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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, C3, C4, C4, C22, C6, C2×C4, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C42.C2, C3⋊Dic3, C3×C12, C3×C12, C62, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×C3⋊Dic3, C6×C12, C6×C12, C12.6Q8, C6.Dic6, C12⋊Dic3, C122, C12.25Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C3⋊S3, Dic6, C22×S3, C42.C2, C2×C3⋊S3, C2×Dic6, C4○D12, C324Q8, C22×C3⋊S3, C12.6Q8, C2×C324Q8, C12.59D6, 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)
(1 272 58 89 285 123 173 211 148 167 101 183)(2 273 59 90 286 124 174 212 149 168 102 184)(3 274 60 91 287 125 175 213 150 157 103 185)(4 275 49 92 288 126 176 214 151 158 104 186)(5 276 50 93 277 127 177 215 152 159 105 187)(6 265 51 94 278 128 178 216 153 160 106 188)(7 266 52 95 279 129 179 205 154 161 107 189)(8 267 53 96 280 130 180 206 155 162 108 190)(9 268 54 85 281 131 169 207 156 163 97 191)(10 269 55 86 282 132 170 208 145 164 98 192)(11 270 56 87 283 121 171 209 146 165 99 181)(12 271 57 88 284 122 172 210 147 166 100 182)(13 223 65 231 36 78 138 254 204 119 246 43)(14 224 66 232 25 79 139 255 193 120 247 44)(15 225 67 233 26 80 140 256 194 109 248 45)(16 226 68 234 27 81 141 257 195 110 249 46)(17 227 69 235 28 82 142 258 196 111 250 47)(18 228 70 236 29 83 143 259 197 112 251 48)(19 217 71 237 30 84 144 260 198 113 252 37)(20 218 72 238 31 73 133 261 199 114 241 38)(21 219 61 239 32 74 134 262 200 115 242 39)(22 220 62 240 33 75 135 263 201 116 243 40)(23 221 63 229 34 76 136 264 202 117 244 41)(24 222 64 230 35 77 137 253 203 118 245 42)
(1 15 179 134)(2 14 180 133)(3 13 169 144)(4 24 170 143)(5 23 171 142)(6 22 172 141)(7 21 173 140)(8 20 174 139)(9 19 175 138)(10 18 176 137)(11 17 177 136)(12 16 178 135)(25 53 241 149)(26 52 242 148)(27 51 243 147)(28 50 244 146)(29 49 245 145)(30 60 246 156)(31 59 247 155)(32 58 248 154)(33 57 249 153)(34 56 250 152)(35 55 251 151)(36 54 252 150)(37 207 78 274)(38 206 79 273)(39 205 80 272)(40 216 81 271)(41 215 82 270)(42 214 83 269)(43 213 84 268)(44 212 73 267)(45 211 74 266)(46 210 75 265)(47 209 76 276)(48 208 77 275)(61 285 194 107)(62 284 195 106)(63 283 196 105)(64 282 197 104)(65 281 198 103)(66 280 199 102)(67 279 200 101)(68 278 201 100)(69 277 202 99)(70 288 203 98)(71 287 204 97)(72 286 193 108)(85 119 157 237)(86 118 158 236)(87 117 159 235)(88 116 160 234)(89 115 161 233)(90 114 162 232)(91 113 163 231)(92 112 164 230)(93 111 165 229)(94 110 166 240)(95 109 167 239)(96 120 168 238)(121 264 187 227)(122 263 188 226)(123 262 189 225)(124 261 190 224)(125 260 191 223)(126 259 192 222)(127 258 181 221)(128 257 182 220)(129 256 183 219)(130 255 184 218)(131 254 185 217)(132 253 186 228)

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

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

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

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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