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G = D4⋊C4order 32 = 25

1st semidirect product of D4 and C4 acting via C4/C2=C2

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

Aliases: D41C4, C2.1D8, C4.11D4, C2.1SD16, C22.8D4, C4⋊C41C2, (C2×C8)⋊2C2, C4.1(C2×C4), (C2×D4).3C2, C2.6(C22⋊C4), (C2×C4).14C22, 2-Sylow(CSO-(4,3)), SmallGroup(32,9)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — D4⋊C4
C1C2C22C2×C4C2×D4 — D4⋊C4
C1C2C4 — D4⋊C4
C1C22C2×C4 — D4⋊C4
C1C2C2C2×C4 — D4⋊C4

Generators and relations for D4⋊C4
 G = < a,b,c | a4=b2=c4=1, bab=cac-1=a-1, cbc-1=ab >

4C2
4C2
2C22
2C22
4C22
4C22
4C4
2C8
2D4
2C23
2C2×C4

Character table of D4⋊C4

 class 12A2B2C2D2E4A4B4C4D8A8B8C8D
 size 11114422442222
ρ111111111111111    trivial
ρ21111-1-111-1-11111    linear of order 2
ρ31111-1-11111-1-1-1-1    linear of order 2
ρ411111111-1-1-1-1-1-1    linear of order 2
ρ51-1-111-11-1i-i-i-iii    linear of order 4
ρ61-1-11-111-1-ii-i-iii    linear of order 4
ρ71-1-11-111-1i-iii-i-i    linear of order 4
ρ81-1-111-11-1-iiii-i-i    linear of order 4
ρ92-2-2200-22000000    orthogonal lifted from D4
ρ10222200-2-2000000    orthogonal lifted from D4
ρ1122-2-20000002-22-2    orthogonal lifted from D8
ρ1222-2-2000000-22-22    orthogonal lifted from D8
ρ132-22-2000000--2-2-2--2    complex lifted from SD16
ρ142-22-2000000-2--2--2-2    complex lifted from SD16

Permutation representations of D4⋊C4
On 16 points - transitive group 16T26
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 9)(2 12)(3 11)(4 10)(5 14)(6 13)(7 16)(8 15)
(1 7 10 14)(2 6 11 13)(3 5 12 16)(4 8 9 15)

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

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

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

G:=TransitiveGroup(16,26);

Matrix representation of D4⋊C4 in GL3(𝔽17) generated by

100
001
0160
,
1600
001
010
,
1300
0125
055
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[16,0,0,0,0,1,0,1,0],[13,0,0,0,12,5,0,5,5] >;

D4⋊C4 in GAP, Magma, Sage, TeX

D_4\rtimes C_4
% in TeX

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

G:=SmallGroup(32,9);
// by ID

G=gap.SmallGroup(32,9);
# by ID

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

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

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