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## G = C12.25Dic6order 288 = 25·32

### 7th non-split extension by C12 of Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C12.25Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C6.Dic6 — C12.25Dic6
 Lower central C32 — C62 — C12.25Dic6
 Upper central C1 — C22 — C42

Generators and relations for C12.25Dic6
G = < a,b,c | a12=b12=1, c2=a6b6, ab=ba, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 508 in 168 conjugacy classes, 77 normal (11 characteristic)
C1, C2, C2 [×2], C3 [×4], C4 [×2], C4 [×6], C22, C6 [×12], C2×C4, C2×C4 [×2], C2×C4 [×4], C32, Dic3 [×16], C12 [×8], C12 [×8], C2×C6 [×4], C42, C4⋊C4 [×6], C3×C6, C3×C6 [×2], C2×Dic3 [×16], C2×C12 [×12], C42.C2, C3⋊Dic3 [×4], C3×C12 [×2], C3×C12 [×2], C62, Dic3⋊C4 [×16], C4⋊Dic3 [×8], C4×C12 [×4], C2×C3⋊Dic3 [×4], C6×C12, C6×C12 [×2], C12.6Q8 [×4], C6.Dic6 [×4], C12⋊Dic3 [×2], C122, C12.25Dic6
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], Q8 [×2], C23, D6 [×12], C2×Q8, C4○D4 [×2], C3⋊S3, Dic6 [×8], C22×S3 [×4], C42.C2, C2×C3⋊S3 [×3], C2×Dic6 [×4], C4○D12 [×8], C324Q8 [×2], C22×C3⋊S3, C12.6Q8 [×4], C2×C324Q8, C12.59D6 [×2], C12.25Dic6

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

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

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

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

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 Q8 D6 C4○D4 Dic6 C4○D12 kernel C12.25Dic6 C6.Dic6 C12⋊Dic3 C122 C4×C12 C3×C12 C2×C12 C3×C6 C12 C6 # reps 1 4 2 1 4 2 12 4 16 32

Matrix representation of C12.25Dic6 in GL4(𝔽13) generated by

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

C12.25Dic6 in GAP, Magma, Sage, TeX

C_{12}._{25}{\rm Dic}_6
% in TeX

G:=Group("C12.25Dic6");
// GroupNames label

G:=SmallGroup(288,727);
// by ID

G=gap.SmallGroup(288,727);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,100,2693,9414]);
// Polycyclic

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

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