Copied to
clipboard

## G = Dic7⋊4M4(2)  order 448 = 26·7

### 2nd semidirect product of Dic7 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — Dic7⋊4M4(2)
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4×Dic7 — C2×C4×Dic7 — Dic7⋊4M4(2)
 Lower central C7 — C2×C14 — Dic7⋊4M4(2)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for Dic74M4(2)
G = < a,b,c,d | a14=c8=d2=1, b2=a7, bab-1=a-1, ac=ca, ad=da, cbc-1=a7b, bd=db, dcd=c5 >

Subgroups: 420 in 126 conjugacy classes, 67 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, Dic7, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊C8, C2×C42, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×C14, C4⋊M4(2), C2×C7⋊C8, C4.Dic7, C4×Dic7, C4×Dic7, C2×C56, C7×M4(2), C22×Dic7, C22×C28, Dic7⋊C8, C2×C4.Dic7, C2×C4×Dic7, C14×M4(2), Dic74M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, D14, C2×C4⋊C4, C2×M4(2), Dic14, C4×D7, C7⋊D4, C22×D7, C4⋊M4(2), Dic7⋊C4, C2×Dic14, C2×C4×D7, C2×C7⋊D4, D7×M4(2), C2×Dic7⋊C4, Dic74M4(2)

Smallest permutation representation of Dic74M4(2)
On 224 points
Generators in S224
(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)
(1 155 8 162)(2 168 9 161)(3 167 10 160)(4 166 11 159)(5 165 12 158)(6 164 13 157)(7 163 14 156)(15 105 22 112)(16 104 23 111)(17 103 24 110)(18 102 25 109)(19 101 26 108)(20 100 27 107)(21 99 28 106)(29 128 36 135)(30 127 37 134)(31 140 38 133)(32 139 39 132)(33 138 40 131)(34 137 41 130)(35 136 42 129)(43 185 50 192)(44 184 51 191)(45 183 52 190)(46 196 53 189)(47 195 54 188)(48 194 55 187)(49 193 56 186)(57 212 64 219)(58 211 65 218)(59 224 66 217)(60 223 67 216)(61 222 68 215)(62 221 69 214)(63 220 70 213)(71 197 78 204)(72 210 79 203)(73 209 80 202)(74 208 81 201)(75 207 82 200)(76 206 83 199)(77 205 84 198)(85 117 92 124)(86 116 93 123)(87 115 94 122)(88 114 95 121)(89 113 96 120)(90 126 97 119)(91 125 98 118)(141 181 148 174)(142 180 149 173)(143 179 150 172)(144 178 151 171)(145 177 152 170)(146 176 153 169)(147 175 154 182)
(1 195 60 99 144 31 124 200)(2 196 61 100 145 32 125 201)(3 183 62 101 146 33 126 202)(4 184 63 102 147 34 113 203)(5 185 64 103 148 35 114 204)(6 186 65 104 149 36 115 205)(7 187 66 105 150 37 116 206)(8 188 67 106 151 38 117 207)(9 189 68 107 152 39 118 208)(10 190 69 108 153 40 119 209)(11 191 70 109 154 41 120 210)(12 192 57 110 141 42 121 197)(13 193 58 111 142 29 122 198)(14 194 59 112 143 30 123 199)(15 172 127 93 76 163 55 217)(16 173 128 94 77 164 56 218)(17 174 129 95 78 165 43 219)(18 175 130 96 79 166 44 220)(19 176 131 97 80 167 45 221)(20 177 132 98 81 168 46 222)(21 178 133 85 82 155 47 223)(22 179 134 86 83 156 48 224)(23 180 135 87 84 157 49 211)(24 181 136 88 71 158 50 212)(25 182 137 89 72 159 51 213)(26 169 138 90 73 160 52 214)(27 170 139 91 74 161 53 215)(28 171 140 92 75 162 54 216)
(1 151)(2 152)(3 153)(4 154)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 114)(58 115)(59 116)(60 117)(61 118)(62 119)(63 120)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 113)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 216)(86 217)(87 218)(88 219)(89 220)(90 221)(91 222)(92 223)(93 224)(94 211)(95 212)(96 213)(97 214)(98 215)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(127 134)(128 135)(129 136)(130 137)(131 138)(132 139)(133 140)(155 171)(156 172)(157 173)(158 174)(159 175)(160 176)(161 177)(162 178)(163 179)(164 180)(165 181)(166 182)(167 169)(168 170)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)

