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## G = D5×C4⋊Q8order 320 = 26·5

### Direct product of D5 and C4⋊Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D5×C4⋊Q8
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C42 — D5×C4⋊Q8
 Lower central C5 — C2×C10 — D5×C4⋊Q8
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for D5×C4⋊Q8
G = < a,b,c,d,e | a5=b2=c4=d4=1, e2=d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 942 in 290 conjugacy classes, 131 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C4 [×14], C22, C22 [×6], C5, C2×C4, C2×C4 [×6], C2×C4 [×27], Q8 [×16], C23, D5 [×4], C10, C10 [×2], C42, C42 [×3], C4⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×7], C2×Q8 [×2], C2×Q8 [×14], Dic5 [×6], Dic5 [×4], C20 [×6], C20 [×4], D10 [×6], C2×C10, C2×C42, C2×C4⋊C4 [×4], C4⋊Q8, C4⋊Q8 [×7], C22×Q8 [×2], Dic10 [×12], C4×D5 [×12], C4×D5 [×8], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C2×C4⋊Q8, C4×Dic5, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×4], C4×C20, C5×C4⋊C4 [×4], C2×Dic10 [×6], C2×C4×D5, C2×C4×D5 [×6], Q8×D5 [×8], Q8×C10 [×2], C202Q8, D5×C42, C20⋊Q8 [×4], D5×C4⋊C4 [×4], Dic5⋊Q8 [×2], C5×C4⋊Q8, C2×Q8×D5 [×2], D5×C4⋊Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×8], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×12], C24, D10 [×7], C4⋊Q8 [×4], C22×D4, C22×Q8 [×2], C22×D5 [×7], C2×C4⋊Q8, D4×D5 [×2], Q8×D5 [×4], C23×D5, C2×D4×D5, C2×Q8×D5 [×2], D5×C4⋊Q8

Smallest permutation representation of D5×C4⋊Q8
On 160 points
Generators in S160
(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)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)(81 86)(82 90)(83 89)(84 88)(85 87)(91 96)(92 100)(93 99)(94 98)(95 97)(101 106)(102 110)(103 109)(104 108)(105 107)(111 116)(112 120)(113 119)(114 118)(115 117)(121 126)(122 130)(123 129)(124 128)(125 127)(131 136)(132 140)(133 139)(134 138)(135 137)(141 146)(142 150)(143 149)(144 148)(145 147)(151 156)(152 160)(153 159)(154 158)(155 157)
(1 84 9 89)(2 85 10 90)(3 81 6 86)(4 82 7 87)(5 83 8 88)(11 91 16 96)(12 92 17 97)(13 93 18 98)(14 94 19 99)(15 95 20 100)(21 101 26 106)(22 102 27 107)(23 103 28 108)(24 104 29 109)(25 105 30 110)(31 111 36 116)(32 112 37 117)(33 113 38 118)(34 114 39 119)(35 115 40 120)(41 121 46 126)(42 122 47 127)(43 123 48 128)(44 124 49 129)(45 125 50 130)(51 131 56 136)(52 132 57 137)(53 133 58 138)(54 134 59 139)(55 135 60 140)(61 141 66 146)(62 142 67 147)(63 143 68 148)(64 144 69 149)(65 145 70 150)(71 151 76 156)(72 152 77 157)(73 153 78 158)(74 154 79 159)(75 155 80 160)
(1 54 14 44)(2 55 15 45)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)(81 131 91 121)(82 132 92 122)(83 133 93 123)(84 134 94 124)(85 135 95 125)(86 136 96 126)(87 137 97 127)(88 138 98 128)(89 139 99 129)(90 140 100 130)(101 151 111 141)(102 152 112 142)(103 153 113 143)(104 154 114 144)(105 155 115 145)(106 156 116 146)(107 157 117 147)(108 158 118 148)(109 159 119 149)(110 160 120 150)
(1 114 14 104)(2 115 15 105)(3 111 11 101)(4 112 12 102)(5 113 13 103)(6 116 16 106)(7 117 17 107)(8 118 18 108)(9 119 19 109)(10 120 20 110)(21 81 31 91)(22 82 32 92)(23 83 33 93)(24 84 34 94)(25 85 35 95)(26 86 36 96)(27 87 37 97)(28 88 38 98)(29 89 39 99)(30 90 40 100)(41 141 51 151)(42 142 52 152)(43 143 53 153)(44 144 54 154)(45 145 55 155)(46 146 56 156)(47 147 57 157)(48 148 58 158)(49 149 59 159)(50 150 60 160)(61 131 71 121)(62 132 72 122)(63 133 73 123)(64 134 74 124)(65 135 75 125)(66 136 76 126)(67 137 77 127)(68 138 78 128)(69 139 79 129)(70 140 80 130)

