D4xF5 - GroupNames

G = D₄xF₅ order 160 = 2⁵·5

Direct product of D₄ and F₅

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

Aliases: D₄xF₅, D₂₀:₃C₄, D₁₀.₁₂C₂³, C₅:(C₄xD₄), C₂₀:(C₂xC₄), C₅:D₄:C₄, D₁₀:(C₂xC₄), C₄:F₅:₂C₂, C₄:₁(C₂xF₅), (C₅xD₄):₃C₄, Dic₅:(C₂xC₄), (C₄xF₅):₃C₂, (D₄xD₅).₃C₂, D₅.₂(C₂xD₄), C₂²:F₅:₃C₂, C₂²:₁(C₂xF₅), (C₂²xF₅):₁C₂, C₂.₉(C₂²xF₅), D₅.₂(C₄oD₄), C₁₀.₈(C₂²xC₄), (C₂xF₅).₃C₂², (C₄xD₅).₁₂C₂², (C₂²xD₅).₁₆C₂², (C₂xC₁₀):(C₂xC₄), Aut(D₂₀), Hol(C₂₀), SmallGroup(160,207)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — D₄xF₅

Chief series

C₁ — C₅ — D₅ — D₁₀ — C₂xF₅ — C₂²xF₅ — D₄xF₅

Lower central

C₅ — C₁₀ — D₄xF₅

Upper central

C₁ — C₂ — D₄

Generators and relations for D₄xF₅
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, C₂xC₄, D₄, D₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, Dic₅, C₂₀, F₅, F₅, D₁₀, D₁₀, D₁₀, C₂xC₁₀, C₄xD₄, C₄xD₅, D₂₀, C₅:D₄, C₅xD₄, C₂xF₅, C₂xF₅, C₂xF₅, C₂²xD₅, C₄xF₅, C₄:F₅, C₂²:F₅, D₄xD₅, C₂²xF₅, D₄xF₅
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂³, C₂²xC₄, C₂xD₄, C₄oD₄, F₅, C₄xD₄, C₂xF₅, C₂²xF₅, D₄xF₅

Character table of D₄xF₅

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 C₄oD₄
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 C₄oD₄
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 C₂xF₅
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 C₂xF₅
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 C₂xF₅
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 D₄xF₅
►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);

D₄xF₅ is a maximal subgroup of
D₄₀:C₄ SD₁₆:F₅ D₁₀.C₂⁴ D₅.2₊¹⁺⁴ D₆₀:₃C₄ C₃:D₄:F₅
D₄xF₅ is a maximal quotient of
C₅:C₈:₈D₄ C₅:C₈:D₄ D₁₀:M₄(2) Dic₅:M₄(2) D₁₀:(C₄:C₄) C₁₀.(C₄xD₄) 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) (C₂xF₅):D₄ C₂.(D₄xF₅) D₆₀:₃C₄ C₃:D₄:F₅

Matrix representation of D₄xF₅ ►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] >;

D₄xF₅ 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 D₄xF₅ in TeX

G = D4xF5 order 160 = 25·5

Direct product of D4 and F5

G = D₄xF₅ order 160 = 2⁵·5

Direct product of D₄ and F₅