G:=sub<Sym(224)| (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), (1,155,8,162)(2,168,9,161)(3,167,10,160)(4,166,11,159)(5,165,12,158)(6,164,13,157)(7,163,14,156)(15,105,22,112)(16,104,23,111)(17,103,24,110)(18,102,25,109)(19,101,26,108)(20,100,27,107)(21,99,28,106)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,185,50,192)(44,184,51,191)(45,183,52,190)(46,196,53,189)(47,195,54,188)(48,194,55,187)(49,193,56,186)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(71,197,78,204)(72,210,79,203)(73,209,80,202)(74,208,81,201)(75,207,82,200)(76,206,83,199)(77,205,84,198)(85,117,92,124)(86,116,93,123)(87,115,94,122)(88,114,95,121)(89,113,96,120)(90,126,97,119)(91,125,98,118)(141,181,148,174)(142,180,149,173)(143,179,150,172)(144,178,151,171)(145,177,152,170)(146,176,153,169)(147,175,154,182), (1,195,60,99,144,31,124,200)(2,196,61,100,145,32,125,201)(3,183,62,101,146,33,126,202)(4,184,63,102,147,34,113,203)(5,185,64,103,148,35,114,204)(6,186,65,104,149,36,115,205)(7,187,66,105,150,37,116,206)(8,188,67,106,151,38,117,207)(9,189,68,107,152,39,118,208)(10,190,69,108,153,40,119,209)(11,191,70,109,154,41,120,210)(12,192,57,110,141,42,121,197)(13,193,58,111,142,29,122,198)(14,194,59,112,143,30,123,199)(15,172,127,93,76,163,55,217)(16,173,128,94,77,164,56,218)(17,174,129,95,78,165,43,219)(18,175,130,96,79,166,44,220)(19,176,131,97,80,167,45,221)(20,177,132,98,81,168,46,222)(21,178,133,85,82,155,47,223)(22,179,134,86,83,156,48,224)(23,180,135,87,84,157,49,211)(24,181,136,88,71,158,50,212)(25,182,137,89,72,159,51,213)(26,169,138,90,73,160,52,214)(27,170,139,91,74,161,53,215)(28,171,140,92,75,162,54,216), (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,113)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,211)(95,212)(96,213)(97,214)(98,215)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,169)(168,170)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)>;

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), (1,155,8,162)(2,168,9,161)(3,167,10,160)(4,166,11,159)(5,165,12,158)(6,164,13,157)(7,163,14,156)(15,105,22,112)(16,104,23,111)(17,103,24,110)(18,102,25,109)(19,101,26,108)(20,100,27,107)(21,99,28,106)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,185,50,192)(44,184,51,191)(45,183,52,190)(46,196,53,189)(47,195,54,188)(48,194,55,187)(49,193,56,186)(57,212,64,219)(58,211,65,218)(59,224,66,217)(60,223,67,216)(61,222,68,215)(62,221,69,214)(63,220,70,213)(71,197,78,204)(72,210,79,203)(73,209,80,202)(74,208,81,201)(75,207,82,200)(76,206,83,199)(77,205,84,198)(85,117,92,124)(86,116,93,123)(87,115,94,122)(88,114,95,121)(89,113,96,120)(90,126,97,119)(91,125,98,118)(141,181,148,174)(142,180,149,173)(143,179,150,172)(144,178,151,171)(145,177,152,170)(146,176,153,169)(147,175,154,182), (1,195,60,99,144,31,124,200)(2,196,61,100,145,32,125,201)(3,183,62,101,146,33,126,202)(4,184,63,102,147,34,113,203)(5,185,64,103,148,35,114,204)(6,186,65,104,149,36,115,205)(7,187,66,105,150,37,116,206)(8,188,67,106,151,38,117,207)(9,189,68,107,152,39,118,208)(10,190,69,108,153,40,119,209)(11,191,70,109,154,41,120,210)(12,192,57,110,141,42,121,197)(13,193,58,111,142,29,122,198)(14,194,59,112,143,30,123,199)(15,172,127,93,76,163,55,217)(16,173,128,94,77,164,56,218)(17,174,129,95,78,165,43,219)(18,175,130,96,79,166,44,220)(19,176,131,97,80,167,45,221)(20,177,132,98,81,168,46,222)(21,178,133,85,82,155,47,223)(22,179,134,86,83,156,48,224)(23,180,135,87,84,157,49,211)(24,181,136,88,71,158,50,212)(25,182,137,89,72,159,51,213)(26,169,138,90,73,160,52,214)(27,170,139,91,74,161,53,215)(28,171,140,92,75,162,54,216), (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,114)(58,115)(59,116)(60,117)(61,118)(62,119)(63,120)(64,121)(65,122)(66,123)(67,124)(68,125)(69,126)(70,113)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(85,216)(86,217)(87,218)(88,219)(89,220)(90,221)(91,222)(92,223)(93,224)(94,211)(95,212)(96,213)(97,214)(98,215)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)(133,140)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,169)(168,170)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210) );

