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## G = C74.C6order 444 = 22·3·37

### The non-split extension by C74 of C6 acting faithfully

Aliases: C74.C6, C372C12, Dic37⋊C3, C37⋊C32C4, C2.(C37⋊C6), (C2×C37⋊C3).C2, SmallGroup(444,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C37 — C74.C6
 Chief series C1 — C37 — C74 — C2×C37⋊C3 — C74.C6
 Lower central C37 — C74.C6
 Upper central C1 — C2

Generators and relations for C74.C6
G = < a,b | a74=1, b6=a37, bab-1=a11 >

Character table of C74.C6

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 37A 37B 37C 37D 37E 37F 74A 74B 74C 74D 74E 74F size 1 1 37 37 37 37 37 37 37 37 37 37 6 6 6 6 6 6 6 6 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 1 ζ3 ζ32 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ65 ζ6 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 ζ32 ζ3 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ6 ζ65 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 1 -i i -1 -1 i i -i -i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 1 i -i -1 -1 -i -i i i 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 4 ρ9 1 -1 ζ3 ζ32 i -i ζ6 ζ65 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 12 ρ10 1 -1 ζ32 ζ3 i -i ζ65 ζ6 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 12 ρ11 1 -1 ζ3 ζ32 -i i ζ6 ζ65 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 12 ρ12 1 -1 ζ32 ζ3 -i i ζ65 ζ6 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 12 ρ13 6 6 0 0 0 0 0 0 0 0 0 0 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 orthogonal lifted from C37⋊C6 ρ14 6 6 0 0 0 0 0 0 0 0 0 0 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 orthogonal lifted from C37⋊C6 ρ15 6 6 0 0 0 0 0 0 0 0 0 0 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 orthogonal lifted from C37⋊C6 ρ16 6 6 0 0 0 0 0 0 0 0 0 0 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 orthogonal lifted from C37⋊C6 ρ17 6 6 0 0 0 0 0 0 0 0 0 0 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 orthogonal lifted from C37⋊C6 ρ18 6 6 0 0 0 0 0 0 0 0 0 0 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 orthogonal lifted from C37⋊C6 ρ19 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 symplectic faithful, Schur index 2 ρ20 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 symplectic faithful, Schur index 2 ρ21 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 symplectic faithful, Schur index 2 ρ22 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 symplectic faithful, Schur index 2 ρ23 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 symplectic faithful, Schur index 2 ρ24 6 -6 0 0 0 0 0 0 0 0 0 0 ζ3736+ζ3727+ζ3726+ζ3711+ζ3710+ζ37 ζ3735+ζ3722+ζ3720+ζ3717+ζ3715+ζ372 ζ3734+ζ3733+ζ3730+ζ377+ζ374+ζ373 ζ3732+ζ3724+ζ3719+ζ3718+ζ3713+ζ375 ζ3731+ζ3729+ζ3723+ζ3714+ζ378+ζ376 ζ3728+ζ3725+ζ3721+ζ3716+ζ3712+ζ379 -ζ3731-ζ3729-ζ3723-ζ3714-ζ378-ζ376 -ζ3728-ζ3725-ζ3721-ζ3716-ζ3712-ζ379 -ζ3736-ζ3727-ζ3726-ζ3711-ζ3710-ζ37 -ζ3735-ζ3722-ζ3720-ζ3717-ζ3715-ζ372 -ζ3734-ζ3733-ζ3730-ζ377-ζ374-ζ373 -ζ3732-ζ3724-ζ3719-ζ3718-ζ3713-ζ375 symplectic faithful, Schur index 2

Smallest permutation representation of C74.C6
On 148 points
Generators in S148
```(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)
(1 76 38 113)(2 103 27 112 48 87 39 140 64 75 11 124)(3 130 16 111 21 98 40 93 53 148 58 135)(4 83 5 110 68 109 41 120 42 147 31 146)(6 137 57 108 14 131 43 100 20 145 51 94)(7 90 46 107 61 142 44 127 9 144 24 105)(8 117 35 106 34 79 45 80 72 143 71 116)(10 97 13 104 54 101 47 134 50 141 17 138)(12 77 65 102 74 123 49 114 28 139 37 86)(15 84 32 99 67 82 52 121 69 136 30 119)(18 91 73 96 60 115 55 128 36 133 23 78)(19 118 62 95 33 126 56 81 25 132 70 89)(22 125 29 92 26 85 59 88 66 129 63 122)```

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

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

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

Matrix representation of C74.C6 in GL7(𝔽1777)

 1776 0 0 0 0 0 0 0 791 182 907 1580 1037 1141 0 636 205 848 1345 469 451 0 1326 1493 353 353 1493 1326 0 451 469 1345 848 205 636 0 1141 1037 1580 907 182 791 0 986 1596 1466 1622 1336 637
,
 1200 0 0 0 0 0 0 0 671 224 902 732 984 947 0 1509 345 391 1100 50 1086 0 964 1590 276 345 1488 765 0 1765 485 1117 854 224 1619 0 1452 928 1726 866 1388 756 0 167 1479 809 1412 1700 20

`G:=sub<GL(7,GF(1777))| [1776,0,0,0,0,0,0,0,791,636,1326,451,1141,986,0,182,205,1493,469,1037,1596,0,907,848,353,1345,1580,1466,0,1580,1345,353,848,907,1622,0,1037,469,1493,205,182,1336,0,1141,451,1326,636,791,637],[1200,0,0,0,0,0,0,0,671,1509,964,1765,1452,167,0,224,345,1590,485,928,1479,0,902,391,276,1117,1726,809,0,732,1100,345,854,866,1412,0,984,50,1488,224,1388,1700,0,947,1086,765,1619,756,20] >;`

C74.C6 in GAP, Magma, Sage, TeX

`C_{74}.C_6`
`% in TeX`

`G:=Group("C74.C6");`
`// GroupNames label`

`G:=SmallGroup(444,1);`
`// by ID`

`G=gap.SmallGroup(444,1);`
`# by ID`

`G:=PCGroup([4,-2,-3,-2,-37,24,6915,2503]);`
`// Polycyclic`

`G:=Group<a,b|a^74=1,b^6=a^37,b*a*b^-1=a^11>;`
`// generators/relations`

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