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G = D4⋊C4  order 32 = 25

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

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

Aliases: D4⋊1C4, C2.1D8, C4.11D4, C2.1SD16, C22.8D4, C4⋊C4⋊1C2, (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

Derived series
C1 — C4 — D4⋊C4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — D4⋊C4
Lower central
C1 — C2 — C4 — D4⋊C4
Upper central
C1 — C22 — C2×C4 — D4⋊C4
Jennings
C1 — C2 — C2 — C2×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 >

C1
C2
C2
C2
4C2
4C2
C4
C4
C22
2C22
2C22
4C22
4C22
4C4
C2×C4
D4
D4
2C8
2D4
2C23
2C2×C4
C4⋊C4
C2×C8
C2×D4
D4⋊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
ρ1122-2-2000000√2-√2√2-√2    orthogonal lifted from D8
ρ1222-2-2000000-√2√2-√2√2    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);
 

D4⋊C4 is a maximal subgroup of
 C23.24D4  C23.36D4  C23.37D4  C4×D8  C4×SD16  SD16⋊C4  C22⋊D8  D4⋊D4  C22⋊SD16  D4.7D4  C4⋊D8  C4⋊SD16  D4.2D4  Q8.D4  C8⋊8D4  C8⋊7D4  C8⋊D4  C8⋊2D4  D4⋊2Q8  D4.Q8  C22.D8  C23.46D4  C23.19D4  C42.78C22  C42.28C22  C42.29C22  C3⋊S3.2D8  C62.3D4  C3⋊S3.5D8  C2.AΓL1(𝔽9)
 D4p⋊C4: D8⋊C4  C6.D8  C2.D24  D20⋊6C4  D20⋊5C4  D20⋊C4  C14.D8  C2.D56 ...
 C2p.D8: D4⋊Q8  C4.4D8  D4⋊Dic3  D4⋊Dic5  D4⋊Dic7  D4⋊Dic11  D4⋊Dic13 ...
D4⋊C4 is a maximal quotient of
 D4⋊C8  C22.SD16  C4.D8  C4.10D8  C22.4Q16  C3⋊S3.2D8  C62.3D4  C3⋊S3.5D8  C2.AΓL1(𝔽9)
 D4p⋊C4: C2.D16  D8⋊2C4  C6.D8  C2.D24  D20⋊6C4  D20⋊5C4  D20⋊C4  C14.D8 ...
 C4p.D4: C2.Q32  D8.C4  M5(2)⋊C2  C8.17D4  D4⋊Dic3  D4⋊Dic5  D4⋊Dic7  D4⋊Dic11 ...

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
 

Export

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

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