177,252)(178,241)(179,242)(180,243)(181,282)(182,283)(183,284)(184,285)(185,286)(186,287)(187,288)(188,277)(189,278)(190,279)(191,280)(192,281), (1,133,256,170,203,85)(2,134,257,171,204,86)(3,135,258,172,193,87)(4,136,259,173,194,88)(5,137,260,174,195,89)(6,138,261,175,196,90)(7,139,262,176,197,91)(8,140,263,177,198,92)(9,141,264,178,199,93)(10,142,253,179,200,94)(11,143,254,180,201,95)(12,144,255,169,202,96)(13,56,208,217,268,189)(14,57,209,218,269,190)(15,58,210,219,270,191)(16,59,211,220,271,192)(17,60,212,221,272,181)(18,49,213,222,273,182)(19,50,214,223,274,183)(20,51,215,224,275,184)(21,52,216,225,276,185)(22,53,205,226,265,186)(23,54,206,227,266,187)(24,55,207,228,267,188)(25,147,62,160,124,251)(26,148,63,161,125,252)(27,149,64,162,126,241)(28,150,65,163,127,242)(29,151,66,164,128,243)(30,152,67,165,129,244)(31,153,68,166,130,245)(32,154,69,167,131,246)(33,155,70,168,132,247)(34,156,71,157,121,248)(35,145,72,158,122,249)(36,146,61,159,123,250)(37,107,80,119,286,236)(38,108,81,120,287,237)(39,97,82,109,288,238)(40,98,83,110,277,239)(41,99,84,111,278,240)(42,100,73,112,279,229)(43,101,74,113,280,230)(44,102,75,114,281,231)(45,103,76,115,282,232)(46,104,77,116,283,233)(47,105,78,117,284,234)(48,106,79,118,285,235), (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), (1,219,176,21)(2,14,177,224)(3,217,178,19)(4,24,179,222)(5,227,180,17)(6,22,169,220)(7,225,170,15)(8,20,171,218)(9,223,172,13)(10,18,173,228)(11,221,174,23)(12,16,175,226)(25,286,166,101)(26,106,167,279)(27,284,168,99)(28,104,157,277)(29,282,158,97)(30,102,159,287)(31,280,160,107)(32,100,161,285)(33,278,162,105)(34,98,163,283)(35,288,164,103)(36,108,165,281)(37,153,113,124)(38,129,114,146)(39,151,115,122)(40,127,116,156)(41,149,117,132)(42,125,118,154)(43,147,119,130)(44,123,120,152)(45,145,109,128)(46,121,110,150)(47,155,111,126)(48,131,112,148)(49,259,267,94)(50,87,268,264)(51,257,269,92)(52,85,270,262)(53,255,271,90)(54,95,272,260)(55,253,273,88)(56,93,274,258)(57,263,275,86)(58,91,276,256)(59,261,265,96)(60,89,266,254)(61,237,244,75)(62,80,245,230)(63,235,246,73)(64,78,247,240)(65,233,248,83)(66,76,249,238)(67,231,250,81)(68,74,251,236)(69,229,252,79)(70,84,241,234)(71,239,242,77)(72,82,243,232)(133,210,197,185)(134,190,198,215)(135,208,199,183)(136,188,200,213)(137,206,201,181)(138,186,202,211)(139,216,203,191)(140,184,204,209)(141,214,193,189)(142,182,194,207)(143,212,195,187)(144,192,196,205) );
G=PermutationGroup([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,61),(7,62),(8,63),(9,64),(10,65),(11,66),(12,67),(13,240),(14,229),(15,230),(16,231),(17,232),(18,233),(19,234),(20,235),(21,236),(22,237),(23,238),(24,239),(25,197),(26,198),(27,199),(28,200),(29,201),(30,202),(31,203),(32,204),(33,193),(34,194),(35,195),(36,196),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,49),(47,50),(48,51),(73,218),(74,219),(75,220),(76,221),(77,222),(78,223),(79,224),(80,225),(81,226),(82,227),(83,228),(84,217),(85,153),(86,154),(87,155),(88,156),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,206),(98,207),(99,208),(100,209),(101,210),(102,211),(103,212),(104,213),(105,214),(106,215),(107,216),(108,205),(109,266),(110,267),(111,268),(112,269),(113,270),(114,271),(115,272),(116,273),(117,274),(118,275),(119,276),(120,265),(121,259),(122,260),(123,261),(124,262),(125,263),(126,264),(127,253),(128,254),(129,255),(130,256),(131,257),(132,258),(133,166),(134,167),(135,168),(136,157),(137,158),(138,159),(139,160),(140,161),(141,162),(142,163),(143,164),(144,165),(169,244),(170,245),(171,246),(172,247),(173,248),(174,249),(175