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## G = C4×D14⋊C4order 448 = 26·7

### Direct product of C4 and D14⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×D14⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — D7×C22×C4 — C4×D14⋊C4
 Lower central C7 — C14 — C4×D14⋊C4
 Upper central C1 — C22×C4 — C2×C42

Generators and relations for C4×D14⋊C4
G = < a,b,c,d | a4=b14=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b7c >

Subgroups: 1156 in 258 conjugacy classes, 107 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, D7, C14, C14, C42, C22⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C23×C4, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C4×C22⋊C4, C4×Dic7, D14⋊C4, C4×C28, C2×C4×D7, C2×C4×D7, C22×Dic7, C22×Dic7, C22×C28, C22×C28, C23×D7, C14.C42, C2×C4×Dic7, C2×D14⋊C4, C2×C4×C28, D7×C22×C4, C4×D14⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D7, D28, C7⋊D4, C22×D7, C4×C22⋊C4, D14⋊C4, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, D7×C42, C42⋊D7, C4×D28, C2×D14⋊C4, C4×C7⋊D4, C4×D14⋊C4

Smallest permutation representation of C4×D14⋊C4
On 224 points
Generators in S224
(1 134 24 120)(2 135 25 121)(3 136 26 122)(4 137 27 123)(5 138 28 124)(6 139 15 125)(7 140 16 126)(8 127 17 113)(9 128 18 114)(10 129 19 115)(11 130 20 116)(12 131 21 117)(13 132 22 118)(14 133 23 119)(29 159 43 144)(30 160 44 145)(31 161 45 146)(32 162 46 147)(33 163 47 148)(34 164 48 149)(35 165 49 150)(36 166 50 151)(37 167 51 152)(38 168 52 153)(39 155 53 154)(40 156 54 141)(41 157 55 142)(42 158 56 143)(57 193 74 178)(58 194 75 179)(59 195 76 180)(60 196 77 181)(61 183 78 182)(62 184 79 169)(63 185 80 170)(64 186 81 171)(65 187 82 172)(66 188 83 173)(67 189 84 174)(68 190 71 175)(69 191 72 176)(70 192 73 177)(85 212 109 199)(86 213 110 200)(87 214 111 201)(88 215 112 202)(89 216 99 203)(90 217 100 204)(91 218 101 205)(92 219 102 206)(93 220 103 207)(94 221 104 208)(95 222 105 209)(96 223 106 210)(97 224 107 197)(98 211 108 198)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 28)(11 27)(12 26)(13 25)(14 24)(29 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 56)(40 55)(41 54)(42 53)(57 72)(58 71)(59 84)(60 83)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(85 101)(86 100)(87 99)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(113 140)(114 139)(115 138)(116 137)(117 136)(118 135)(119 134)(120 133)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(141 157)(142 156)(143 155)(144 168)(145 167)(146 166)(147 165)(148 164)(149 163)(150 162)(151 161)(152 160)(153 159)(154 158)(169 186)(170 185)(171 184)(172 183)(173 196)(174 195)(175 194)(176 193)(177 192)(178 191)(179 190)(180 189)(181 188)(182 187)(197 220)(198 219)(199 218)(200 217)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 224)(208 223)(209 222)(210 221)
(1 219 55 181)(2 220 56 182)(3 221 43 169)(4 222 44 170)(5 223 45 171)(6 224 46 172)(7 211 47 173)(8 212 48 174)(9 213 49 175)(10 214 50 176)(11 215 51 177)(12 216 52 178)(13 217 53 179)(14 218 54 180)(15 197 32 187)(16 198 33 188)(17 199 34 189)(18 200 35 190)(19 201 36 191)(20 202 37 192)(21 203 38 193)(22 204 39 194)(23 205 40 195)(24 206 41 196)(25 207 42 183)(26 208 29 184)(27 209 30 185)(28 210 31 186)(57 131 99 153)(58 132 100 154)(59 133 101 141)(60 134 102 142)(61 135 103 143)(62 136 104 144)(63 137 105 145)(64 138 106 146)(65 139 107 147)(66 140 108 148)(67 127 109 149)(68 128 110 150)(69 129 111 151)(70 130 112 152)(71 114 86 165)(72 115 87 166)(73 116 88 167)(74 117 89 168)(75 118 90 155)(76 119 91 156)(77 120 92 157)(78 121 93 158)(79 122 94 159)(80 123 95 160)(81 124 96 161)(82 125 97 162)(83 126 98 163)(84 113 85 164)

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

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

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

136 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 7A 7B 7C 14A ··· 14U 28A ··· 28BT order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 14 14 14 14 1 ··· 1 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D7 C4○D4 D14 C4×D7 D28 C7⋊D4 C4○D28 kernel C4×D14⋊C4 C14.C42 C2×C4×Dic7 C2×D14⋊C4 C2×C4×C28 D7×C22×C4 D14⋊C4 C2×C4×D7 C2×C28 C2×C42 C2×C14 C22×C4 C2×C4 C2×C4 C2×C4 C22 # reps 1 2 1 2 1 1 16 8 4 3 4 9 36 12 12 24

Matrix representation of C4×D14⋊C4 in GL4(𝔽29) generated by

 17 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 19 10 0 0 19 7
,
 1 0 0 0 0 1 0 0 0 0 22 28 0 0 19 7
,
 12 0 0 0 0 1 0 0 0 0 21 6 0 0 23 8
G:=sub<GL(4,GF(29))| [17,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,19,19,0,0,10,7],[1,0,0,0,0,1,0,0,0,0,22,19,0,0,28,7],[12,0,0,0,0,1,0,0,0,0,21,23,0,0,6,8] >;

C4×D14⋊C4 in GAP, Magma, Sage, TeX

C_4\times D_{14}\rtimes C_4
% in TeX

G:=Group("C4xD14:C4");
// GroupNames label

G:=SmallGroup(448,472);
// by ID

G=gap.SmallGroup(448,472);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,58,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^14=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations

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