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

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

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

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

G:=TransitiveGroup(16,41);

C4.D4 is a maximal subgroup of
C2≀C4  C23.D4  M4(2).8C22  D44D4  D4.9D4  C23.F5  C4.S3≀C2  (C2×C62).C4  Dic13.4D4
 C4p.D4: D4.3D4  D4.4D4  C12.46D4  C12.D4  C20.46D4  C20.D4  C28.46D4  C28.D4 ...
C4.D4 is a maximal quotient of
C23.F5  C4.S3≀C2  (C2×C62).C4  Dic13.4D4
 C4.D4p: C4.D8  C12.46D4  C20.46D4  C28.46D4  C44.46D4  C52.46D4 ...
 (C2×C4).D2p: C23⋊C8  C42.C22  C4.6Q16  C22.C42  C12.D4  C20.D4  C28.D4  C44.D4 ...

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

Export

Subgroup lattice of C4.D4 in TeX
Character table of C4.D4 in TeX

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