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## G = C12.57D12order 288 = 25·32

### 24th non-split extension by C12 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.57D12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C2×C32⋊4C8 — C12.57D12
 Lower central C32 — C3×C6 — C12.57D12
 Upper central C1 — C2×C4 — C42

Generators and relations for C12.57D12
G = < a,b,c | a12=b12=1, c2=a9, ab=ba, cac-1=a5, cbc-1=b-1 >

Subgroups: 244 in 114 conjugacy classes, 81 normal (23 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C4, C22, C6 [×12], C8 [×2], C2×C4 [×3], C32, C12 [×16], C12 [×4], C2×C6 [×4], C42, C2×C8 [×2], C3×C6 [×3], C3⋊C8 [×8], C2×C12 [×12], C4⋊C8, C3×C12 [×2], C3×C12 [×2], C3×C12, C62, C2×C3⋊C8 [×8], C4×C12 [×4], C324C8 [×2], C6×C12 [×3], C12⋊C8 [×4], C2×C324C8 [×2], C122, C12.57D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D4, Q8, Dic3 [×8], D6 [×4], C4⋊C4, C2×C8, M4(2), C3⋊S3, C3⋊C8 [×8], Dic6 [×4], D12 [×4], C2×Dic3 [×4], C4⋊C8, C3⋊Dic3 [×2], C2×C3⋊S3, C2×C3⋊C8 [×4], C4.Dic3 [×4], C4⋊Dic3 [×4], C324C8 [×2], C324Q8, C12⋊S3, C2×C3⋊Dic3, C12⋊C8 [×4], C2×C324C8, C12.58D6, C12⋊Dic3, C12.57D12

Smallest permutation representation of C12.57D12
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)
(1 195 35 168 71 283 106 271 134 110 48 224)(2 196 36 157 72 284 107 272 135 111 37 225)(3 197 25 158 61 285 108 273 136 112 38 226)(4 198 26 159 62 286 97 274 137 113 39 227)(5 199 27 160 63 287 98 275 138 114 40 228)(6 200 28 161 64 288 99 276 139 115 41 217)(7 201 29 162 65 277 100 265 140 116 42 218)(8 202 30 163 66 278 101 266 141 117 43 219)(9 203 31 164 67 279 102 267 142 118 44 220)(10 204 32 165 68 280 103 268 143 119 45 221)(11 193 33 166 69 281 104 269 144 120 46 222)(12 194 34 167 70 282 105 270 133 109 47 223)(13 146 256 182 173 230 80 208 121 93 247 56)(14 147 257 183 174 231 81 209 122 94 248 57)(15 148 258 184 175 232 82 210 123 95 249 58)(16 149 259 185 176 233 83 211 124 96 250 59)(17 150 260 186 177 234 84 212 125 85 251 60)(18 151 261 187 178 235 73 213 126 86 252 49)(19 152 262 188 179 236 74 214 127 87 241 50)(20 153 263 189 180 237 75 215 128 88 242 51)(21 154 264 190 169 238 76 216 129 89 243 52)(22 155 253 191 170 239 77 205 130 90 244 53)(23 156 254 192 171 240 78 206 131 91 245 54)(24 145 255 181 172 229 79 207 132 92 246 55)
(1 23 10 20 7 17 4 14)(2 16 11 13 8 22 5 19)(3 21 12 18 9 15 6 24)(25 243 34 252 31 249 28 246)(26 248 35 245 32 242 29 251)(27 241 36 250 33 247 30 244)(37 259 46 256 43 253 40 262)(38 264 47 261 44 258 41 255)(39 257 48 254 45 263 42 260)(49 203 58 200 55 197 52 194)(50 196 59 193 56 202 53 199)(51 201 60 198 57 195 54 204)(61 129 70 126 67 123 64 132)(62 122 71 131 68 128 65 125)(63 127 72 124 69 121 66 130)(73 102 82 99 79 108 76 105)(74 107 83 104 80 101 77 98)(75 100 84 97 81 106 78 103)(85 159 94 168 91 165 88 162)(86 164 95 161 92 158 89 167)(87 157 96 166 93 163 90 160)(109 187 118 184 115 181 112 190)(110 192 119 189 116 186 113 183)(111 185 120 182 117 191 114 188)(133 178 142 175 139 172 136 169)(134 171 143 180 140 177 137 174)(135 176 144 173 141 170 138 179)(145 226 154 223 151 220 148 217)(146 219 155 228 152 225 149 222)(147 224 156 221 153 218 150 227)(205 287 214 284 211 281 208 278)(206 280 215 277 212 286 209 283)(207 285 216 282 213 279 210 288)(229 273 238 270 235 267 232 276)(230 266 239 275 236 272 233 269)(231 271 240 268 237 265 234 274)```

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 8A ··· 8H 12A ··· 12AV order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 ··· 2 18 ··· 18 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C4 C8 S3 D4 Q8 Dic3 D6 M4(2) C3⋊C8 Dic6 D12 C4.Dic3 kernel C12.57D12 C2×C32⋊4C8 C122 C6×C12 C3×C12 C4×C12 C3×C12 C3×C12 C2×C12 C2×C12 C3×C6 C12 C12 C12 C6 # reps 1 2 1 4 8 4 1 1 8 4 2 16 8 8 16

Matrix representation of C12.57D12 in GL4(𝔽73) generated by

 72 1 0 0 72 0 0 0 0 0 27 0 0 0 0 27
,
 59 7 0 0 66 66 0 0 0 0 72 72 0 0 1 0
,
 72 0 0 0 72 1 0 0 0 0 61 18 0 0 30 12
`G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,27,0,0,0,0,27],[59,66,0,0,7,66,0,0,0,0,72,1,0,0,72,0],[72,72,0,0,0,1,0,0,0,0,61,30,0,0,18,12] >;`

C12.57D12 in GAP, Magma, Sage, TeX

`C_{12}._{57}D_{12}`
`% in TeX`

`G:=Group("C12.57D12");`
`// GroupNames label`

`G:=SmallGroup(288,279);`
`// by ID`

`G=gap.SmallGroup(288,279);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,64,100,2693,9414]);`
`// Polycyclic`

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

׿
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