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## G = C42.188D14order 448 = 26·7

### 8th non-split extension by C42 of D14 acting via D14/D7=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C42.188D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×C4○D28 — C42.188D14
 Lower central C7 — C14 — C42.188D14
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C42.188D14
G = < a,b,c,d | a4=b4=c14=1, d2=a2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 1204 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C4×C4○D4, C4×Dic7, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, D7×C42, C42⋊D7, Dic74D4, Dic73Q8, D28⋊C4, C2×C4×Dic7, C7×C42⋊C2, C2×C4○D28, C42.188D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, D14, C23×C4, C2×C4○D4, C4×D7, C22×D7, C4×C4○D4, C2×C4×D7, C23×D7, D7×C22×C4, D7×C4○D4, C42.188D14

Smallest permutation representation of C42.188D14
On 224 points
Generators in S224
```(1 130 70 118)(2 131 57 119)(3 132 58 120)(4 133 59 121)(5 134 60 122)(6 135 61 123)(7 136 62 124)(8 137 63 125)(9 138 64 126)(10 139 65 113)(11 140 66 114)(12 127 67 115)(13 128 68 116)(14 129 69 117)(15 158 77 210)(16 159 78 197)(17 160 79 198)(18 161 80 199)(19 162 81 200)(20 163 82 201)(21 164 83 202)(22 165 84 203)(23 166 71 204)(24 167 72 205)(25 168 73 206)(26 155 74 207)(27 156 75 208)(28 157 76 209)(29 184 154 43)(30 185 141 44)(31 186 142 45)(32 187 143 46)(33 188 144 47)(34 189 145 48)(35 190 146 49)(36 191 147 50)(37 192 148 51)(38 193 149 52)(39 194 150 53)(40 195 151 54)(41 196 152 55)(42 183 153 56)(85 221 105 171)(86 222 106 172)(87 223 107 173)(88 224 108 174)(89 211 109 175)(90 212 110 176)(91 213 111 177)(92 214 112 178)(93 215 99 179)(94 216 100 180)(95 217 101 181)(96 218 102 182)(97 219 103 169)(98 220 104 170)
(1 91 8 104)(2 112 9 85)(3 93 10 106)(4 100 11 87)(5 95 12 108)(6 102 13 89)(7 97 14 110)(15 194 84 187)(16 54 71 47)(17 196 72 189)(18 56 73 49)(19 184 74 191)(20 44 75 51)(21 186 76 193)(22 46 77 53)(23 188 78 195)(24 48 79 55)(25 190 80 183)(26 50 81 43)(27 192 82 185)(28 52 83 45)(29 155 36 200)(30 208 37 163)(31 157 38 202)(32 210 39 165)(33 159 40 204)(34 198 41 167)(35 161 42 206)(57 92 64 105)(58 99 65 86)(59 94 66 107)(60 101 67 88)(61 96 68 109)(62 103 69 90)(63 98 70 111)(113 222 120 179)(114 173 121 216)(115 224 122 181)(116 175 123 218)(117 212 124 169)(118 177 125 220)(119 214 126 171)(127 174 134 217)(128 211 135 182)(129 176 136 219)(130 213 137 170)(131 178 138 221)(132 215 139 172)(133 180 140 223)(141 156 148 201)(142 209 149 164)(143 158 150 203)(144 197 151 166)(145 160 152 205)(146 199 153 168)(147 162 154 207)
(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)
(1 162 70 200)(2 199 57 161)(3 160 58 198)(4 197 59 159)(5 158 60 210)(6 209 61 157)(7 156 62 208)(8 207 63 155)(9 168 64 206)(10 205 65 167)(11 166 66 204)(12 203 67 165)(13 164 68 202)(14 201 69 163)(15 134 77 122)(16 121 78 133)(17 132 79 120)(18 119 80 131)(19 130 81 118)(20 117 82 129)(21 128 83 116)(22 115 84 127)(23 140 71 114)(24 113 72 139)(25 138 73 126)(26 125 74 137)(27 136 75 124)(28 123 76 135)(29 111 154 91)(30 90 141 110)(31 109 142 89)(32 88 143 108)(33 107 144 87)(34 86 145 106)(35 105 146 85)(36 98 147 104)(37 103 148 97)(38 96 149 102)(39 101 150 95)(40 94 151 100)(41 99 152 93)(42 92 153 112)(43 213 184 177)(44 176 185 212)(45 211 186 175)(46 174 187 224)(47 223 188 173)(48 172 189 222)(49 221 190 171)(50 170 191 220)(51 219 192 169)(52 182 193 218)(53 217 194 181)(54 180 195 216)(55 215 196 179)(56 178 183 214)```

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 4W ··· 4AD 7A 7B 7C 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28AP order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 14 14 14 14 1 1 1 1 2 ··· 2 7 ··· 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D7 C4○D4 D14 D14 D14 D14 C4×D7 D7×C4○D4 kernel C42.188D14 D7×C42 C42⋊D7 Dic7⋊4D4 Dic7⋊3Q8 D28⋊C4 C2×C4×Dic7 C7×C42⋊C2 C2×C4○D28 C4○D28 C42⋊C2 Dic7 C42 C22⋊C4 C4⋊C4 C22×C4 C2×C4 C2 # reps 1 2 2 4 2 2 1 1 1 16 3 8 6 6 6 3 24 12

Matrix representation of C42.188D14 in GL4(𝔽29) generated by

 1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 12
,
 17 0 0 0 0 17 0 0 0 0 28 19 0 0 6 1
,
 7 19 0 0 10 19 0 0 0 0 28 0 0 0 6 1
,
 20 20 0 0 25 9 0 0 0 0 12 4 0 0 0 17
`G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[17,0,0,0,0,17,0,0,0,0,28,6,0,0,19,1],[7,10,0,0,19,19,0,0,0,0,28,6,0,0,0,1],[20,25,0,0,20,9,0,0,0,0,12,0,0,0,4,17] >;`

C42.188D14 in GAP, Magma, Sage, TeX

`C_4^2._{188}D_{14}`
`% in TeX`

`G:=Group("C4^2.188D14");`
`// GroupNames label`

`G:=SmallGroup(448,975);`
`// by ID`

`G=gap.SmallGroup(448,975);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,297,80,18822]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^14=1,d^2=a^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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