C2wrC4 - GroupNames

G = C₂wrC₄ order 64 = 2⁶

Wreath product of C₂ by C₄

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₂ wr C₄, AΣL₁(F₁₆), C₂⁴:₁C₄, C₂³.₁D₄, (C₂xC₄).₁D₄, C₂³:C₄:₁C₂, C₂²:C₄:₁C₄, C₄.D₄:₅C₂, C₂³.₁(C₂xC₄), C₂²wrC₂.₁C₂, C₂.₆(C₂³:C₄), (C₂xD₄).₁C₂², C₂².₉(C₂²:C₄), 2-Sylow(AGammaL(1,16)), SmallGroup(64,32)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂wrC₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂xD₄ — C₂²wrC₂ — C₂wrC₄

Lower central

C₁ — C₂ — C₂² — C₂³ — C₂wrC₄

Upper central

C₁ — C₂ — C₂² — C₂xD₄ — C₂wrC₄

Jennings

C₁ — C₂ — C₂² — C₂xD₄ — C₂wrC₄

Generators and relations for C₂wrC₄
G = < a,b,c,d,e | a²=b²=c²=d²=e⁴=1, ab=ba, ac=ca, ad=da, eae^-1=abcd, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 129 in 47 conjugacy classes, 13 normal (all characteristic)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, C₂²:C₄, C₂³:C₄, C₂wrC₄

C₁

C₂

₂C₂

₄C₂

C₂²

₂C₂²

₂C₄

₂C₂²

₄C₂²

₄C₄

₄C₂²

₈C₄

C₂³

C₂xC₄

₂C₂xC₄

₂C₂³

₄D₄

₄C₂³

₄C₈

₄C₂³

₄C₂xC₄

C₂xD₄

C₂²:C₄

C₂⁴

₂C₂²:C₄

₂C₂xD₄

₂M₄(2)

C₂³:C₄

C₄.D₄

C₂²wrC₂

C₂wrC₄

Character table of C₂wrC₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	8A	8B
size	1	1	2	4	4	4	4	4	8	8	8	8	8
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	1	1	1	1	1	-1	1	-1	i	-1	-i	i	-i	linear of order 4
ρ₆	1	1	1	-1	-1	-1	1	-1	i	1	-i	-i	i	linear of order 4
ρ₇	1	1	1	1	1	-1	1	-1	-i	-1	i	-i	i	linear of order 4
ρ₈	1	1	1	-1	-1	-1	1	-1	-i	1	i	i	-i	linear of order 4
ρ₉	2	2	2	0	0	2	-2	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	0	0	-2	-2	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	4	-4	0	2	-2	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₂	4	-4	0	-2	2	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₁₃	4	4	-4	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₂³:C₄

Permutation representations of C₂wrC₄
►On 8 points - transitive group 8T27

Generators in S₈

(2 7)

(2 7)(3 8)

(2 7)(4 5)

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

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

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

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

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

G:=TransitiveGroup(8,27);

►On 8 points - transitive group 8T28

Generators in S₈

(1 5)(4 7)

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

(1 4)(5 7)

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

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

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

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

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

G:=TransitiveGroup(8,28);

►On 16 points - transitive group 16T130

Generators in S₁₆

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

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

(1 15)(3 13)(5 12)(7 10)

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

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

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

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

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

G:=TransitiveGroup(16,130);

►On 16 points - transitive group 16T157

Generators in S₁₆

(2 13)(3 8)(7 9)(10 14)

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

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

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

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

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

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

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

G:=TransitiveGroup(16,157);

►On 16 points - transitive group 16T158

Generators in S₁₆

(2 8)(4 9)(6 14)(11 16)

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

(2 16)(4 14)(6 9)(8 11)

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

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

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

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

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

G:=TransitiveGroup(16,158);

►On 16 points - transitive group 16T159

Generators in S₁₆

(1 6)(2 13)(3 14)(7 11)(8 12)(10 16)

(1 6)(2 11)(4 15)(5 9)(7 13)(10 16)

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

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

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

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

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

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

G:=TransitiveGroup(16,159);

►On 16 points - transitive group 16T166

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,166);

►On 16 points - transitive group 16T170

Generators in S₁₆

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

(2 16)(3 13)(5 12)(8 11)

(2 16)(4 14)(6 9)(8 11)

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

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

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

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

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

G:=TransitiveGroup(16,170);

►On 16 points - transitive group 16T171

Generators in S₁₆

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

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

(1 6)(4 8)(10 12)(14 16)

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

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

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

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

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

G:=TransitiveGroup(16,171);

►On 16 points - transitive group 16T172

Generators in S₁₆

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

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

(2 16)(4 14)(6 9)(8 11)

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

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

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

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

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

G:=TransitiveGroup(16,172);

C₂wrC₄ is a maximal subgroup of
D₄wrC₂ C₄²:₄D₄ C₄²:₅D₄ C₄²:₆D₄ C₂⁴:Dic₃ C₅:C₂wrC₄ C₂⁴:₂F₅ C₂⁴:F₅
(C₂xD₄).D_2p: C₄oC₂wrC₄ C₂⁴.₃₆D₄ C₂wrC₄:C₂ C₃:C₂wrC₄ C₂³.₃D₁₂ C₂⁴:₅Dic₃ C₅:₃C₂wrC₄ C₂³.₃D₂₀ ...
C₂wrC₄ is a maximal quotient of
C₂⁴:C₈ C₂³.₂M₄(2) C₂³.Q₁₆ C₂.₇C₂wrC₄ C₄².D₄ C₄².₂D₄ C₄².₃D₄ C₄².₄D₄ C₅:C₂wrC₄
C₂³.D_4p: C₂⁴.D₄ C₂³.₃D₁₂ C₂³.₃D₂₀ C₂³.₃D₂₈ ...
(C₂xC₄).D_4p: C₂.C₂wrC₄ C₃:C₂wrC₄ C₅:₃C₂wrC₄ C₇:C₂wrC₄ ...
(C₂³xC_2p):C₄: C₂⁴.₅D₄ C₂⁴:₅Dic₃ C₂⁴:₂Dic₅ C₂⁴:₂F₅ C₂⁴:Dic₇ ...

Polynomial with Galois group C₂wrC₄ over Q

action	f(x)	Disc(f)
8T27	x⁸-4x⁷-x⁶+17x⁵-6x⁴-21x³+6x²+8x+1	3⁴·5⁶·1021
8T28	x⁸-9x⁶+24x⁴-20x²+5	2⁸·5⁵·101²

Matrix representation of C₂wrC₄ ►in GL₄(Z) generated by

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

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

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

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

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

G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,0,-1,0,0,-1,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[0,0,1,0,0,0,0,-1,1,0,0,0,0,1,0,0] >;

C₂wrC₄ in GAP, Magma, Sage, TeX

C_2\wr C_4

% in TeX

G:=Group("C2wrC4");

// GroupNames label

G:=SmallGroup(64,32);

// by ID

G=gap.SmallGroup(64,32);

# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,297,255,1444]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;

// generators/relations

Export

Subgroup lattice of C₂wrC₄ in TeX
Character table of C₂wrC₄ in TeX

G = C2wrC4 order 64 = 26

Wreath product of C2 by C4

G = C₂wrC₄ order 64 = 2⁶

Wreath product of C₂ by C₄