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## G = C2×C12⋊Dic3order 288 = 25·32

### Direct product of C2 and C12⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C12⋊Dic3
 Lower central C32 — C3×C6 — C2×C12⋊Dic3
 Upper central C1 — C23 — C22×C4

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 [×3], C2 [×4], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C6 [×28], C2×C4 [×6], C2×C4 [×8], C23, C32, Dic3 [×16], C12 [×16], C2×C6 [×28], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×32], C2×C12 [×24], C22×C6 [×4], C2×C4⋊C4, C3⋊Dic3 [×4], C3×C12 [×4], C62, C62 [×6], C4⋊Dic3 [×16], C22×Dic3 [×8], C22×C12 [×4], C2×C3⋊Dic3 [×4], C2×C3⋊Dic3 [×4], C6×C12 [×6], C2×C62, C2×C4⋊Dic3 [×4], C12⋊Dic3 [×4], C22×C3⋊Dic3 [×2], C2×C6×C12, C2×C12⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×16], D6 [×12], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3⋊S3, Dic6 [×8], D12 [×8], C2×Dic3 [×24], C22×S3 [×4], C2×C4⋊C4, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], C4⋊Dic3 [×16], C2×Dic6 [×4], C2×D12 [×4], C22×Dic3 [×4], C324Q8 [×2], C12⋊S3 [×2], C2×C3⋊Dic3 [×6], C22×C3⋊S3, C2×C4⋊Dic3 [×4], C12⋊Dic3 [×4], C2×C324Q8, C2×C12⋊S3, C22×C3⋊Dic3, C2×C12⋊Dic3

Smallest permutation representation of C2×C12⋊Dic3
Regular action on 288 points
Generators in S288
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 49)(9 50)(10 51)(11 52)(12 53)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 169)(23 170)(24 171)(25 225)(26 226)(27 227)(28 228)(29 217)(30 218)(31 219)(32 220)(33 221)(34 222)(35 223)(36 224)(37 71)(38 72)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(73 94)(74 95)(75 96)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)(97 184)(98 185)(99 186)(100 187)(101 188)(102 189)(103 190)(104 191)(105 192)(106 181)(107 182)(108 183)(109 158)(110 159)(111 160)(112 161)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 157)(121 287)(122 288)(123 277)(124 278)(125 279)(126 280)(127 281)(128 282)(129 283)(130 284)(131 285)(132 286)(133 194)(134 195)(135 196)(136 197)(137 198)(138 199)(139 200)(140 201)(141 202)(142 203)(143 204)(144 193)(145 208)(146 209)(147 210)(148 211)(149 212)(150 213)(151 214)(152 215)(153 216)(154 205)(155 206)(156 207)(229 253)(230 254)(231 255)(232 256)(233 257)(234 258)(235 259)(236 260)(237 261)(238 262)(239 263)(240 264)(241 268)(242 269)(243 270)(244 271)(245 272)(246 273)(247 274)(248 275)(249 276)(250 265)(251 266)(252 267)
(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 42 81 217 183 201)(2 43 82 218 184 202)(3 44 83 219 185 203)(4 45 84 220 186 204)(5 46 73 221 187 193)(6 47 74 222 188 194)(7 48 75 223 189 195)(8 37 76 224 190 196)(9 38 77 225 191 197)(10 39 78 226 192 198)(11 40 79 227 181 199)(12 41 80 228 182 200)(13 118 236 208 242 127)(14 119 237 209 243 128)(15 120 238 210 244 129)(16 109 239 211 245 130)(17 110 240 212 246 131)(18 111 229 213 247 132)(19 112 230 214 248 121)(20 113 231 215 249 122)(21 114 232 216 250 123)(22 115 233 205 251 124)(23 116 234 206 252 125)(24 117 235 207 241 126)(25 104 136 50 72 86)(26 105 137 51 61 87)(27 106 138 52 62 88)(28 107 139 53 63 89)(29 108 140 54 64 90)(30 97 141 55 65 91)(31 98 142 56 66 92)(32 99 143 57 67 93)(33 100 144 58 68 94)(34 101 133 59 69 95)(35 102 134 60 70 96)(36 103 135 49 71 85)(145 269 281 172 167 260)(146 270 282 173 168 261)(147 271 283 174 157 262)(148 272 284 175 158 263)(149 273 285 176 159 264)(150 274 286 177 160 253)(151 275 287 178 161 254)(152 276 288 179 162 255)(153 265 277 180 163 256)(154 266 278 169 164 257)(155 267 279 170 165 258)(156 268 280 171 166 259)
(1 117 217 241)(2 116 218 252)(3 115 219 251)(4 114 220 250)(5 113 221 249)(6 112 222 248)(7 111 223 247)(8 110 224 246)(9 109 225 245)(10 120 226 244)(11 119 227 243)(12 118 228 242)(13 182 208 41)(14 181 209 40)(15 192 210 39)(16 191 211 38)(17 190 212 37)(18 189 213 48)(19 188 214 47)(20 187 215 46)(21 186 216 45)(22 185 205 44)(23 184 206 43)(24 183 207 42)(25 272 50 158)(26 271 51 157)(27 270 52 168)(28 269 53 167)(29 268 54 166)(30 267 55 165)(31 266 56 164)(32 265 57 163)(33 276 58 162)(34 275 59 161)(35 274 60 160)(36 273 49 159)(61 174 105 147)(62 173 106 146)(63 172 107 145)(64 171 108 156)(65 170 97 155)(66 169 98 154)(67 180 99 153)(68 179 100 152)(69 178 101 151)(70 177 102 150)(71 176 103 149)(72 175 104 148)(73 122 193 231)(74 121 194 230)(75 132 195 229)(76 131 196 240)(77 130 197 239)(78 129 198 238)(79 128 199 237)(80 127 200 236)(81 126 201 235)(82 125 202 234)(83 124 203 233)(84 123 204 232)(85 285 135 264)(86 284 136 263)(87 283 137 262)(88 282 138 261)(89 281 139 260)(90 280 140 259)(91 279 141 258)(92 278 142 257)(93 277 143 256)(94 288 144 255)(95 287 133 254)(96 286 134 253)

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

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

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

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

׿
×
𝔽