Copied to
clipboard

G = C4≀C2order 32 = 25

Wreath product of C4 by C2

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

Aliases: C4C2, D42C4, Q82C4, C423C2, C4.17D4, C22.3D4, M4(2)⋊4C2, C4.3(C2×C4), C4○D4.1C2, C2.8(C22⋊C4), (C2×C4).16C22, 2-Sylow(GL(2,5)), SmallGroup(32,11)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C4≀C2
C1C2C4C2×C4C4○D4 — C4≀C2
C1C2C4 — C4≀C2
C1C4C2×C4 — C4≀C2
C1C2C2C2×C4 — C4≀C2

Generators and relations for C4≀C2
 G = < a,b,c | a4=b2=c4=1, bab=a-1, ac=ca, cbc-1=a-1b >

2C2
4C2
2C4
2C22
2C4
2C4
2C2×C4
2C2×C4
2C8
2D4

Character table of C4≀C2

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B
 size 11241122222444
ρ111111111111111    trivial
ρ2111111-11-1-1-11-1-1    linear of order 2
ρ3111-11111111-1-1-1    linear of order 2
ρ4111-111-11-1-1-1-111    linear of order 2
ρ511-1-1-1-1-i1ii-i1-ii    linear of order 4
ρ611-1-1-1-1i1-i-ii1i-i    linear of order 4
ρ711-11-1-1-i1ii-i-1i-i    linear of order 4
ρ811-11-1-1i1-i-ii-1-ii    linear of order 4
ρ922-20220-2000000    orthogonal lifted from D4
ρ102220-2-20-2000000    orthogonal lifted from D4
ρ112-200-2i2i-1-i01-i-1+i1+i000    complex faithful
ρ122-2002i-2i-1+i01+i-1-i1-i000    complex faithful
ρ132-200-2i2i1+i0-1+i1-i-1-i000    complex faithful
ρ142-2002i-2i1-i0-1-i1+i-1+i000    complex faithful

Permutation representations of C4≀C2
On 8 points - transitive group 8T17
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 8)(2 7)(3 6)(4 5)
(1 3)(2 4)(5 6 7 8)

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

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

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

G:=TransitiveGroup(8,17);

On 16 points - transitive group 16T28
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 4)(2 3)(5 7)(9 10)(11 12)(14 16)
(1 8 11 16)(2 5 12 13)(3 6 9 14)(4 7 10 15)

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

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

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

G:=TransitiveGroup(16,28);

On 16 points - transitive group 16T42
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6)(2 5)(3 8)(4 7)(9 13)(10 16)(11 15)(12 14)
(1 15 3 13)(2 16 4 14)(5 9)(6 10)(7 11)(8 12)

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

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

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

G:=TransitiveGroup(16,42);

Polynomial with Galois group C4≀C2 over ℚ
actionf(x)Disc(f)
8T17x8-4x7+14x5-8x4-12x3+7x2+2x-1212·413

Matrix representation of C4≀C2 in GL2(𝔽5) generated by

30
02
,
02
30
,
20
01
G:=sub<GL(2,GF(5))| [3,0,0,2],[0,3,2,0],[2,0,0,1] >;

C4≀C2 in GAP, Magma, Sage, TeX

C_4\wr C_2
% in TeX

G:=Group("C4wrC2");
// GroupNames label

G:=SmallGroup(32,11);
// by ID

G=gap.SmallGroup(32,11);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,157,72,58]);
// Polycyclic

G:=Group<a,b,c|a^4=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^-1*b>;
// generators/relations

׿
×
𝔽