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## G = D4×F5order 160 = 25·5

### Direct product of D4 and F5

Aliases: D4×F5, D203C4, D10.12C23, C5⋊(C4×D4), C20⋊(C2×C4), C5⋊D4⋊C4, D10⋊(C2×C4), C4⋊F52C2, C41(C2×F5), (C5×D4)⋊3C4, Dic5⋊(C2×C4), (C4×F5)⋊3C2, (D4×D5).3C2, D5.2(C2×D4), C22⋊F53C2, C221(C2×F5), (C22×F5)⋊1C2, C2.9(C22×F5), D5.2(C4○D4), C10.8(C22×C4), (C2×F5).3C22, (C4×D5).12C22, (C22×D5).16C22, (C2×C10)⋊(C2×C4), Aut(D20), Hol(C20), SmallGroup(160,207)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4×F5
 Chief series C1 — C5 — D5 — D10 — C2×F5 — C22×F5 — D4×F5
 Lower central C5 — C10 — D4×F5
 Upper central C1 — C2 — D4

Generators and relations for D4×F5
G = < a,b,c,d | a4=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 332 in 94 conjugacy classes, 38 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, D4, D4, C23, D5, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, F5, F5, D10, D10, D10, C2×C10, C4×D4, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C2×F5, C2×F5, C22×D5, C4×F5, C4⋊F5, C22⋊F5, D4×D5, C22×F5, D4×F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×F5, C22×F5, D4×F5

Character table of D4×F5

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5 10A 10B 10C 20 size 1 1 2 2 5 5 10 10 2 5 5 5 5 10 10 10 10 10 10 10 4 4 8 8 8 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 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 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 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 linear of order 2 ρ5 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 ρ6 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 ρ7 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 ρ8 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 ρ9 1 1 1 -1 -1 -1 -1 1 -1 -i i i -i i 1 i -i -i i -i 1 1 1 -1 -1 linear of order 4 ρ10 1 1 -1 -1 -1 -1 1 1 1 -i i i -i -i -1 -i i -i i i 1 1 -1 -1 1 linear of order 4 ρ11 1 1 -1 -1 -1 -1 1 1 1 i -i -i i i -1 i -i i -i -i 1 1 -1 -1 1 linear of order 4 ρ12 1 1 1 -1 -1 -1 -1 1 -1 i -i -i i -i 1 -i i i -i i 1 1 1 -1 -1 linear of order 4 ρ13 1 1 -1 1 -1 -1 1 -1 -1 -i i i -i i 1 -i i i -i -i 1 1 -1 1 -1 linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 1 -i i i -i -i -1 i -i i -i i 1 1 1 1 1 linear of order 4 ρ15 1 1 1 1 -1 -1 -1 -1 1 i -i -i i i -1 -i i -i i -i 1 1 1 1 1 linear of order 4 ρ16 1 1 -1 1 -1 -1 1 -1 -1 i -i -i i -i 1 i -i -i i i 1 1 -1 1 -1 linear of order 4 ρ17 2 -2 0 0 2 -2 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ18 2 -2 0 0 2 -2 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 2 -2 0 0 0 orthogonal lifted from D4 ρ19 2 -2 0 0 -2 2 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 2 -2 0 0 0 complex lifted from C4○D4 ρ20 2 -2 0 0 -2 2 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 2 -2 0 0 0 complex lifted from C4○D4 ρ21 4 4 4 -4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 -4 4 0 0 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 orthogonal lifted from C2×F5 ρ23 4 4 -4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 -1 orthogonal lifted from C2×F5 ρ24 4 4 4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ25 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 orthogonal faithful

Permutation representations of D4×F5
On 20 points - transitive group 20T42
Generators in S20
(1 11 6 16)(2 12 7 17)(3 13 8 18)(4 14 9 19)(5 15 10 20)
(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)

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

G:=Group( (1,11,6,16)(2,12,7,17)(3,13,8,18)(4,14,9,19)(5,15,10,20), (11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,6)(2,8,5,9)(3,10,4,7)(11,16)(12,18,15,19)(13,20,14,17) );

G=PermutationGroup([[(1,11,6,16),(2,12,7,17),(3,13,8,18),(4,14,9,19),(5,15,10,20)], [(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,6),(2,8,5,9),(3,10,4,7),(11,16),(12,18,15,19),(13,20,14,17)]])

G:=TransitiveGroup(20,42);

Matrix representation of D4×F5 in GL6(𝔽41)

 40 39 0 0 0 0 1 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [40,1,0,0,0,0,39,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D4×F5 in GAP, Magma, Sage, TeX

D_4\times F_5
% in TeX

G:=Group("D4xF5");
// GroupNames label

G:=SmallGroup(160,207);
// by ID

G=gap.SmallGroup(160,207);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,188,2309,599]);
// Polycyclic

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

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