C2^4:D5 - GroupNames

G = C₂⁴⋊D₅ order 160 = 2⁵·5

The semidirect product of C₂⁴ and D₅ acting faithfully

non-abelian, soluble, monomial

Aliases: C₂⁴⋊D₅, C₂⁴⋊C₅⋊C₂, SmallGroup(160,234)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂⁴⋊D₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₂⁴⋊D₅

Lower central

C₂⁴⋊C₅ — C₂⁴⋊D₅

Upper central

C₁

Generators and relations for C₂⁴⋊D₅
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁵=f²=1, fdf=ab=ba, ac=ca, ebe^-1=fbf=ad=da, eae^-1=d, af=fa, bc=cb, bd=db, cd=dc, ece^-1=abd, fcf=bcd, ede^-1=abcd, fef=e^-1 >

C₁

₅C₂

₂₀C₂

₁₆C₅

₅C₂²

₁₀C₄

₁₀C₂²

₁₀C₄

₁₀C₂²

₁₆D₅

₅C₂³

₅C₂×C₄

₅C₂³

₅C₂×C₄

₅C₂³

₁₀D₄

C₂⁴

₅C₂×D₄

₅C₂²⋊C₄

₅C₂×D₄

₅C₂²⋊C₄

₅C₂×D₄

₅C₂²≀C₂

C₂⁴⋊C₅

C₂⁴⋊D₅

Character table of C₂⁴⋊D₅

class	1	2A	2B	2C	2D	4A	4B	4C	5A	5B
size	1	5	5	5	20	20	20	20	32	32
ρ₁	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	-1	-1	-1	-1	1	1	linear of order 2
ρ₃	2	2	2	2	0	0	0	0	^-1+√5/₂	^-1-√5/₂	orthogonal lifted from D₅
ρ₄	2	2	2	2	0	0	0	0	^-1-√5/₂	^-1+√5/₂	orthogonal lifted from D₅
ρ₅	5	-3	1	1	-1	1	1	-1	0	0	orthogonal faithful
ρ₆	5	-3	1	1	1	-1	-1	1	0	0	orthogonal faithful
ρ₇	5	1	-3	1	1	1	-1	-1	0	0	orthogonal faithful
ρ₈	5	1	-3	1	-1	-1	1	1	0	0	orthogonal faithful
ρ₉	5	1	1	-3	1	-1	1	-1	0	0	orthogonal faithful
ρ₁₀	5	1	1	-3	-1	1	-1	1	0	0	orthogonal faithful

Permutation representations of C₂⁴⋊D₅
►On 10 points - transitive group 10T15

Generators in S₁₀

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

(3 10)(4 6)

(3 10)(5 7)

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

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

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

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

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

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

G:=TransitiveGroup(10,15);

►On 10 points - transitive group 10T16

Generators in S₁₀

(2 9)(4 6)

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

(1 8)(2 9)

(1 8)(3 10)

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

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

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

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

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

G:=TransitiveGroup(10,16);

►On 16 points: primitive - transitive group 16T415

Generators in S₁₆

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

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

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

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

(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)

(2 4)(5 6)(7 9)(10 11)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,415);

►On 20 points - transitive group 20T38

Generators in S₂₀

(2 11)(3 18)(4 10)(8 17)(9 12)(13 19)

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

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

(1 15)(2 17)(3 9)(7 16)(8 11)(12 18)

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

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

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

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

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

G:=TransitiveGroup(20,38);

►On 20 points - transitive group 20T39

Generators in S₂₀

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,39);

►On 20 points - transitive group 20T43

Generators in S₂₀

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

(1 11)(2 12)(9 17)(10 18)

(1 11)(3 13)(6 19)(9 17)

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

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

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

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

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

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

G:=TransitiveGroup(20,43);

►On 20 points - transitive group 20T45

Generators in S₂₀

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

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,45);

C₂⁴⋊D₅ is a maximal subgroup of C₂⁴⋊F₅ F₁₆⋊C₂ C₂⁴⋊D₁₅
C₂⁴⋊D₅ is a maximal quotient of 2_-¹⁺⁴.D₅ 2_-¹⁺⁴⋊D₅ C₂⁵.D₅ C₂⁴⋊D₁₅

Polynomial with Galois group C₂⁴⋊D₅ over ℚ

action	f(x)	Disc(f)
10T15	x¹⁰-2x⁹-14x⁸+26x⁷+62x⁶-110x⁵-85x⁴+170x³+8x²-84x+27	2¹⁰·3²·67²·401⁴
10T16	x¹⁰+2x⁹-103x⁸+212x⁷+1931x⁶-9457x⁵+14669x⁴-5930x³-3032x²+503x+83	37⁴·67²·281²·401⁵·3581²·6343²

Matrix representation of C₂⁴⋊D₅ ►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

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
-1	0	0	0	0
0	0	0	0	-1
0	0	0	-1	0
0	0	-1	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],[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,-1,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0] >;

C₂⁴⋊D₅ in GAP, Magma, Sage, TeX

C_2^4\rtimes D_5

% in TeX

G:=Group("C2^4:D5");

// GroupNames label

G:=SmallGroup(160,234);

// by ID

G=gap.SmallGroup(160,234);

# by ID

G:=PCGroup([6,-2,-5,-2,2,2,2,97,542,278,729,3304,1810,1805,191]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂⁴⋊D₅ in TeX
Character table of C₂⁴⋊D₅ in TeX

G = C24⋊D5 order 160 = 25·5

The semidirect product of C24 and D5 acting faithfully

G = C₂⁴⋊D₅ order 160 = 2⁵·5

The semidirect product of C₂⁴ and D₅ acting faithfully