C2^4:C5 - GroupNames

G = C₂⁴⋊C₅ order 80 = 2⁴·5

The semidirect product of C₂⁴ and C₅ acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C₂⁴⋊C₅, SmallGroup(80,49)

Series: Derived ►Chief ►Lower central ►Upper central

C₁ — C₂⁴ — C₂⁴⋊C₅

C₁ — C₂⁴ — C₂⁴⋊C₅

C₂⁴ — C₂⁴⋊C₅

C₁

Generators and relations for C₂⁴⋊C₅
G = < a,b,c,d,e | a²=b²=c²=d²=e⁵=1, ab=ba, ac=ca, ad=da, eae^-1=abc, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, ede^-1=a >

C₁

₅C₂

₁₆C₅

₅C₂²

₅C₂³

C₂⁴

C₂⁴⋊C₅

Character table of C₂⁴⋊C₅

class	1	2A	2B	2C	5A	5B	5C	5D
size	1	5	5	5	16	16	16	16
ρ₁	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	ζ₅²	ζ₅	ζ₅⁴	ζ₅³	linear of order 5
ρ₃	1	1	1	1	ζ₅³	ζ₅⁴	ζ₅	ζ₅²	linear of order 5
ρ₄	1	1	1	1	ζ₅⁴	ζ₅²	ζ₅³	ζ₅	linear of order 5
ρ₅	1	1	1	1	ζ₅	ζ₅³	ζ₅²	ζ₅⁴	linear of order 5
ρ₆	5	-3	1	1	0	0	0	0	orthogonal faithful
ρ₇	5	1	-3	1	0	0	0	0	orthogonal faithful
ρ₈	5	1	1	-3	0	0	0	0	orthogonal faithful

Permutation representations of C₂⁴⋊C₅
►On 10 points - transitive group 10T8

Generators in S₁₀

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

(4 6)(5 7)

(1 8)(4 6)

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

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

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

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

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

G:=TransitiveGroup(10,8);

►On 16 points: primitive - transitive group 16T178

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,178);

►On 20 points - transitive group 20T17

Generators in S₂₀

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

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

(1 15)(3 9)(5 20)(6 14)(7 16)(12 18)

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

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

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

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

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

G:=TransitiveGroup(20,17);

►On 20 points - transitive group 20T23

Generators in S₂₀

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

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

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

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

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

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

G:=TransitiveGroup(20,23);

C₂⁴⋊C₅ is a maximal subgroup of C₂⁴⋊D₅ F₁₆
C₂⁴⋊C₅ is a maximal quotient of 2_-¹⁺⁴⋊C₅ C₂⁴⋊C₂₅

Polynomial with Galois group C₂⁴⋊C₅ over ℚ

action	f(x)	Disc(f)
10T8	x¹⁰-20x⁸+149x⁶-519x⁴+851x²-529	2¹⁰·11⁸·23⁶

Matrix representation of 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

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],[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] >;

C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_2^4\rtimes C_5

% in TeX

G:=Group("C2^4:C5");

// GroupNames label

G:=SmallGroup(80,49);

// by ID

G=gap.SmallGroup(80,49);

# by ID

G:=PCGroup([5,-5,-2,2,2,2,401,677,1103,1879]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₂⁴⋊C₅ in TeX
Character table of C₂⁴⋊C₅ in TeX

G = C24⋊C5 order 80 = 24·5

The semidirect product of C24 and C5 acting faithfully

G = C₂⁴⋊C₅ order 80 = 2⁴·5

The semidirect product of C₂⁴ and C₅ acting faithfully