G:=sub<Sym(160)| (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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(81,86)(82,90)(83,89)(84,88)(85,87)(91,96)(92,100)(93,99)(94,98)(95,97)(101,106)(102,110)(103,109)(104,108)(105,107)(111,116)(112,120)(113,119)(114,118)(115,117)(121,126)(122,130)(123,129)(124,128)(125,127)(131,136)(132,140)(133,139)(134,138)(135,137)(141,146)(142,150)(143,149)(144,148)(145,147)(151,156)(152,160)(153,159)(154,158)(155,157), (1,84,9,89)(2,85,10,90)(3,81,6,86)(4,82,7,87)(5,83,8,88)(11,91,16,96)(12,92,17,97)(13,93,18,98)(14,94,19,99)(15,95,20,100)(21,101,26,106)(22,102,27,107)(23,103,28,108)(24,104,29,109)(25,105,30,110)(31,111,36,116)(32,112,37,117)(33,113,38,118)(34,114,39,119)(35,115,40,120)(41,121,46,126)(42,122,47,127)(43,123,48,128)(44,124,49,129)(45,125,50,130)(51,131,56,136)(52,132,57,137)(53,133,58,138)(54,134,59,139)(55,135,60,140)(61,141,66,146)(62,142,67,147)(63,143,68,148)(64,144,69,149)(65,145,70,150)(71,151,76,156)(72,152,77,157)(73,153,78,158)(74,154,79,159)(75,155,80,160), (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150), (1,114,14,104)(2,115,15,105)(3,111,11,101)(4,112,12,102)(5,113,13,103)(6,116,16,106)(7,117,17,107)(8,118,18,108)(9,119,19,109)(10,120,20,110)(21,81,31,91)(22,82,32,92)(23,83,33,93)(24,84,34,94)(25,85,35,95)(26,86,36,96)(27,87,37,97)(28,88,38,98)(29,89,39,99)(30,90,40,100)(41,141,51,151)(42,142,52,152)(43,143,53,153)(44,144,54,154)(45,145,55,155)(46,146,56,156)(47,147,57,157)(48,148,58,158)(49,149,59,159)(50,150,60,160)(61,131,71,121)(62,132,72,122)(63,133,73,123)(64,134,74,124)(65,135,75,125)(66,136,76,126)(67,137,77,127)(68,138,78,128)(69,139,79,129)(70,140,80,130)>;

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), (1,8)(2,7)(3,6)(4,10)(5,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)(81,86)(82,90)(83,89)(84,88)(85,87)(91,96)(92,100)(93,99)(94,98)(95,97)(101,106)(102,110)(103,109)(104,108)(105,107)(111,116)(112,120)(113,119)(114,118)(115,117)(121,126)(122,130)(123,129)(124,128)(125,127)(131,136)(132,140)(133,139)(134,138)(135,137)(141,146)(142,150)(143,149)(144,148)(145,147)(151,156)(152,160)(153,159)(154,158)(155,157), (1,84,9,89)(2,85,10,90)(3,81,6,86)(4,82,7,87)(5,83,8,88)(11,91,16,96)(12,92,17,97)(13,93,18,98)(14,94,19,99)(15,95,20,100)(21,101,26,106)(22,102,27,107)(23,103,28,108)(24,104,29,109)(25,105,30,110)(31,111,36,116)(32,112,37,117)(33,113,38,118)(34,114,39,119)(35,115,40,120)(41,121,46,126)(42,122,47,127)(43,123,48,128)(44,124,49,129)(45,125,50,130)(51,131,56,136)(52,132,57,137)(53,133,58,138)(54,134,59,139)(55,135,60,140)(61,141,66,146)(62,142,67,147)(63,143,68,148)(64,144,69,149)(65,145,70,150)(71,151,76,156)(72,152,77,157)(73,153,78,158)(74,154,79,159)(75,155,80,160), (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70)(81,131,91,121)(82,132,92,122)(83,133,93,123)(84,134,94,124)(85,135,95,125)(86,136,96,126)(87,137,97,127)(88,138,98,128)(89,139,99,129)(90,140,100,130)(101,151,111,141)(102,152,112,142)(103,153,113,143)(104,154,114,144)(105,155,115,145)(106,156,116,146)(107,157,117,147)(108,158,118,148)(109,159,119,149)(110,160,120,150), (1,114,14,104)(2,115,15,105)(3,111,11,101)(4,112,12,102)(5,113,13,103)(6,116,16,106)(7,117,17,107)(8,118,18,108)(9,119,19,109)(10,120,20,110)(21,81,31,91)(22,82,32,92)(23,83,33,93)(24,84,34,94)(25,85,35,95)(26,86,36,96)(27,87,37,97)(28,88,38,98)(29,89,39,99)(30,90,40,100)(41,141,51,151)(42,142,52,152)(43,143,53,153)(44,144,54,154)(45,145,55,155)(46,146,56,156)(47,147,57,157)(48,148,58,158)(49,149,59,159)(50,150,60,160)(61,131,71,121)(62,132,72,122)(63,133,73,123)(64,134,74,124)(65,135,75,125)(66,136,76,126)(67,137,77,127)(68,138,78,128)(69,139,79,129)(70,140,80,130) );