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)], [(1,155,8,162),(2,168,9,161),(3,167,10,160),(4,166,11,159),(5,165,12,158),(6,164,13,157),(7,163,14,156),(15,105,22,112),(16,104,23,111),(17,103,24,110),(18,102,25,109),(19,101,26,108),(20,100,27,107),(21,99,28,106),(29,128,36,135),(30,127,37,134),(31,140,38,133),(32,139,39,132),(33,138,40,131),(34,137,41,130),(35,136,42,129),(43,185,50,192),(44,184,51,191),(45,183,52,190),(46,196,53,189),(47,195,54,188),(48,194,55,187),(49,193,56,186),(57,212,64,219),(58,211,65,218),(59,224,66,217),(60,223,67,216),(61,222,68,215),(62,221,69,214),(63,220,70,213),(71,197,78,204),(72,210,79,203),(73,209,80,202),(74,208,81,201),(75,207,82,200),(76,206,83,199),(77,205,84,198),(85,117,92,124),(86,116,93,123),(87,115,94,122),(88,114,95,121),(89,113,96,120),(90,126,97,119),(91,125,98,118),(141,181,148,174),(142,180,149,173),(143,179,150,172),(144,178,151,171),(145,177,152,170),(146,176,153,169),(147,175,154,182)], [(1,195,60,99,144,31,124,200),(2,196,61,100,145,32,125,201),(3,183,62,101,146,33,126,202),(4,184,63,102,147,34,113,203),(5,185,64,103,148,35,114,204),(6,186,65,104,149,36,115,205),(7,187,66,105,150,37,116,206),(8,188,67,106,151,38,117,207),(9,189,68,107,152,39,118,208),(10,190,69,108,153,40,119,209),(11,191,70,109,154,41,120,210),(12,192,57,110,141,42,121,197),(13,193,58,111,142,29,122,198),(14,194,59,112,143,30,123,199),(15,172,127,93,76,163,55,217),(16,173,128,94,77,164,56,218),(17,174,129,95,78,165,43,219),(18,175,130,96,79,166,44,220),(19,176,131,97,80,167,45,221),(20,177,132,98,81,168,46,222),(21,178,133,85,82,155,47,223),(22,179,134,86,83,156,48,224),(23,180,135,87,84,157,49,211),(24,181,136,88,71,158,50,212),(25,182,137,89,72,159,51,213),(26,169,138,90,73,160,52,214),(27,170,139,91,74,161,53,215),(28,171,140,92,75,162,54,216)], [(1,151),(2,152),(3,153),(4,154),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,114),(58,115),(59,116),(60,117),(61,118),(62,119),(63,120),(64,121),(65,122),(66,123),(67,124),(68,125),(69,126),(70,113),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(85,216),(86,217),(87,218),(88,219),(89,220),(90,221),(91,222),(92,223),(93,224),(94,211),(95,212),(96,213),(97,214),(98,215),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(127,134),(128,135),(129,136),(130,137),(131,138),(132,139),(133,140),(155,171),(156,172),(157,173),(158,174),(159,175),(160,176),(161,177),(162,178),(163,179),(164,180),(165,181),(166,182),(167,169),(168,170),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)]])

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 1 1 1 1 2 2 14 ··· 14 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 C4 D4 Q8 D7 M4(2) D14 D14 Dic14 C4×D7 C7⋊D4 C4×D7 D7×M4(2) kernel Dic7⋊4M4(2) Dic7⋊C8 C2×C4.Dic7 C2×C4×Dic7 C14×M4(2) C4×Dic7 C22×Dic7 C2×C28 C2×C28 C2×M4(2) Dic7 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 1 1 1 4 4 2 2 3 8 6 3 12 6 12 6 12

Matrix representation of Dic74M4(2) in GL4(𝔽113) generated by

 0 112 0 0 1 89 0 0 0 0 112 0 0 0 0 112
,
 73 83 0 0 27 40 0 0 0 0 98 0 0 0 98 15
,
 17 8 0 0 105 96 0 0 0 0 1 111 0 0 8 112
,
 112 0 0 0 0 112 0 0 0 0 1 0 0 0 1 112
G:=sub<GL(4,GF(113))| [0,1,0,0,112,89,0,0,0,0,112,0,0,0,0,112],[73,27,0,0,83,40,0,0,0,0,98,98,0,0,0,15],[17,105,0,0,8,96,0,0,0,0,1,8,0,0,111,112],[112,0,0,0,0,112,0,0,0,0,1,1,0,0,0,112] >;

Dic74M4(2) in GAP, Magma, Sage, TeX

{\rm Dic}_7\rtimes_4M_4(2)
% in TeX

G:=Group("Dic7:4M4(2)");
// GroupNames label

G:=SmallGroup(448,652);
// by ID

G=gap.SmallGroup(448,652);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,422,387,58,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^14=c^8=d^2=1,b^2=a^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

׿
×
𝔽