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G = C2≀C4order 64 = 26

Wreath product of C2 by C4

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

Aliases: C2C4, AΣL1(𝔽16), C241C4, C23.1D4, (C2×C4).1D4, C23⋊C41C2, C22⋊C41C4, C4.D45C2, C23.1(C2×C4), C22≀C2.1C2, C2.6(C23⋊C4), (C2×D4).1C22, C22.9(C22⋊C4), 2-Sylow(AGammaL(1,16)), SmallGroup(64,32)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2≀C4
C1C2C22C23C2×D4C22≀C2 — C2≀C4
C1C2C22C23 — C2≀C4
C1C2C22C2×D4 — C2≀C4
C1C2C22C2×D4 — C2≀C4

Generators and relations for C2≀C4
 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 >

2C2
4C2
4C2
4C2
4C2
2C22
2C22
2C22
2C4
2C22
4C22
4C4
4C22
4C22
4C22
4C22
4C22
4C22
4C22
8C4
2C2×C4
2C23
4D4
4D4
4D4
4C23
4C23
4C8
4C23
4C2×C4
2C22⋊C4
2C22⋊C4
2C2×D4
2M4(2)

Character table of C2≀C4

 class 12A2B2C2D2E2F4A4B4C4D8A8B
 size 1124444488888
ρ11111111111111    trivial
ρ2111-1-11111-11-1-1    linear of order 2
ρ311111111-11-1-1-1    linear of order 2
ρ4111-1-1111-1-1-111    linear of order 2
ρ511111-11-1i-1-ii-i    linear of order 4
ρ6111-1-1-11-1i1-i-ii    linear of order 4
ρ711111-11-1-i-1i-ii    linear of order 4
ρ8111-1-1-11-1-i1ii-i    linear of order 4
ρ9222002-2-200000    orthogonal lifted from D4
ρ1022200-2-2200000    orthogonal lifted from D4
ρ114-402-200000000    orthogonal faithful
ρ124-40-2200000000    orthogonal faithful
ρ1344-40000000000    orthogonal lifted from C23⋊C4

Permutation representations of C2≀C4
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 12)(7 11)(8 14)
(2 13)(3 8)(4 9)(5 15)(7 11)(12 14)
(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)| (2,13)(3,12)(7,11)(8,14), (2,13)(3,8)(4,9)(5,15)(7,11)(12,14), (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( (2,13)(3,12)(7,11)(8,14), (2,13)(3,8)(4,9)(5,15)(7,11)(12,14), (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([(2,13),(3,12),(7,11),(8,14)], [(2,13),(3,8),(4,9),(5,15),(7,11),(12,14)], [(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(16,157);

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

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

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

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

C2≀C4 is a maximal subgroup of
D4≀C2  C424D4  C425D4  C426D4  C24⋊Dic3  C5⋊C2≀C4  C242F5  C24⋊F5
 (C2×D4).D2p: C4○C2≀C4  C24.36D4  C2≀C4⋊C2  C3⋊C2≀C4  C23.3D12  C245Dic3  C53C2≀C4  C23.3D20 ...
C2≀C4 is a maximal quotient of
C24⋊C8  C23.2M4(2)  C23.Q16  C2.7C2≀C4  C42.D4  C42.2D4  C42.3D4  C42.4D4  C5⋊C2≀C4
 C23.D4p: C24.D4  C23.3D12  C23.3D20  C23.3D28 ...
 (C2×C4).D4p: C2.C2≀C4  C3⋊C2≀C4  C53C2≀C4  C7⋊C2≀C4 ...
 (C23×C2p)⋊C4: C24.5D4  C245Dic3  C242Dic5  C242F5  C24⋊Dic7 ...

Polynomial with Galois group C2≀C4 over ℚ
actionf(x)Disc(f)
8T27x8-4x7-x6+17x5-6x4-21x3+6x2+8x+134·56·1021
8T28x8-9x6+24x4-20x2+528·55·1012

Matrix representation of C2≀C4 in GL4(ℤ) generated by

-1000
0-100
000-1
00-10
,
0100
1000
0001
0010
,
1000
0100
00-10
000-1
,
-1000
0-100
00-10
000-1
,
0010
0001
1000
0-100
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] >;

C2≀C4 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 C2≀C4 in TeX
Character table of C2≀C4 in TeX

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