C2xC2^4:C5 - GroupNames

G = C₂×C₂⁴⋊C₅ order 160 = 2⁵·5

Direct product of C₂ and C₂⁴⋊C₅

direct product, metabelian, soluble, monomial, A-group

Aliases: C₂×C₂⁴⋊C₅, C₂ ≀C₅, AΣL₁(𝔽₃₂), C₂⁵⋊C₅, C₂⁴⋊C₁₀, SmallGroup(160,235)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂×C₂⁴⋊C₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂×C₂⁴⋊C₅

Lower central

C₂⁴ — C₂×C₂⁴⋊C₅

Upper central

C₁ — C₂

Generators and relations for C₂×C₂⁴⋊C₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e²=f⁵=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcd, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, fef^-1=b >

Subgroups: 408 in 82 conjugacy classes, 6 normal (all characteristic)
C₁, C₂, C₂, C₂², C₅, C₂³, C₁₀, C₂⁴, C₂⁴, C₂⁵, C₂⁴⋊C₅, C₂×C₂⁴⋊C₅
Quotients: C₁, C₂, C₅, C₁₀, C₂⁴⋊C₅, C₂×C₂⁴⋊C₅

Character table of C₂×C₂⁴⋊C₅

class	1	2A	2B	2C	2D	2E	2F	2G	5A	5B	5C	5D	10A	10B	10C	10D
size	1	1	5	5	5	5	5	5	16	16	16	16	16	16	16	16
ρ₁	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	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	ζ₅⁴	ζ₅³	ζ₅²	ζ₅	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	linear of order 5
ρ₄	1	1	1	1	1	1	1	1	ζ₅	ζ₅²	ζ₅³	ζ₅⁴	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	linear of order 5
ρ₅	1	-1	1	1	1	-1	-1	-1	ζ₅	ζ₅²	ζ₅³	ζ₅⁴	-ζ₅⁴	-ζ₅²	-ζ₅³	-ζ₅	linear of order 10
ρ₆	1	-1	1	1	1	-1	-1	-1	ζ₅²	ζ₅⁴	ζ₅	ζ₅³	-ζ₅³	-ζ₅⁴	-ζ₅	-ζ₅²	linear of order 10
ρ₇	1	1	1	1	1	1	1	1	ζ₅³	ζ₅	ζ₅⁴	ζ₅²	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	linear of order 5
ρ₈	1	1	1	1	1	1	1	1	ζ₅²	ζ₅⁴	ζ₅	ζ₅³	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	linear of order 5
ρ₉	1	-1	1	1	1	-1	-1	-1	ζ₅³	ζ₅	ζ₅⁴	ζ₅²	-ζ₅²	-ζ₅	-ζ₅⁴	-ζ₅³	linear of order 10
ρ₁₀	1	-1	1	1	1	-1	-1	-1	ζ₅⁴	ζ₅³	ζ₅²	ζ₅	-ζ₅	-ζ₅³	-ζ₅²	-ζ₅⁴	linear of order 10
ρ₁₁	5	5	1	-3	1	1	1	-3	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₁₂	5	-5	1	-3	1	-1	-1	3	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₃	5	-5	1	1	-3	3	-1	-1	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₄	5	-5	-3	1	1	-1	3	-1	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₅	5	5	-3	1	1	1	-3	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅
ρ₁₆	5	5	1	1	-3	-3	1	1	0	0	0	0	0	0	0	0	orthogonal lifted from C₂⁴⋊C₅

Permutation representations of C₂×C₂⁴⋊C₅
►On 10 points - transitive group 10T14

Generators in S₁₀

(1 8)(2 9)(3 10)(4 6)(5 7)

(1 8)(3 10)

(1 8)(2 9)(4 6)(5 7)

(3 10)(4 6)

(2 9)(4 6)

(1 2 3 4 5)(6 7 8 9 10)

G:=sub<Sym(10)| (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(3,10), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (2,9)(4,6), (1,2,3,4,5)(6,7,8,9,10)>;

G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(3,10), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (2,9)(4,6), (1,2,3,4,5)(6,7,8,9,10) );

G=PermutationGroup([[(1,8),(2,9),(3,10),(4,6),(5,7)], [(1,8),(3,10)], [(1,8),(2,9),(4,6),(5,7)], [(3,10),(4,6)], [(2,9),(4,6)], [(1,2,3,4,5),(6,7,8,9,10)]])

G:=TransitiveGroup(10,14);

►On 20 points - transitive group 20T40

Generators in S₂₀

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

(1 11)(2 17)(3 8)(6 16)(7 12)(13 18)

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

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

(2 12)(3 18)(4 9)(7 17)(8 13)(14 19)

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

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

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

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

G:=TransitiveGroup(20,40);

►On 20 points - transitive group 20T41

Generators in S₂₀

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

(1 11)(2 17)(3 8)(6 16)(7 12)(13 18)

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

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

(2 12)(3 18)(4 9)(7 17)(8 13)(14 19)

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

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

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

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

G:=TransitiveGroup(20,41);

►On 20 points - transitive group 20T44

Generators in S₂₀

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

(1 13)(3 15)(8 17)(10 19)

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

(3 15)(4 11)(6 20)(10 19)

(2 14)(4 11)(6 20)(9 18)

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

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

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

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

G:=TransitiveGroup(20,44);

►On 20 points - transitive group 20T46

Generators in S₂₀

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,46);

C₂×C₂⁴⋊C₅ is a maximal subgroup of C₂⁵.D₅
C₂×C₂⁴⋊C₅ is a maximal quotient of 2_-¹⁺⁴.C₁₀

Polynomial with Galois group C₂×C₂⁴⋊C₅ over ℚ

action	f(x)	Disc(f)
10T14	x¹⁰-12x⁸+51x⁶-96x⁴+80x²-23	2¹⁰·11⁸·23

Matrix representation of C₂×C₂⁴⋊C₅ ►in GL₅(ℤ)

-1	0	0	0	0
0	-1	0	0	0
0	0	-1	0	0
0	0	0	-1	0
0	0	0	0	-1

-1	0	0	0	0
0	-1	0	0	0
0	0	-1	0	0
0	0	0	1	0
0	0	0	0	-1

-1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	-1

-1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	-1	0
0	0	0	0	1

-1	0	0	0	0
0	-1	0	0	0
0	0	1	0	0
0	0	0	-1	0
0	0	0	0	-1

0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1
1	0	0	0	0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C₂×C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes C_5

% in TeX

G:=Group("C2xC2^4:C5");

// GroupNames label

G:=SmallGroup(160,235);

// by ID

G=gap.SmallGroup(160,235);

# by ID

G:=PCGroup([6,-2,-5,-2,2,2,2,728,1089,1660,2711]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^5=1,a*b=b*a,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,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,f*e*f^-1=b>;

// generators/relations

Export

Character table of C₂×C₂⁴⋊C₅ in TeX

G = C2×C24⋊C5 order 160 = 25·5

Direct product of C2 and C24⋊C5

G = C₂×C₂⁴⋊C₅ order 160 = 2⁵·5

Direct product of C₂ and C₂⁴⋊C₅