C2wrC2^2 - GroupNames

G = C₂≀C₂² order 64 = 2⁶

Wreath product of C₂ by C₂²

p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C₂ ≀C₂², C₂³⋊D₄, C₂⁴⋊₁C₂², C₂³.₃C₂³, 2₊¹⁺⁴⋊₂C₂, (C₂×C₄)⋊D₄, C₂³⋊C₄⋊₃C₂, C₂²≀C₂⋊₁C₂, C₂.₁₈C₂²≀C₂, C₂²⋊C₄⋊₁C₂², (C₂×D₄).₉C₂², C₂².₁₆(C₂×D₄), 2-Sylow(A₈), Hol(C₂×C₄), SmallGroup(64,138)

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

Derived series

C₁ — C₂³ — C₂≀C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂×D₄ — 2₊¹⁺⁴ — C₂≀C₂²

Lower central

C₁ — C₂ — C₂³ — C₂≀C₂²

Upper central

C₁ — C₂ — C₂³ — C₂≀C₂²

Jennings

C₁ — C₂ — C₂³ — C₂≀C₂²

Generators and relations for C₂≀C₂²
G = < a,b,c,d,e | a²=b²=c²=d⁴=e²=1, ab=ba, ac=ca, dad^-1=eae=abc, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 225 in 99 conjugacy classes, 27 normal (6 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₂≀C₂²
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₂≀C₂²

Character table of C₂≀C₂²

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F
size	1	1	2	2	2	4	4	4	4	4	4	4	4	8	8	8
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	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	linear of order 2
ρ₅	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	linear of order 2
ρ₇	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	linear of order 2
ρ₉	2	2	-2	-2	2	0	0	2	0	0	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	-2	-2	0	-2	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	-2	2	0	0	-2	0	0	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	2	-2	0	0	0	-2	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	-2	2	-2	0	0	0	2	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	-2	-2	0	2	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	4	-4	0	0	0	-2	0	0	0	2	0	0	0	0	0	0	orthogonal faithful
ρ₁₆	4	-4	0	0	0	2	0	0	0	-2	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₂≀C₂²
►On 8 points - transitive group 8T29

Generators in S₈

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

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

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

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

(2 4)(5 7)

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

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

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

G:=TransitiveGroup(8,29);

►On 8 points - transitive group 8T31

Generators in S₈

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

(2 3)(5 7)

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

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

(6 8)

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

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

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

G:=TransitiveGroup(8,31);

►On 16 points - transitive group 16T127

Generators in S₁₆

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

(1 4)(5 7)(10 12)(14 16)

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

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

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

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

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

G:=TransitiveGroup(16,127);

►On 16 points - transitive group 16T128

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,128);

►On 16 points - transitive group 16T129

Generators in S₁₆

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

(1 15)(3 13)(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)

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

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

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

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

G:=TransitiveGroup(16,129);

►On 16 points - transitive group 16T147

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,147);

►On 16 points - transitive group 16T149

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,149);

►On 16 points - transitive group 16T150

Generators in S₁₆

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

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

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

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

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

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

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

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

G:=TransitiveGroup(16,150);

C₂≀C₂² is a maximal subgroup of
C₂≀A₄ C₂³⋊S₄
C₂⁴⋊D_2p: D₄≀C₂ C₄²⋊₅D₄ C₂⁴⋊C₂³ C₂⁴⋊₆D₆ C₂⁴⋊D₆ C₂⁴⋊₂D₁₀ C₂⁴⋊D₁₄ ...
(C₂²×C_2p)⋊D₄: C₄²⋊₄D₄ C₄²⋊₆D₄ C₂³.₇C₂⁴ C₂³.₉C₂⁴ C₂³⋊D₁₂ 2₊¹⁺⁴⋊₇S₃ C₂³⋊D₂₀ 2₊¹⁺⁴⋊₂D₅ ...
C₂≀C₂² is a maximal quotient of
C₂³⋊SD₁₆ C₄⋊C₄.D₄ (C₂×C₄)⋊D₈ (C₂×C₄)⋊SD₁₆ C₂⁴.₉D₄ C₂³⋊₂SD₁₆ C₂³⋊Q₁₆ C₄⋊C₄.₆D₄ Q₈⋊D₄⋊C₂ (C₂×C₄)⋊Q₁₆ C₂⁴.₁₂D₄ 2₊¹⁺⁴⋊₂C₄ C₂⁴.₂₂D₄ C₂⁴⋊₂Q₈ C₂⁴.₃₃D₄ C₂⁴.₁₈₂C₂³ C₄².₁₃D₄ C₄².₁₄D₄ C₄².₁₅D₄ C₄².₁₆D₄ C₄².₁₇D₄ Q₈≀C₂
C₂³⋊D_4p: C₂³⋊D₈ C₂³⋊D₁₂ C₂³⋊D₂₀ C₂³⋊D₂₈ ...
C₂⁴⋊D_2p: C₂⁴⋊D₄ D₄≀C₂ C₄²⋊₅D₄ C₂⁴⋊₆D₆ C₂⁴⋊₂D₁₀ C₂⁴⋊D₁₄ ...
(C₂²×C_2p)⋊D₄: C₂⁵.C₂² C₄²⋊₄D₄ C₄²⋊₆D₄ 2₊¹⁺⁴⋊₇S₃ 2₊¹⁺⁴⋊₂D₅ 2₊¹⁺⁴⋊₂D₇ ...

Polynomial with Galois group C₂≀C₂² over ℚ

action	f(x)	Disc(f)
8T29	x⁸-x⁶+x²+1	2¹⁴·7⁴
8T31	x⁸-4x⁷-4x⁶+18x⁵+8x⁴-20x³-3x²+6x-1	2¹²·5⁴·761

Matrix representation of C₂≀C₂² ►in GL₄(ℤ) generated by

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

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

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

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

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

C₂≀C₂² in GAP, Magma, Sage, TeX

C_2\wr C_2^2

% in TeX

G:=Group("C2wrC2^2");

// GroupNames label

G:=SmallGroup(64,138);

// by ID

G=gap.SmallGroup(64,138);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,121,362,255,730]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂≀C₂² in TeX

G = C2≀C22 order 64 = 26

Wreath product of C2 by C22

G = C₂≀C₂² order 64 = 2⁶

Wreath product of C₂ by C₂²