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G = C4⋊D4order 32 = 25

The semidirect product of C4 and D4 acting via D4/C22=C2

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

Aliases: C42D4, C221D4, C23.2C22, C22.11C23, C4⋊C42C2, (C2×D4)⋊2C2, C2.5(C2×D4), C22⋊C43C2, (C22×C4)⋊4C2, C2.4(C4○D4), (C2×C4).20C22, SmallGroup(32,28)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊D4
C1C2C22C23C22×C4 — C4⋊D4
C1C22 — C4⋊D4
C1C22 — C4⋊D4
C1C22 — C4⋊D4

Generators and relations for C4⋊D4
 G = < a,b,c | a4=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

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

Character table of C4⋊D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F
 size 11112244222244
ρ111111111111111    trivial
ρ21111-1-11-111-1-11-1    linear of order 2
ρ31111-1-1-1111-1-1-11    linear of order 2
ρ4111111-1-11111-1-1    linear of order 2
ρ51111111-1-1-1-1-1-11    linear of order 2
ρ61111-1-111-1-111-1-1    linear of order 2
ρ71111-1-1-1-1-1-11111    linear of order 2
ρ8111111-11-1-1-1-11-1    linear of order 2
ρ92-22-20000002-200    orthogonal lifted from D4
ρ102-22-2000000-2200    orthogonal lifted from D4
ρ1122-2-2-2200000000    orthogonal lifted from D4
ρ1222-2-22-200000000    orthogonal lifted from D4
ρ132-2-220000-2i2i0000    complex lifted from C4○D4
ρ142-2-2200002i-2i0000    complex lifted from C4○D4

Permutation representations of C4⋊D4
On 16 points - transitive group 16T34
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 15 9)(2 5 16 12)(3 8 13 11)(4 7 14 10)
(1 4)(2 3)(5 11)(6 10)(7 9)(8 12)(13 16)(14 15)

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

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

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

G:=TransitiveGroup(16,34);

On 16 points - transitive group 16T43
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 6 9 14)(2 5 10 13)(3 8 11 16)(4 7 12 15)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)

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

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

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

G:=TransitiveGroup(16,43);

Matrix representation of C4⋊D4 in GL4(𝔽5) generated by

0100
4000
0021
0003
,
1000
0400
0042
0041
,
1000
0400
0010
0014
G:=sub<GL(4,GF(5))| [0,4,0,0,1,0,0,0,0,0,2,0,0,0,1,3],[1,0,0,0,0,4,0,0,0,0,4,4,0,0,2,1],[1,0,0,0,0,4,0,0,0,0,1,1,0,0,0,4] >;

C4⋊D4 in GAP, Magma, Sage, TeX

C_4\rtimes D_4
% in TeX

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

G:=SmallGroup(32,28);
// by ID

G=gap.SmallGroup(32,28);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,101,46,302]);
// Polycyclic

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

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