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G = C12.9Dic6order 288 = 25·32

9th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.9Dic6, C62.111D4, (C3×C6).35D8, C12.24(C4×S3), C324C83C4, (C2×C12).85D6, (C3×C12).11Q8, (C3×C6).15Q16, C6.21(D4⋊S3), C327(C2.D8), C33(C6.Q16), (C6×C12).52C22, C6.10(C3⋊Q16), C2.1(C327D8), C4.1(C324Q8), C6.16(Dic3⋊C4), C2.1(C327Q16), C12⋊Dic3.12C2, C2.3(C6.Dic6), C22.12(C327D4), C4.11(C4×C3⋊S3), C4⋊C4.1(C3⋊S3), (C3×C4⋊C4).17S3, (C3×C6).37(C4⋊C4), (C3×C12).46(C2×C4), (C32×C4⋊C4).4C2, (C2×C6).87(C3⋊D4), (C2×C324C8).5C2, (C2×C4).34(C2×C3⋊S3), SmallGroup(288,282)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C12.9Dic6
C1C3C32C3×C6C62C6×C12C2×C324C8 — C12.9Dic6
C32C3×C6C3×C12 — C12.9Dic6
C1C22C2×C4C4⋊C4

Generators and relations for C12.9Dic6
 G = < a,b,c | a12=b12=1, c2=a9b6, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >

Subgroups: 332 in 108 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×4], C12 [×8], C12 [×4], C2×C6 [×4], C4⋊C4, C4⋊C4, C2×C8, C3×C6 [×3], C3⋊C8 [×8], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C2.D8, C3⋊Dic3, C3×C12 [×2], C3×C12, C62, C2×C3⋊C8 [×4], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C324C8 [×2], C2×C3⋊Dic3, C6×C12, C6×C12, C6.Q16 [×4], C2×C324C8, C12⋊Dic3, C32×C4⋊C4, C12.9Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, D6 [×4], C4⋊C4, D8, Q16, C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C2.D8, C2×C3⋊S3, Dic3⋊C4 [×4], D4⋊S3 [×4], C3⋊Q16 [×4], C324Q8, C4×C3⋊S3, C327D4, C6.Q16 [×4], C6.Dic6, C327D8, C327Q16, C12.9Dic6

Smallest permutation representation of C12.9Dic6
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)
(1 194 35 18 280 97 268 179 147 38 231 160)(2 201 36 13 281 104 269 174 148 45 232 167)(3 196 25 20 282 99 270 169 149 40 233 162)(4 203 26 15 283 106 271 176 150 47 234 157)(5 198 27 22 284 101 272 171 151 42 235 164)(6 193 28 17 285 108 273 178 152 37 236 159)(7 200 29 24 286 103 274 173 153 44 237 166)(8 195 30 19 287 98 275 180 154 39 238 161)(9 202 31 14 288 105 276 175 155 46 239 168)(10 197 32 21 277 100 265 170 156 41 240 163)(11 204 33 16 278 107 266 177 145 48 229 158)(12 199 34 23 279 102 267 172 146 43 230 165)(49 225 65 256 250 190 141 211 85 127 75 110)(50 220 66 263 251 185 142 206 86 122 76 117)(51 227 67 258 252 192 143 213 87 129 77 112)(52 222 68 253 241 187 144 208 88 124 78 119)(53 217 69 260 242 182 133 215 89 131 79 114)(54 224 70 255 243 189 134 210 90 126 80 109)(55 219 71 262 244 184 135 205 91 121 81 116)(56 226 72 257 245 191 136 212 92 128 82 111)(57 221 61 264 246 186 137 207 93 123 83 118)(58 228 62 259 247 181 138 214 94 130 84 113)(59 223 63 254 248 188 139 209 95 125 73 120)(60 218 64 261 249 183 140 216 96 132 74 115)
(1 134 265 51 7 140 271 57)(2 139 266 56 8 133 272 50)(3 144 267 49 9 138 273 55)(4 137 268 54 10 143 274 60)(5 142 269 59 11 136 275 53)(6 135 270 52 12 141 276 58)(13 263 42 131 19 257 48 125)(14 256 43 124 20 262 37 130)(15 261 44 129 21 255 38 123)(16 254 45 122 22 260 39 128)(17 259 46 127 23 253 40 121)(18 264 47 132 24 258 41 126)(25 241 146 75 31 247 152 81)(26 246 147 80 32 252 153 74)(27 251 148 73 33 245 154 79)(28 244 149 78 34 250 155 84)(29 249 150 83 35 243 156 77)(30 242 151 76 36 248 145 82)(61 231 90 277 67 237 96 283)(62 236 91 282 68 230 85 288)(63 229 92 287 69 235 86 281)(64 234 93 280 70 240 87 286)(65 239 94 285 71 233 88 279)(66 232 95 278 72 238 89 284)(97 221 157 216 103 227 163 210)(98 226 158 209 104 220 164 215)(99 219 159 214 105 225 165 208)(100 224 160 207 106 218 166 213)(101 217 161 212 107 223 167 206)(102 222 162 205 108 228 168 211)(109 194 186 176 115 200 192 170)(110 199 187 169 116 193 181 175)(111 204 188 174 117 198 182 180)(112 197 189 179 118 203 183 173)(113 202 190 172 119 196 184 178)(114 195 191 177 120 201 185 171)

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

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

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

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L8A8B8C8D12A···12X
order122233334444446···6888812···12
size11112222224436362···2181818184···4

54 irreducible representations

dim1111122222222244
type+++++-+++--+-
imageC1C2C2C2C4S3Q8D4D6D8Q16Dic6C4×S3C3⋊D4D4⋊S3C3⋊Q16
kernelC12.9Dic6C2×C324C8C12⋊Dic3C32×C4⋊C4C324C8C3×C4⋊C4C3×C12C62C2×C12C3×C6C3×C6C12C12C2×C6C6C6
# reps1111441142288844

Matrix representation of C12.9Dic6 in GL6(𝔽73)

0720000
110000
0072200
0072100
0000720
0000072
,
2700000
0270000
00602400
00721300
00004343
00003013
,
41630000
22320000
00324100
0016000
0000027
0000270

G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,60,72,0,0,0,0,24,13,0,0,0,0,0,0,43,30,0,0,0,0,43,13],[41,22,0,0,0,0,63,32,0,0,0,0,0,0,32,16,0,0,0,0,41,0,0,0,0,0,0,0,0,27,0,0,0,0,27,0] >;

C12.9Dic6 in GAP, Magma, Sage, TeX

C_{12}._9{\rm Dic}_6
% in TeX

G:=Group("C12.9Dic6");
// GroupNames label

G:=SmallGroup(288,282);
// by ID

G=gap.SmallGroup(288,282);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^9*b^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations

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