,250),(176,251),(177,252),(178,241),(179,242),(180,243),(181,282),(182,283),(183,284),(184,285),(185,286),(186,287),(187,288),(188,277),(189,278),(190,279),(191,280),(192,281)], [(1,133,256,170,203,85),(2,134,257,171,204,86),(3,135,258,172,193,87),(4,136,259,173,194,88),(5,137,260,174,195,89),(6,138,261,175,196,90),(7,139,262,176,197,91),(8,140,263,177,198,92),(9,141,264,178,199,93),(10,142,253,179,200,94),(11,143,254,180,201,95),(12,144,255,169,202,96),(13,56,208,217,268,189),(14,57,209,218,269,190),(15,58,210,219,270,191),(16,59,211,220,271,192),(17,60,212,221,272,181),(18,49,213,222,273,182),(19,50,214,223,274,183),(20,51,215,224,275,184),(21,52,216,225,276,185),(22,53,205,226,265,186),(23,54,206,227,266,187),(24,55,207,228,267,188),(25,147,62,160,124,251),(26,148,63,161,125,252),(27,149,64,162,126,241),(28,150,65,163,127,242),(29,151,66,164,128,243),(30,152,67,165,129,244),(31,153,68,166,130,245),(32,154,69,167,131,246),(33,155,70,168,132,247),(34,156,71,157,121,248),(35,145,72,158,122,249),(36,146,61,159,123,250),(37,107,80,119,286,236),(38,108,81,120,287,237),(39,97,82,109,288,238),(40,98,83,110,277,239),(41,99,84,111,278,240),(42,100,73,112,279,229),(43,101,74,113,280,230),(44,102,75,114,281,231),(45,103,76,115,282,232),(46,104,77,116,283,233),(47,105,78,117,284,234),(48,106,79,118,285,235)], [(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)], [(1,219,176,21),(2,14,177,224),(3,217,178,19),(4,24,179,222),(5,227,180,17),(6,22,169,220),(7,225,170,15),(8,20,171,218),(9,223,172,13),(10,18,173,228),(11,221,174,23),(12,16,175,226),(25,286,166,101),(26,106,167,279),(27,284,168,99),(28,104,157,277),(29,282,158,97),(30,102,159,287),(31,280,160,107),(32,100,161,285),(33,278,162,105),(34,98,163,283),(35,288,164,103),(36,108,165,281),(37,153,113,124),(38,129,114,146),(39,151,115,122),(40,127,116,156),(41,149,117,132),(42,125,118,154),(43,147,119,130),(44,123,120,152),(45,145,109,128),(46,121,110,150),(47,155,111,126),(48,131,112,148),(49,259,267,94),(50,87,268,264),(51,257,269,92),(52,85,270,262),(53,255,271,90),(54,95,272,260),(55,253,273,88),(56,93,274,258),(57,263,275,86),(58,91,276,256),(59,261,265,96),(60,89,266,254),(61,237,244,75),(62,80,245,230),(63,235,246,73),(64,78,247,240),(65,233,248,83),(66,76,249,238),(67,231,250,81),(68,74,251,236),(69,229,252,79),(70,84,241,234),(71,239,242,77),(72,82,243,232),(133,210,197,185),(134,190,198,215),(135,208,199,183),(136,188,200,213),(137,206,201,181),(138,186,202,211),(139,216,203,191),(140,184,204,209),(141,214,193,189),(142,182,194,207),(143,212,195,187),(144,192,196,205)]])
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 | D6 | D6 | Dic6 | C4×S3 | C3⋊D4 |
| kernel | C2×C6.Dic6 | C6.Dic6 | C22×C3⋊Dic3 | C2×C6×C12 | C2×C3⋊Dic3 | C22×C12 | C62 | C62 | C2×C12 | C22×C6 | C2×C6 | C2×C6 | C2×C6 |
| # reps | 1 | 4 | 2 | 1 | 8 | 4 | 2 | 2 | 8 | 4 | 16 | 16 | 16 |
Matrix representation of C2×C6.Dic6 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 2 | 11 |
| 7 | 10 | 0 | 0 | 0 | 0 |
| 3 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[11,5,0,0,0,0,0,7,0,0,0,0,0,0,3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,2,0,0,0,0,11,11],[7,3,0,0,0,0,10,6,0,0,0,0,0,0,0,3,0,0,0,0,9,0,0,0,0,0,0,0,5,8,0,0,0,0,0,8] >;
C2×C6.Dic6 in GAP, Magma, Sage, TeX
C_2\times C_6.{\rm Dic}_6 % in TeX
G:=Group("C2xC6.Dic6"); // GroupNames label
G:=SmallGroup(288,780);
// by ID
G=gap.SmallGroup(288,780);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^12=1,d^2=b^3*c^6,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=b^3*c^-1>;
// generators/relations