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## G = C42⋊2F5order 320 = 26·5

### 2nd semidirect product of C42 and F5 acting via F5/C5=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42⋊2F5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×D20 — D10.D4 — C42⋊2F5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42⋊2F5
 Upper central C1 — C2 — C22 — C2×C4 — C42

Generators and relations for C422F5
G = < a,b,c,d | a4=b4=c5=d4=1, ab=ba, ac=ca, dad-1=a-1b, bc=cb, dbd-1=a2b-1, dcd-1=c3 >

Subgroups: 474 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C2×D4, C2×Q8, Dic5, C20, F5, D10, C2×C10, C23⋊C4, C4.4D4, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C423C4, D10⋊C4, C4×C20, C22⋊F5, C2×Dic10, C2×D20, D10.D4, C4.D20, C422F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, F5, C23⋊C4, C2×F5, C423C4, C22⋊F5, D10.D4, C422F5

Character table of C422F5

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 5 10A 10B 10C 20A 20B 20C 20D 20E 20F 20G 20H 20I 20J 20K 20L size 1 1 2 20 20 4 4 4 40 40 40 40 40 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 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 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 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 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 ρ4 1 1 1 1 1 -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 ρ5 1 1 1 -1 -1 1 1 1 i -1 -i i -i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 -1 -1 -1 1 -1 -i 1 -i i i 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 linear of order 4 ρ7 1 1 1 -1 -1 1 1 1 -i -1 i -i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 4 ρ8 1 1 1 -1 -1 -1 1 -1 i 1 i -i -i 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 linear of order 4 ρ9 2 2 2 2 -2 0 -2 0 0 0 0 0 0 2 2 2 2 0 -2 -2 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 2 0 -2 0 0 0 0 0 0 2 2 2 2 0 -2 -2 0 0 0 0 -2 -2 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 0 -4 4 -4 0 0 0 0 0 -1 -1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 orthogonal lifted from C2×F5 ρ12 4 4 -4 0 0 0 0 0 0 0 0 0 0 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ13 4 4 4 0 0 4 4 4 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 2ζ4ζ54+2ζ4ζ52+ζ4 -√5 √5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 √5 -√5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ43ζ52+2ζ43ζ5+ζ43 orthogonal lifted from D10.D4 ρ15 4 4 4 0 0 0 -4 0 0 0 0 0 0 -1 -1 -1 -1 -√5 1 1 -√5 √5 √5 -√5 1 1 -√5 √5 √5 orthogonal lifted from C22⋊F5 ρ16 4 4 4 0 0 0 -4 0 0 0 0 0 0 -1 -1 -1 -1 √5 1 1 √5 -√5 -√5 √5 1 1 √5 -√5 -√5 orthogonal lifted from C22⋊F5 ρ17 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 2ζ4ζ53+2ζ4ζ5+ζ4 -√5 √5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 √5 -√5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ43ζ54+2ζ43ζ53+ζ43 orthogonal lifted from D10.D4 ρ18 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 2ζ43ζ52+2ζ43ζ5+ζ43 √5 -√5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 -√5 √5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ4ζ53+2ζ4ζ5+ζ4 orthogonal lifted from D10.D4 ρ19 4 4 -4 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 2ζ43ζ54+2ζ43ζ53+ζ43 √5 -√5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 -√5 √5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ4ζ54+2ζ4ζ52+ζ4 orthogonal lifted from D10.D4 ρ20 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 4 0 0 -4 2i 0 0 -2i -2i -2i -2i 0 0 2i 2i 2i complex lifted from C42⋊3C4 ρ21 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 -1 -√5 √5 1 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 complex faithful ρ22 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 -1 -√5 √5 1 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 complex faithful ρ23 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 -1 √5 -√5 1 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 complex faithful ρ24 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 -1 √5 -√5 1 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 complex faithful ρ25 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 -1 -√5 √5 1 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 complex faithful ρ26 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 -1 √5 -√5 1 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 2ζ43ζ54+2ζ43ζ53+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ52+2ζ43ζ5+ζ43 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 complex faithful ρ27 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 4 0 0 -4 -2i 0 0 2i 2i 2i 2i 0 0 -2i -2i -2i complex lifted from C42⋊3C4 ρ28 4 -4 0 0 0 -2i 0 2i 0 0 0 0 0 -1 -√5 √5 1 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 2ζ4ζ53+2ζ4ζ5+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ54+2ζ4ζ52+ζ4 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 complex faithful ρ29 4 -4 0 0 0 2i 0 -2i 0 0 0 0 0 -1 √5 -√5 1 ζ43ζ54+ζ43ζ5+ζ43-ζ54+ζ5 2ζ43ζ52+2ζ43ζ5+ζ43 2ζ4ζ53+2ζ4ζ5+ζ4 ζ4ζ54+ζ4ζ5+ζ4-ζ54+ζ5 ζ4ζ53+ζ4ζ52+ζ4-ζ53+ζ52 ζ4ζ53+ζ4ζ52+ζ4+ζ53-ζ52 ζ4ζ54+ζ4ζ5+ζ4+ζ54-ζ5 2ζ4ζ54+2ζ4ζ52+ζ4 2ζ43ζ54+2ζ43ζ53+ζ43 ζ43ζ54+ζ43ζ5+ζ43+ζ54-ζ5 ζ43ζ53+ζ43ζ52+ζ43+ζ53-ζ52 ζ43ζ53+ζ43ζ52+ζ43-ζ53+ζ52 complex faithful

