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## G = C6.4Dic12order 288 = 25·32

### 4th non-split extension by C6 of Dic12 acting via Dic12/C24=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C6.4Dic12
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12⋊Dic3 — C6.4Dic12
 Lower central C32 — C3×C6 — C3×C12 — C6.4Dic12
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C6.4Dic12
G = < a,b,c | a6=b24=1, c2=a3b12, ab=ba, cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 484 in 126 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×3], C22, C6 [×12], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×12], C12 [×8], C2×C6 [×4], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C24 [×4], Dic6 [×12], C2×Dic3 [×8], C2×C12 [×4], Q8⋊C4, C3⋊Dic3 [×3], C3×C12 [×2], C62, C4⋊Dic3 [×4], C2×C24 [×4], C2×Dic6 [×4], C3×C24, C324Q8 [×2], C324Q8, C2×C3⋊Dic3 [×2], C6×C12, C2.Dic12 [×4], C12⋊Dic3, C6×C24, C2×C324Q8, C6.4Dic12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, SD16, Q16, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], Q8⋊C4, C2×C3⋊S3, C24⋊C2 [×4], Dic12 [×4], D6⋊C4 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C2.Dic12 [×4], C242S3, C325Q16, C6.11D12, C6.4Dic12

Smallest permutation representation of C6.4Dic12
Regular action on 288 points
Generators in S288
(1 176 258 76 200 144)(2 177 259 77 201 121)(3 178 260 78 202 122)(4 179 261 79 203 123)(5 180 262 80 204 124)(6 181 263 81 205 125)(7 182 264 82 206 126)(8 183 241 83 207 127)(9 184 242 84 208 128)(10 185 243 85 209 129)(11 186 244 86 210 130)(12 187 245 87 211 131)(13 188 246 88 212 132)(14 189 247 89 213 133)(15 190 248 90 214 134)(16 191 249 91 215 135)(17 192 250 92 216 136)(18 169 251 93 193 137)(19 170 252 94 194 138)(20 171 253 95 195 139)(21 172 254 96 196 140)(22 173 255 73 197 141)(23 174 256 74 198 142)(24 175 257 75 199 143)(25 119 287 153 51 226)(26 120 288 154 52 227)(27 97 265 155 53 228)(28 98 266 156 54 229)(29 99 267 157 55 230)(30 100 268 158 56 231)(31 101 269 159 57 232)(32 102 270 160 58 233)(33 103 271 161 59 234)(34 104 272 162 60 235)(35 105 273 163 61 236)(36 106 274 164 62 237)(37 107 275 165 63 238)(38 108 276 166 64 239)(39 109 277 167 65 240)(40 110 278 168 66 217)(41 111 279 145 67 218)(42 112 280 146 68 219)(43 113 281 147 69 220)(44 114 282 148 70 221)(45 115 283 149 71 222)(46 116 284 150 72 223)(47 117 285 151 49 224)(48 118 286 152 50 225)
(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)
(1 284 88 235)(2 222 89 271)(3 282 90 233)(4 220 91 269)(5 280 92 231)(6 218 93 267)(7 278 94 229)(8 240 95 265)(9 276 96 227)(10 238 73 287)(11 274 74 225)(12 236 75 285)(13 272 76 223)(14 234 77 283)(15 270 78 221)(16 232 79 281)(17 268 80 219)(18 230 81 279)(19 266 82 217)(20 228 83 277)(21 288 84 239)(22 226 85 275)(23 286 86 237)(24 224 87 273)(25 243 165 141)(26 128 166 254)(27 241 167 139)(28 126 168 252)(29 263 145 137)(30 124 146 250)(31 261 147 135)(32 122 148 248)(33 259 149 133)(34 144 150 246)(35 257 151 131)(36 142 152 244)(37 255 153 129)(38 140 154 242)(39 253 155 127)(40 138 156 264)(41 251 157 125)(42 136 158 262)(43 249 159 123)(44 134 160 260)(45 247 161 121)(46 132 162 258)(47 245 163 143)(48 130 164 256)(49 211 105 175)(50 186 106 198)(51 209 107 173)(52 184 108 196)(53 207 109 171)(54 182 110 194)(55 205 111 169)(56 180 112 216)(57 203 113 191)(58 178 114 214)(59 201 115 189)(60 176 116 212)(61 199 117 187)(62 174 118 210)(63 197 119 185)(64 172 120 208)(65 195 97 183)(66 170 98 206)(67 193 99 181)(68 192 100 204)(69 215 101 179)(70 190 102 202)(71 213 103 177)(72 188 104 200)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 6A ··· 6L 8A 8B 8C 8D 12A ··· 12P 24A ··· 24AF order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 2 2 2 36 36 36 36 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + - image C1 C2 C2 C2 C4 S3 D4 D4 D6 SD16 Q16 C4×S3 C3⋊D4 D12 C24⋊C2 Dic12 kernel C6.4Dic12 C12⋊Dic3 C6×C24 C2×C32⋊4Q8 C32⋊4Q8 C2×C24 C3×C12 C62 C2×C12 C3×C6 C3×C6 C12 C12 C2×C6 C6 C6 # reps 1 1 1 1 4 4 1 1 4 2 2 8 8 8 16 16

Matrix representation of C6.4Dic12 in GL6(𝔽73)

 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 12 0 0 0 0 67 0 0 0 0 0 0 0 43 43 0 0 0 0 30 13
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 58 4 0 0 0 0 17 15 0 0 0 0 0 0 8 26 0 0 0 0 34 65

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,43,30,0,0,0,0,43,13],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,58,17,0,0,0,0,4,15,0,0,0,0,0,0,8,34,0,0,0,0,26,65] >;

C6.4Dic12 in GAP, Magma, Sage, TeX

C_6._4{\rm Dic}_{12}
% in TeX

G:=Group("C6.4Dic12");
// GroupNames label

G:=SmallGroup(288,291);
// by ID

G=gap.SmallGroup(288,291);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,85,92,422,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=a^3*b^12,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations

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