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## G = C2×C6.Dic6order 288 = 25·32

### Direct product of C2 and C6.Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C6.Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — C2×C6.Dic6
 Lower central C32 — C3×C6 — C2×C6.Dic6
 Upper central C1 — C23 — C22×C4

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 [×3], C2 [×4], C3 [×4], C4 [×8], C22, C22 [×6], C6 [×28], C2×C4 [×2], C2×C4 [×12], C23, C32, Dic3 [×24], C12 [×8], C2×C6 [×28], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×40], C2×C12 [×8], C2×C12 [×8], C22×C6 [×4], C2×C4⋊C4, C3⋊Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C62 [×6], Dic3⋊C4 [×16], C22×Dic3 [×8], C22×C12 [×4], C2×C3⋊Dic3 [×8], C2×C3⋊Dic3 [×2], C6×C12 [×2], C6×C12 [×2], C2×C62, C2×Dic3⋊C4 [×4], C6.Dic6 [×4], C22×C3⋊Dic3 [×2], C2×C6×C12, C2×C6.Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×12], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C3⋊S3, Dic6 [×8], C4×S3 [×8], C3⋊D4 [×8], C22×S3 [×4], C2×C4⋊C4, C2×C3⋊S3 [×3], Dic3⋊C4 [×16], C2×Dic6 [×4], S3×C2×C4 [×4], C2×C3⋊D4 [×4], C324Q8 [×2], C4×C3⋊S3 [×2], C327D4 [×2], C22×C3⋊S3, C2×Dic3⋊C4 [×4], C6.Dic6 [×4], C2×C324Q8, C2×C4×C3⋊S3, C2×C327D4, C2×C6.Dic6

Smallest permutation representation of C2×C6.Dic6
Regular action on 288 points
Generators in S288
(1 163)(2 164)(3 165)(4 166)(5 167)(6 168)(7 157)(8 158)(9 159)(10 160)(11 161)(12 162)(13 275)(14 276)(15 265)(16 266)(17 267)(18 268)(19 269)(20 270)(21 271)(22 272)(23 273)(24 274)(25 171)(26 172)(27 173)(28 174)(29 175)(30 176)(31 177)(32 178)(33 179)(34 180)(35 169)(36 170)(37 92)(38 93)(39 94)(40 95)(41 96)(42 85)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 75)(50 76)(51 77)(52 78)(53 79)(54 80)(55 81)(56 82)(57 83)(58 84)(59 73)(60 74)(61 236)(62 237)(63 238)(64 239)(65 240)(66 229)(67 230)(68 231)(69 232)(70 233)(71 234)(72 235)(97 243)(98 244)(99 245)(100 246)(101 247)(102 248)(103 249)(104 250)(105 251)(106 252)(107 241)(108 242)(109 150)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 145)(117 146)(118 147)(119 148)(120 149)(121 189)(122 190)(123 191)(124 192)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 228)(134 217)(135 218)(136 219)(137 220)(138 221)(139 222)(140 223)(141 224)(142 225)(143 226)(144 227)(193 283)(194 284)(195 285)(196 286)(197 287)(198 288)(199 277)(200 278)(201 279)(202 280)(203 281)(204 282)(205 254)(206 255)(207 256)(208 257)(209 258)(210 259)(211 260)(212 261)(213 262)(214 263)(215 264)(216 253)
(1 151 131 68 266 26)(2 152 132 69 267 27)(3 153 121 70 268 28)(4 154 122 71 269 29)(5 155 123 72 270 30)(6 156 124 61 271 31)(7 145 125 62 272 32)(8 146 126 63 273 33)(9 147 127 64 274 34)(10 148 128 65 275 35)(11 149 129 66 276 36)(12 150 130 67 265 25)(13 169 160 119 184 240)(14 170 161 120 185 229)(15 171 162 109 186 230)(16 172 163 110 187 231)(17 173 164 111 188 232)(18 174 165 112 189 233)(19 175 166 113 190 234)(20 176 167 114 191 235)(21 177 168 115 192 236)(22 178 157 116 181 237)(23 179 158 117 182 238)(24 180 159 118 183 239)(37 57 104 218 199 253)(38 58 105 219 200 254)(39 59 106 220 201 255)(40 60 107 221 202 256)(41 49 108 222 203 257)(42 50 97 223 204 258)(43 51 98 224 193 259)(44 52 99 225 194 260)(45 53 100 226 195 261)(46 54 101 227 196 262)(47 55 102 228 197 263)(48 56 103 217 198 264)(73 252 137 279 206 94)(74 241 138 280 207 95)(75 242 139 281 208 96)(76 243 140 282 209 85)(77 244 141 283 210 86)(78 245 142 284 211 87)(79 246 143 285 212 88)(80 247 144 286 213 89)(81 248 133 287 214 90)(82 249 134 288 215 91)(83 250 135 277 216 92)(84 251 136 278 205 93)
(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)
(1 213 62 241)(2 246 63 206)(3 211 64 251)(4 244 65 216)(5 209 66 249)(6 242 67 214)(7 207 68 247)(8 252 69 212)(9 205 70 245)(10 250 71 210)(11 215 72 243)(12 248 61 208)(13 199 113 51)(14 56 114 204)(15 197 115 49)(16 54 116 202)(17 195 117 59)(18 52 118 200)(19 193 119 57)(20 50 120 198)(21 203 109 55)(22 60 110 196)(23 201 111 53)(24 58 112 194)(25 133 124 96)(26 89 125 138)(27 143 126 94)(28 87 127 136)(29 141 128 92)(30 85 129 134)(31 139 130 90)(32 95 131 144)(33 137 132 88)(34 93 121 142)(35 135 122 86)(36 91 123 140)(37 175 224 184)(38 189 225 180)(39 173 226 182)(40 187 227 178)(41 171 228 192)(42 185 217 176)(43 169 218 190)(44 183 219 174)(45 179 220 188)(46 181 221 172)(47 177 222 186)(48 191 223 170)(73 267 285 146)(74 151 286 272)(75 265 287 156)(76 149 288 270)(77 275 277 154)(78 147 278 268)(79 273 279 152)(80 145 280 266)(81 271 281 150)(82 155 282 276)(83 269 283 148)(84 153 284 274)(97 161 264 235)(98 240 253 166)(99 159 254 233)(100 238 255 164)(101 157 256 231)(102 236 257 162)(103 167 258 229)(104 234 259 160)(105 165 260 239)(106 232 261 158)(107 163 262 237)(108 230 263 168)

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

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

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

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

׿
×
𝔽