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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77),(81,86),(82,90),(83,89),(84,88),(85,87),(91,96),(92,100),(93,99),(94,98),(95,97),(101,106),(102,110),(103,109),(104,108),(105,107),(111,116),(112,120),(113,119),(114,118),(115,117),(121,126),(122,130),(123,129),(124,128),(125,127),(131,136),(132,140),(133,139),(134,138),(135,137),(141,146),(142,150),(143,149),(144,148),(145,147),(151,156),(152,160),(153,159),(154,158),(155,157)], [(1,84,9,89),(2,85,10,90),(3,81,6,86),(4,82,7,87),(5,83,8,88),(11,91,16,96),(12,92,17,97),(13,93,18,98),(14,94,19,99),(15,95,20,100),(21,101,26,106),(22,102,27,107),(23,103,28,108),(24,104,29,109),(25,105,30,110),(31,111,36,116),(32,112,37,117),(33,113,38,118),(34,114,39,119),(35,115,40,120),(41,121,46,126),(42,122,47,127),(43,123,48,128),(44,124,49,129),(45,125,50,130),(51,131,56,136),(52,132,57,137),(53,133,58,138),(54,134,59,139),(55,135,60,140),(61,141,66,146),(62,142,67,147),(63,143,68,148),(64,144,69,149),(65,145,70,150),(71,151,76,156),(72,152,77,157),(73,153,78,158),(74,154,79,159),(75,155,80,160)], [(1,54,14,44),(2,55,15,45),(3,51,11,41),(4,52,12,42),(5,53,13,43),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70),(81,131,91,121),(82,132,92,122),(83,133,93,123),(84,134,94,124),(85,135,95,125),(86,136,96,126),(87,137,97,127),(88,138,98,128),(89,139,99,129),(90,140,100,130),(101,151,111,141),(102,152,112,142),(103,153,113,143),(104,154,114,144),(105,155,115,145),(106,156,116,146),(107,157,117,147),(108,158,118,148),(109,159,119,149),(110,160,120,150)], [(1,114,14,104),(2,115,15,105),(3,111,11,101),(4,112,12,102),(5,113,13,103),(6,116,16,106),(7,117,17,107),(8,118,18,108),(9,119,19,109),(10,120,20,110),(21,81,31,91),(22,82,32,92),(23,83,33,93),(24,84,34,94),(25,85,35,95),(26,86,36,96),(27,87,37,97),(28,88,38,98),(29,89,39,99),(30,90,40,100),(41,141,51,151),(42,142,52,152),(43,143,53,153),(44,144,54,154),(45,145,55,155),(46,146,56,156),(47,147,57,157),(48,148,58,158),(49,149,59,159),(50,150,60,160),(61,131,71,121),(62,132,72,122),(63,133,73,123),(64,134,74,124),(65,135,75,125),(66,136,76,126),(67,137,77,127),(68,138,78,128),(69,139,79,129),(70,140,80,130)])

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 5 5 5 5 2 ··· 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 8 ··· 8

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 D5 D10 D10 D10 D4×D5 Q8×D5 kernel D5×C4⋊Q8 C20⋊2Q8 D5×C42 C20⋊Q8 D5×C4⋊C4 Dic5⋊Q8 C5×C4⋊Q8 C2×Q8×D5 C4×D5 C4×D5 C4⋊Q8 C42 C4⋊C4 C2×Q8 C4 C4 # reps 1 1 1 4 4 2 1 2 4 8 2 2 8 4 4 8

Matrix representation of D5×C4⋊Q8 in GL6(𝔽41)

 6 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 6 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 14 0 0 0 0 14 34 0 0 0 0 0 0 21 39 0 0 0 0 16 20
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 27 0 0 0 0 27 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 27 7 0 0 0 0 7 14 0 0 0 0 0 0 20 2 0 0 0 0 26 21

G:=sub<GL(6,GF(41))| [6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,6,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,14,0,0,0,0,14,34,0,0,0,0,0,0,21,16,0,0,0,0,39,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,27,0,0,0,0,27,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,27,7,0,0,0,0,7,14,0,0,0,0,0,0,20,26,0,0,0,0,2,21] >;

D5×C4⋊Q8 in GAP, Magma, Sage, TeX

D_5\times C_4\rtimes Q_8
% in TeX

G:=Group("D5xC4:Q8");
// GroupNames label

G:=SmallGroup(320,1395);
// by ID

G=gap.SmallGroup(320,1395);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,100,570,185,80,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=d^4=1,e^2=d^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