D4xF5 - GroupNames

G = D₄×F₅ order 160 = 2⁵·5

Direct product of D₄ and F₅

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄×F₅, D₂₀⋊₃C₄, D₁₀.₁₂C₂³, C₅⋊(C₄×D₄), C₂₀⋊(C₂×C₄), C₅⋊D₄⋊C₄, D₁₀⋊(C₂×C₄), C₄⋊F₅⋊₂C₂, C₄⋊₁(C₂×F₅), (C₅×D₄)⋊₃C₄, Dic₅⋊(C₂×C₄), (C₄×F₅)⋊₃C₂, (D₄×D₅).₃C₂, D₅.₂(C₂×D₄), C₂²⋊F₅⋊₃C₂, C₂²⋊₁(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);

D₄×F₅ is a maximal subgroup of
D₄₀⋊C₄ SD₁₆⋊F₅ D₁₀.C₂⁴ D₅.2₊¹⁺⁴ D₆₀⋊₃C₄ C₃⋊D₄⋊F₅
D₄×F₅ is a maximal quotient of
C₅⋊C₈⋊₈D₄ C₅⋊C₈⋊D₄ D₁₀⋊M₄(2) Dic₅⋊M₄(2) D₁₀⋊(C₄⋊C₄) C₁₀.(C₄×D₄) D₂₀⋊₂C₈ D₁₀⋊₂M₄(2) C₂₀⋊M₄(2) C₄⋊C₄⋊₅F₅ C₂₀⋊(C₄⋊C₄) D₄₀⋊C₄ D₈⋊₅F₅ D₈⋊F₅ SD₁₆⋊F₅ SD₁₆⋊₃F₅ SD₁₆⋊₂F₅ Dic₂₀⋊C₄ Q₁₆⋊₅F₅ Q₁₆⋊F₅ C₅⋊C₈⋊₇D₄ C₂₀⋊₂M₄(2) (C₂×F₅)⋊D₄ C₂.(D₄×F₅) D₆₀⋊₃C₄ C₃⋊D₄⋊F₅

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

Export

Character table of D₄×F₅ in TeX

G = D4×F5 order 160 = 25·5

Direct product of D4 and F5

G = D₄×F₅ order 160 = 2⁵·5

Direct product of D₄ and F₅