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G = D5×S4  order 240 = 24·3·5

Direct product of D5 and S4

direct product, non-abelian, soluble, monomial

Aliases: D5×S4, A4⋊1D10, C5⋊S4⋊C2, (C5×S4)⋊C2, C5⋊1(C2×S4), (D5×A4)⋊C2, (C2×C10)⋊D6, C22⋊(S3×D5), (C5×A4)⋊C22, (C22×D5)⋊S3, SmallGroup(240,194)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×A4 — D5×S4
Chief series
C1 — C22 — C2×C10 — C5×A4 — D5×A4 — D5×S4
Lower central
C5×A4 — D5×S4
Upper central
C1

Generators and relations for D5×S4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 468 in 66 conjugacy classes, 13 normal (all characteristic)
C1, C2 [×5], C3, C4 [×2], C22, C22 [×6], C5, S3 [×2], C6, C2×C4, D4 [×4], C23 [×2], D5, D5 [×2], C10 [×2], A4, D6, C15, C2×D4, Dic5, C20, D10 [×5], C2×C10, C2×C10, S4, S4, C2×A4, C5×S3, C3×D5, D15, C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5, C22×D5, C2×S4, S3×D5, C5×A4, D4×D5, C5×S4, C5⋊S4, D5×A4, D5×S4
Quotients: C1, C2 [×3], C22, S3, D5, D6, D10, S4, C2×S4, S3×D5, D5×S4

Character table of D5×S4

 class 12A2B2C2D2E34A4B5A5B610A10B10C10D15A15B20A20B
 size 135615308630224066121216161212
ρ111111111111111111111    trivial
ρ211-11-1-111-111-111111111    linear of order 2
ρ311-1-1-111-1111-111-1-111-1-1    linear of order 2
ρ4111-11-11-1-111111-1-111-1-1    linear of order 2
ρ522-20-20-1002212200-1-100    orthogonal lifted from D6
ρ6222020-10022-12200-1-100    orthogonal lifted from S3
ρ7220-2002-20-1+√5/2-1-√5/20-1+√5/2-1-√5/21-√5/21+√5/2-1+√5/2-1-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ8220200220-1-√5/2-1+√5/20-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ9220200220-1+√5/2-1-√5/20-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ10220-2002-20-1-√5/2-1+√5/20-1-√5/2-1+√5/21+√5/21-√5/2-1-√5/2-1+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ113-1-311-10-11330-1-11100-1-1    orthogonal lifted from C2×S4
ρ123-131-110-1-1330-1-11100-1-1    orthogonal lifted from S4
ρ133-13-1-1-1011330-1-1-1-10011    orthogonal lifted from S4
ρ143-1-3-11101-1330-1-1-1-10011    orthogonal lifted from C2×S4
ρ15440000-200-1-√5-1+√50-1-√5-1+√5001+√5/21-√5/200    orthogonal lifted from S3×D5
ρ16440000-200-1+√5-1-√50-1+√5-1-√5001-√5/21+√5/200    orthogonal lifted from S3×D5
ρ176-202000-20-3-3√5/2-3+3√5/201+√5/21-√5/2-1-√5/2-1+√5/2001+√5/21-√5/2    orthogonal faithful
ρ186-20-200020-3+3√5/2-3-3√5/201-√5/21+√5/21-√5/21+√5/200-1+√5/2-1-√5/2    orthogonal faithful
ρ196-202000-20-3+3√5/2-3-3√5/201-√5/21+√5/2-1+√5/2-1-√5/2001-√5/21+√5/2    orthogonal faithful
ρ206-20-200020-3-3√5/2-3+3√5/201+√5/21-√5/21+√5/21-√5/200-1-√5/2-1+√5/2    orthogonal faithful

Permutation representations of D5×S4
►On 20 points - transitive group 20T69
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

(1 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 20)(17 19)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)

(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)

(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)

(6 11)(7 12)(8 13)(9 14)(10 15)
 
