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## G = C12⋊6Dic6order 288 = 25·32

### 1st semidirect product of C12 and Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12⋊6Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C2×C32⋊4Q8 — C12⋊6Dic6
 Lower central C32 — C62 — C12⋊6Dic6
 Upper central C1 — C22 — C42

Generators and relations for C126Dic6
G = < a,b,c | a12=b12=1, c2=b6, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 684 in 204 conjugacy classes, 101 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×6], C4 [×4], C22, C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C32, Dic3 [×16], C12 [×24], C2×C6 [×4], C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6, C3×C6 [×2], Dic6 [×16], C2×Dic3 [×16], C2×C12 [×12], C4⋊Q8, C3⋊Dic3 [×4], C3×C12 [×6], C62, C4⋊Dic3 [×16], C4×C12 [×4], C2×Dic6 [×8], C324Q8 [×4], C2×C3⋊Dic3 [×4], C6×C12, C6×C12 [×2], C122Q8 [×4], C12⋊Dic3 [×4], C122, C2×C324Q8 [×2], C126Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], Q8 [×4], C23, D6 [×12], C2×D4, C2×Q8 [×2], C3⋊S3, Dic6 [×16], D12 [×8], C22×S3 [×4], C4⋊Q8, C2×C3⋊S3 [×3], C2×Dic6 [×8], C2×D12 [×4], C324Q8 [×4], C12⋊S3 [×2], C22×C3⋊S3, C122Q8 [×4], C2×C324Q8 [×2], C2×C12⋊S3, C126Dic6

Smallest permutation representation of C126Dic6
Regular action on 288 points
Generators in S288
(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)
(1 177 75 187 121 141 103 201 85 153 115 165)(2 178 76 188 122 142 104 202 86 154 116 166)(3 179 77 189 123 143 105 203 87 155 117 167)(4 180 78 190 124 144 106 204 88 156 118 168)(5 169 79 191 125 133 107 193 89 145 119 157)(6 170 80 192 126 134 108 194 90 146 120 158)(7 171 81 181 127 135 97 195 91 147 109 159)(8 172 82 182 128 136 98 196 92 148 110 160)(9 173 83 183 129 137 99 197 93 149 111 161)(10 174 84 184 130 138 100 198 94 150 112 162)(11 175 73 185 131 139 101 199 95 151 113 163)(12 176 74 186 132 140 102 200 96 152 114 164)(13 220 283 268 27 211 45 262 57 248 69 240)(14 221 284 269 28 212 46 263 58 249 70 229)(15 222 285 270 29 213 47 264 59 250 71 230)(16 223 286 271 30 214 48 253 60 251 72 231)(17 224 287 272 31 215 37 254 49 252 61 232)(18 225 288 273 32 216 38 255 50 241 62 233)(19 226 277 274 33 205 39 256 51 242 63 234)(20 227 278 275 34 206 40 257 52 243 64 235)(21 228 279 276 35 207 41 258 53 244 65 236)(22 217 280 265 36 208 42 259 54 245 66 237)(23 218 281 266 25 209 43 260 55 246 67 238)(24 219 282 267 26 210 44 261 56 247 68 239)
(1 227 103 257)(2 226 104 256)(3 225 105 255)(4 224 106 254)(5 223 107 253)(6 222 108 264)(7 221 97 263)(8 220 98 262)(9 219 99 261)(10 218 100 260)(11 217 101 259)(12 228 102 258)(13 196 45 172)(14 195 46 171)(15 194 47 170)(16 193 48 169)(17 204 37 180)(18 203 38 179)(19 202 39 178)(20 201 40 177)(21 200 41 176)(22 199 42 175)(23 198 43 174)(24 197 44 173)(25 184 67 150)(26 183 68 149)(27 182 69 148)(28 181 70 147)(29 192 71 146)(30 191 72 145)(31 190 61 156)(32 189 62 155)(33 188 63 154)(34 187 64 153)(35 186 65 152)(36 185 66 151)(49 168 287 144)(50 167 288 143)(51 166 277 142)(52 165 278 141)(53 164 279 140)(54 163 280 139)(55 162 281 138)(56 161 282 137)(57 160 283 136)(58 159 284 135)(59 158 285 134)(60 157 286 133)(73 237 95 208)(74 236 96 207)(75 235 85 206)(76 234 86 205)(77 233 87 216)(78 232 88 215)(79 231 89 214)(80 230 90 213)(81 229 91 212)(82 240 92 211)(83 239 93 210)(84 238 94 209)(109 269 127 249)(110 268 128 248)(111 267 129 247)(112 266 130 246)(113 265 131 245)(114 276 132 244)(115 275 121 243)(116 274 122 242)(117 273 123 241)(118 272 124 252)(119 271 125 251)(120 270 126 250)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A ··· 4F 4G 4H 4I 4J 6A ··· 6L 12A ··· 12AV order 1 2 2 2 3 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 ··· 2 36 36 36 36 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 S3 D4 Q8 D6 Dic6 D12 kernel C12⋊6Dic6 C12⋊Dic3 C122 C2×C32⋊4Q8 C4×C12 C3×C12 C3×C12 C2×C12 C12 C12 # reps 1 4 1 2 4 2 4 12 32 16

Matrix representation of C126Dic6 in GL6(𝔽13)

 3 0 0 0 0 0 6 9 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 4 0 0 0 0 0 11 10
,
 10 0 0 0 0 0 7 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 7 7
,
 2 2 0 0 0 0 5 11 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 8 2 0 0 0 0 0 5

G:=sub<GL(6,GF(13))| [3,6,0,0,0,0,0,9,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,4,11,0,0,0,0,0,10],[10,7,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,7,0,0,0,0,0,7],[2,5,0,0,0,0,2,11,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;

C126Dic6 in GAP, Magma, Sage, TeX

C_{12}\rtimes_6{\rm Dic}_6
% in TeX

G:=Group("C12:6Dic6");
// GroupNames label

G:=SmallGroup(288,726);
// by ID

G=gap.SmallGroup(288,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,58,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

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