Copied to
clipboard

G = (C2×D4).7F5order 320 = 26·5

4th non-split extension by C2×D4 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D4).7F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — (C2×D4).7F5
 Lower central C5 — C2×C10 — (C2×D4).7F5
 Upper central C1 — C22 — C2×D4

Generators and relations for (C2×D4).7F5
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=ab-1, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 586 in 158 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×6], C22, C22 [×2], C22 [×8], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5, C10, C10 [×2], C10 [×3], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], Dic5, C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×5], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], (C22×C8)⋊C2, C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], C22.F5 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, D10⋊C8 [×2], C23.2F5 [×2], C22×C5⋊C8, C2×C22.F5, C2×D42D5, (C2×D4).7F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8○D4 [×2], C2×F5 [×3], (C22×C8)⋊C2, C22⋊F5 [×2], C22×F5, D4.F5 [×2], C2×C22⋊F5, (C2×D4).7F5

Smallest permutation representation of (C2×D4).7F5
On 160 points
Generators in S160
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 115)(18 116)(19 117)(20 118)(21 119)(22 120)(23 113)(24 114)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 158)(34 159)(35 160)(36 153)(37 154)(38 155)(39 156)(40 157)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 97)(56 98)(65 108)(66 109)(67 110)(68 111)(69 112)(70 105)(71 106)(72 107)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)(129 152)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)
(1 7 5 3)(2 64 6 60)(4 58 8 62)(9 154 13 158)(10 34 14 38)(11 156 15 160)(12 36 16 40)(17 95 21 91)(18 83 22 87)(19 89 23 93)(20 85 24 81)(25 111 29 107)(26 65 30 69)(27 105 31 109)(28 67 32 71)(33 76 37 80)(35 78 39 74)(41 147 45 151)(42 129 46 133)(43 149 47 145)(44 131 48 135)(49 97 53 101)(50 52 54 56)(51 99 55 103)(57 63 61 59)(66 123 70 127)(68 125 72 121)(73 155 77 159)(75 157 79 153)(82 115 86 119)(84 117 88 113)(90 118 94 114)(92 120 96 116)(98 100 102 104)(106 124 110 128)(108 126 112 122)(130 139 134 143)(132 141 136 137)(138 152 142 148)(140 146 144 150)
(1 104)(2 97)(3 98)(4 99)(5 100)(6 101)(7 102)(8 103)(9 108)(10 109)(11 110)(12 111)(13 112)(14 105)(15 106)(16 107)(17 44)(18 45)(19 46)(20 47)(21 48)(22 41)(23 42)(24 43)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 64)(50 57)(51 58)(52 59)(53 60)(54 61)(55 62)(56 63)(65 76)(66 77)(67 78)(68 79)(69 80)(70 73)(71 74)(72 75)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 137)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)
(1 12 132 143 38)(2 144 13 39 133)(3 40 137 134 14)(4 135 33 15 138)(5 16 136 139 34)(6 140 9 35 129)(7 36 141 130 10)(8 131 37 11 142)(17 65 128 88 51)(18 81 66 52 121)(19 53 82 122 67)(20 123 54 68 83)(21 69 124 84 55)(22 85 70 56 125)(23 49 86 126 71)(24 127 50 72 87)(25 116 90 109 102)(26 110 117 103 91)(27 104 111 92 118)(28 93 97 119 112)(29 120 94 105 98)(30 106 113 99 95)(31 100 107 96 114)(32 89 101 115 108)(41 149 73 63 157)(42 64 150 158 74)(43 159 57 75 151)(44 76 160 152 58)(45 145 77 59 153)(46 60 146 154 78)(47 155 61 79 147)(48 80 156 148 62)
(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)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8H 8I 8J 8K 8L 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 10 10 10 10 10 10 20 20 size 1 1 1 1 2 2 4 20 4 5 5 5 5 10 10 20 4 10 ··· 10 20 20 20 20 4 4 4 8 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C8○D4 F5 C2×F5 C2×F5 C22⋊F5 D4.F5 kernel (C2×D4).7F5 D10⋊C8 C23.2F5 C22×C5⋊C8 C2×C22.F5 C2×D4⋊2D5 C2×Dic10 C2×C5⋊D4 D4×C10 C2×Dic5 C10 C2×D4 C2×C4 C23 C22 C2 # reps 1 2 2 1 1 1 2 4 2 4 8 1 1 2 4 2

Matrix representation of (C2×D4).7F5 in GL6(𝔽41)

 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 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 16 9 0 0 0 0 0 0 19 0 38 38 0 0 3 22 3 0 0 0 0 3 22 3 0 0 38 38 0 19
,
 40 4 0 0 0 0 0 1 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 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 3 0 0 0 0 0 0 3 0 0 0 0 0 0 17 34 30 4 0 0 37 11 7 24 0 0 37 13 30 26 0 0 17 13 28 24

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,16,0,0,0,0,0,9,0,0,0,0,0,0,19,3,0,38,0,0,0,22,3,38,0,0,38,3,22,0,0,0,38,0,3,19],[40,0,0,0,0,0,4,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,17,37,37,17,0,0,34,11,13,13,0,0,30,7,30,28,0,0,4,24,26,24] >;

(C2×D4).7F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)._7F_5
% in TeX

G:=Group("(C2xD4).7F5");
// GroupNames label

G:=SmallGroup(320,1113);
// by ID

G=gap.SmallGroup(320,1113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=1,e^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