Smallest permutation representation of C422F5
On 80 points
Generators in S80
```(1 51 11 41)(2 52 12 42)(3 53 13 43)(4 54 14 44)(5 55 15 45)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 31 26 36)(22 33 30 39)(23 35 29 37)(24 32 28 40)(25 34 27 38)(41 61 56 71)(42 63 60 74)(43 65 59 72)(44 62 58 75)(45 64 57 73)(46 66 51 76)(47 68 55 79)(48 70 54 77)(49 67 53 80)(50 69 52 78)```

`G:=sub<Sym(80)| (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,56,71)(42,63,60,74)(43,65,59,72)(44,62,58,75)(45,64,57,73)(46,66,51,76)(47,68,55,79)(48,70,54,77)(49,67,53,80)(50,69,52,78)>;`

`G:=Group( (1,51,11,41)(2,52,12,42)(3,53,13,43)(4,54,14,44)(5,55,15,45)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (2,3,5,4)(7,8,10,9)(11,16)(12,18,15,19)(13,20,14,17)(21,31,26,36)(22,33,30,39)(23,35,29,37)(24,32,28,40)(25,34,27,38)(41,61,56,71)(42,63,60,74)(43,65,59,72)(44,62,58,75)(45,64,57,73)(46,66,51,76)(47,68,55,79)(48,70,54,77)(49,67,53,80)(50,69,52,78) );`

`G=PermutationGroup([[(1,51,11,41),(2,52,12,42),(3,53,13,43),(4,54,14,44),(5,55,15,45),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)], [(1,26,6,21),(2,27,7,22),(3,28,8,23),(4,29,9,24),(5,30,10,25),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(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)], [(2,3,5,4),(7,8,10,9),(11,16),(12,18,15,19),(13,20,14,17),(21,31,26,36),(22,33,30,39),(23,35,29,37),(24,32,28,40),(25,34,27,38),(41,61,56,71),(42,63,60,74),(43,65,59,72),(44,62,58,75),(45,64,57,73),(46,66,51,76),(47,68,55,79),(48,70,54,77),(49,67,53,80),(50,69,52,78)]])`

Matrix representation of C422F5 in GL4(𝔽41) generated by

 26 9 7 29 12 38 21 19 22 34 19 2 39 20 32 17
,
 3 6 7 40 1 4 7 8 33 34 37 40 1 34 35 38
,
 40 40 40 40 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 40 40 40 40
`G:=sub<GL(4,GF(41))| [26,12,22,39,9,38,34,20,7,21,19,32,29,19,2,17],[3,1,33,1,6,4,34,34,7,7,37,35,40,8,40,38],[40,1,0,0,40,0,1,0,40,0,0,1,40,0,0,0],[1,0,0,40,0,0,1,40,0,0,0,40,0,1,0,40] >;`

C422F5 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_2F_5`
`% in TeX`

`G:=Group("C4^2:2F5");`
`// GroupNames label`

`G:=SmallGroup(320,192);`
`// by ID`

`G=gap.SmallGroup(320,192);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,555,675,297,136,1684,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,b*c=c*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^3>;`
`// generators/relations`

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