Copied to
clipboard

G = C4.D4order 32 = 25

1st non-split extension by C4 of D4 acting via D4/C22=C2

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

Aliases: C4.9D4, C23.C4, M4(2)⋊3C2, (C2×D4).2C2, (C2×C4).1C22, C22.3(C2×C4), C2.4(C22⋊C4), SmallGroup(32,7)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4.D4
C1C2C4C2×C4C2×D4 — C4.D4
C1C2C22 — C4.D4
C1C2C2×C4 — C4.D4
C1C2C2C2×C4 — C4.D4

Generators and relations for C4.D4
 G = < a,b,c | a4=1, b4=a2, c2=a, bab-1=a-1, ac=ca, cbc-1=a-1b3 >

2C2
4C2
4C2
2C22
2C22
4C22
4C22
2C8
2C8
2D4
2D4

Character table of C4.D4

 class 12A2B2C2D4A4B8A8B8C8D
 size 11244224444
ρ111111111111    trivial
ρ2111-1-111-11-11    linear of order 2
ρ3111-1-1111-11-1    linear of order 2
ρ41111111-1-1-1-1    linear of order 2
ρ5111-11-1-1-i-iii    linear of order 4
ρ61111-1-1-1i-i-ii    linear of order 4
ρ71111-1-1-1-iii-i    linear of order 4
ρ8111-11-1-1ii-i-i    linear of order 4
ρ922-200-220000    orthogonal lifted from D4
ρ1022-2002-20000    orthogonal lifted from D4
ρ114-4000000000    orthogonal faithful

Permutation representations of C4.D4
On 8 points - transitive group 8T16
Generators in S8
(1 3 5 7)(2 8 6 4)
(1 2 3 4 5 6 7 8)
(1 4 3 2 5 8 7 6)

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

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

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

G:=TransitiveGroup(8,16);

On 16 points - transitive group 16T36
Generators in S16
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10 3 16 5 14 7 12)(2 11 8 13 6 15 4 9)

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

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

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

G:=TransitiveGroup(16,36);

On 16 points - transitive group 16T41
Generators in S16
(1 12 5 16)(2 9 6 13)(3 14 7 10)(4 11 8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 15 12 4 5 11 16 8)(2 7 9 10 6 3 13 14)

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

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

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

G:=TransitiveGroup(16,41);

Polynomial with Galois group C4.D4 over ℚ
actionf(x)Disc(f)
8T16x8-2x7-12x6+21x5+35x4-39x3-37x2+3x+157·118·192

Matrix representation of C4.D4 in GL4(ℤ) generated by

0100
-1000
0001
00-10
,
0010
000-1
0100
1000
,
0010
0001
0100
-1000
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,0,-1,0,0,1,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,-1,0,0],[0,0,0,-1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C4.D4 in GAP, Magma, Sage, TeX

C_4.D_4
% in TeX

G:=Group("C4.D4");
// GroupNames label

G:=SmallGroup(32,7);
// by ID

G=gap.SmallGroup(32,7);
# by ID

G:=PCGroup([5,-2,2,-2,2,-2,40,61,302,248,58]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^4=a^2,c^2=a,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^3>;
// generators/relations

׿
×
𝔽