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## G = D7×C2×C16order 448 = 26·7

### Direct product of C2×C16 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C2×C16
 Chief series C1 — C7 — C14 — C28 — C56 — C8×D7 — D7×C2×C8 — D7×C2×C16
 Lower central C7 — D7×C2×C16
 Upper central C1 — C2×C16

Generators and relations for D7×C2×C16
G = < a,b,c,d | a2=b16=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 308 in 98 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C16, C16, C2×C8, C2×C8, C22×C4, Dic7, C28, D14, C2×C14, C2×C16, C2×C16, C22×C8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×C16, C7⋊C16, C112, C8×D7, C2×C7⋊C8, C2×C56, C2×C4×D7, D7×C16, C2×C7⋊C16, C2×C112, D7×C2×C8, D7×C2×C16
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C16, C2×C8, C22×C4, D14, C2×C16, C22×C8, C4×D7, C22×D7, C22×C16, C8×D7, C2×C4×D7, D7×C16, D7×C2×C8, D7×C2×C16

Smallest permutation representation of D7×C2×C16
On 224 points
Generators in S224
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 124)(34 125)(35 126)(36 127)(37 128)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(81 192)(82 177)(83 178)(84 179)(85 180)(86 181)(87 182)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 196)(98 197)(99 198)(100 199)(101 200)(102 201)(103 202)(104 203)(105 204)(106 205)(107 206)(108 207)(109 208)(110 193)(111 194)(112 195)(129 164)(130 165)(131 166)(132 167)(133 168)(134 169)(135 170)(136 171)(137 172)(138 173)(139 174)(140 175)(141 176)(142 161)(143 162)(144 163)(145 221)(146 222)(147 223)(148 224)(149 209)(150 210)(151 211)(152 212)(153 213)(154 214)(155 215)(156 216)(157 217)(158 218)(159 219)(160 220)
(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)
(1 205 177 222 40 133 71)(2 206 178 223 41 134 72)(3 207 179 224 42 135 73)(4 208 180 209 43 136 74)(5 193 181 210 44 137 75)(6 194 182 211 45 138 76)(7 195 183 212 46 139 77)(8 196 184 213 47 140 78)(9 197 185 214 48 141 79)(10 198 186 215 33 142 80)(11 199 187 216 34 143 65)(12 200 188 217 35 144 66)(13 201 189 218 36 129 67)(14 202 190 219 37 130 68)(15 203 191 220 38 131 69)(16 204 192 221 39 132 70)(17 61 110 86 150 119 172)(18 62 111 87 151 120 173)(19 63 112 88 152 121 174)(20 64 97 89 153 122 175)(21 49 98 90 154 123 176)(22 50 99 91 155 124 161)(23 51 100 92 156 125 162)(24 52 101 93 157 126 163)(25 53 102 94 158 127 164)(26 54 103 95 159 128 165)(27 55 104 96 160 113 166)(28 56 105 81 145 114 167)(29 57 106 82 146 115 168)(30 58 107 83 147 116 169)(31 59 108 84 148 117 170)(32 60 109 85 149 118 171)
(1 79)(2 80)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 49)(30 50)(31 51)(32 52)(33 178)(34 179)(35 180)(36 181)(37 182)(38 183)(39 184)(40 185)(41 186)(42 187)(43 188)(44 189)(45 190)(46 191)(47 192)(48 177)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 167)(98 168)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(129 193)(130 194)(131 195)(132 196)(133 197)(134 198)(135 199)(136 200)(137 201)(138 202)(139 203)(140 204)(141 205)(142 206)(143 207)(144 208)(145 153)(146 154)(147 155)(148 156)(149 157)(150 158)(151 159)(152 160)(209 217)(210 218)(211 219)(212 220)(213 221)(214 222)(215 223)(216 224)

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

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

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

160 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A ··· 8H 8I ··· 8P 14A ··· 14I 16A ··· 16P 16Q ··· 16AF 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 8 ··· 8 14 ··· 14 16 ··· 16 16 ··· 16 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 1 1 7 7 7 7 1 1 1 1 7 7 7 7 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2

160 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 C16 D7 D14 D14 C4×D7 C4×D7 C8×D7 C8×D7 D7×C16 kernel D7×C2×C16 D7×C16 C2×C7⋊C16 C2×C112 D7×C2×C8 C8×D7 C2×C7⋊C8 C2×C4×D7 C4×D7 C2×Dic7 C22×D7 D14 C2×C16 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 4 2 2 8 4 4 32 3 6 3 6 6 12 12 48

Matrix representation of D7×C2×C16 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 78 0 0 0 0 98 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 80 1 0 0 32 112
,
 1 0 0 0 0 112 0 0 0 0 9 88 0 0 71 104
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[78,0,0,0,0,98,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,80,32,0,0,1,112],[1,0,0,0,0,112,0,0,0,0,9,71,0,0,88,104] >;

D7×C2×C16 in GAP, Magma, Sage, TeX

D_7\times C_2\times C_{16}
% in TeX

G:=Group("D7xC2xC16");
// GroupNames label

G:=SmallGroup(448,433);
// by ID

G=gap.SmallGroup(448,433);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,58,80,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^16=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