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G = C4○D4order 16 = 24

Central product of C4 and D4

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C4D4, C4Q8, D42C2, Q82C2, C4.5C22, C2.3C23, C22.C22, (C2×C4)⋊3C2, Pauli group, SmallGroup(16,13)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4○D4
C1C2C4C2×C4 — C4○D4
C1C2 — C4○D4
C1C4 — C4○D4
C1C2 — C4○D4

Generators and relations for C4○D4
 G = < a,b,c | a4=c2=1, b2=a2, ab=ba, ac=ca, cbc=a2b >

2C2
2C2
2C2

Character table of C4○D4

 class 12A2B2C2D4A4B4C4D4E
 size 1122211222
ρ11111111111    trivial
ρ2111-11-1-1-1-11    linear of order 2
ρ311-11-111-1-11    linear of order 2
ρ41111-1-1-11-1-1    linear of order 2
ρ511-1-11111-1-1    linear of order 2
ρ611-1-1-1-1-1111    linear of order 2
ρ7111-1-111-11-1    linear of order 2
ρ811-111-1-1-11-1    linear of order 2
ρ92-2000-2i2i000    complex faithful
ρ102-20002i-2i000    complex faithful

Permutation representations of C4○D4
On 8 points - transitive group 8T11
Generators in S8
(1 2 3 4)(5 6 7 8)
(1 4 3 2)(5 6 7 8)
(1 6)(2 7)(3 8)(4 5)

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

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

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

G:=TransitiveGroup(8,11);

Regular action on 16 points - transitive group 16T11
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 13 3 15)(2 14 4 16)(5 11 7 9)(6 12 8 10)
(1 11)(2 12)(3 9)(4 10)(5 13)(6 14)(7 15)(8 16)

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

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

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

G:=TransitiveGroup(16,11);

Polynomial with Galois group C4○D4 over ℚ
actionf(x)Disc(f)
8T11x8-2x7-7x6+16x5+4x4-18x3+2x2+4x-1212·54·132

Matrix representation of C4○D4 in GL2(𝔽5) generated by

20
02
,
13
14
,
13
04
G:=sub<GL(2,GF(5))| [2,0,0,2],[1,1,3,4],[1,0,3,4] >;

C4○D4 in GAP, Magma, Sage, TeX

C_4\circ D_4
% in TeX

G:=Group("C4oD4");
// GroupNames label

G:=SmallGroup(16,13);
// by ID

G=gap.SmallGroup(16,13);
# by ID

G:=PCGroup([4,-2,2,2,-2,81,34]);
// Polycyclic

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

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