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (6,11)(7,12)(8,13)(9,14)(10,15)>;
 
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,10)(7,9)(11,15)(12,14)(16,20)(17,19), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (6,11,16)(7,12,17)(8,13,18)(9,14,19)(10,15,20), (6,11)(7,12)(8,13)(9,14)(10,15) );
 
G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,10),(7,9),(11,15),(12,14),(16,20),(17,19)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,11),(7,12),(8,13),(9,14),(10,15)], [(6,11,16),(7,12,17),(8,13,18),(9,14,19),(10,15,20)], [(6,11),(7,12),(8,13),(9,14),(10,15)])
 
G:=TransitiveGroup(20,69);
 

►On 30 points - transitive group 30T54
Generators in S30
(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)

(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)

(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)(21 30)(22 26)(23 27)(24 28)(25 29)
 
G:=sub<Sym(30)| (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29)>;
 
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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29) );
 
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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,17),(18,20),(21,24),(22,23),(26,27),(28,30)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19),(21,30),(22,26),(23,27),(24,28),(25,29)])
 
G:=TransitiveGroup(30,54);
 

►On 30 points - transitive group 30T59
Generators in S30
(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)

(1 12)(2 11)(3 15)(4 14)(5 13)(6 28)(7 27)(8 26)(9 30)(10 29)(16 23)(17 22)(18 21)(19 25)(20 24)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)

(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)
 
G:=sub<Sym(30)| (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30)>;
 
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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30) );
 
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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,28),(7,27),(8,26),(9,30),(10,29),(16,23),(17,22),(18,21),(19,25),(20,24)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,26),(17,27),(18,28),(19,29),(20,30)])
 
G:=TransitiveGroup(30,59);
 

►On 30 points - transitive group 30T62
Generators in S30
(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)

(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)

(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)
 
G:=sub<Sym(30)| (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30)>;
 
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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30) );
 
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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,17),(18,20),(21,24),(22,23),(26,27),(28,30)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,26),(17,27),(18,28),(19,29),(20,30)])
 
G:=TransitiveGroup(30,62);
 

►On 30 points - transitive group 30T63
Generators in S30
(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)

(1 12)(2 11)(3 15)(4 14)(5 13)(6 28)(7 27)(8 26)(9 30)(10 29)(16 23)(17 22)(18 21)(19 25)(20 24)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)

(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)(21 30)(22 26)(23 27)(24 28)(25 29)
 
G:=sub<Sym(30)| (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29)>;
 
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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29) );
 
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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,28),(7,27),(8,26),(9,30),(10,29),(16,23),(17,22),(18,21),(19,25),(20,24)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19),(21,30),(22,26),(23,27),(24,28),(25,29)])
 
G:=TransitiveGroup(30,63);
 

D5×S4 is a maximal quotient of
 CSU2(𝔽3)⋊D5  Dic5.6S4  Dic5.7S4  GL2(𝔽3)⋊D5  D10.1S4  D10.2S4  A4⋊Dic10  Dic5⋊2S4  Dic5⋊S4  D10⋊S4  A4⋊D20

Matrix representation of D5×S4 ►in GL5(𝔽61)

01000
6017000
00100
00010
00001
,
060000
600000
006000
000600
000060
,
10000
01000
000601
000600
001600
,
10000
01000
000160
001060
000060
,
10000
01000
00001
00100
00010
,
600000
060000
00010
00100
00001

G:=sub<GL(5,GF(61))| [0,60,0,0,0,1,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;
 

D5×S4 in GAP, Magma, Sage, TeX 

D_5\times S_4
 
% in TeX
 
G:=Group("D5xS4");
 
// GroupNames label
 
G:=SmallGroup(240,194);
 
// by ID
 
G=gap.SmallGroup(240,194);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-5,-2,2,80,1155,1810,916,1091,1637]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^3=f^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

Export

Character table of D5×S4 in TeX

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