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G = C4⋊1D4  order 32 = 25

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

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

Aliases: C4⋊1D4, C42⋊6C2, C23.5C22, C22.17C23, (C2×D4)⋊3C2, C2.9(C2×D4), (C2×C4).23C22, SmallGroup(32,34)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C4⋊1D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4⋊1D4
Lower central
C1 — C22 — C4⋊1D4
Upper central
C1 — C22 — C4⋊1D4
Jennings
C1 — C22 — C4⋊1D4

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

Subgroups: 90 in 54 conjugacy classes, 26 normal (4 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, D4, C23, C42, C2×D4, C4⋊1D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4⋊1D4

Character table of C4⋊1D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F
 size 11114444222222
ρ111111111111111    trivial
ρ21111-111-1-11-1-1-11    linear of order 2
ρ311111-11-1-1-111-1-1    linear of order 2
ρ41111-1-1111-1-1-11-1    linear of order 2
ρ51111-11-11-1-111-1-1    linear of order 2
ρ6111111-1-11-1-1-11-1    linear of order 2
ρ71111-1-1-1-1111111    linear of order 2
ρ811111-1-11-11-1-1-11    linear of order 2
ρ92-22-20000002-200    orthogonal lifted from D4
ρ102-2-2200002000-20    orthogonal lifted from D4
ρ112-22-2000000-2200    orthogonal lifted from D4
ρ1222-2-200000-20002    orthogonal lifted from D4
ρ1322-2-2000002000-2    orthogonal lifted from D4
ρ142-2-220000-200020    orthogonal lifted from D4

Permutation representations of C4⋊1D4
►On 16 points - transitive group 16T51
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

(1 5)(2 8)(3 7)(4 6)(9 15)(10 14)(11 13)(12 16)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,14,11)(2,7,15,12)(3,8,16,9)(4,5,13,10), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,6,14,11)(2,7,15,12)(3,8,16,9)(4,5,13,10), (1,5)(2,8)(3,7)(4,6)(9,15)(10,14)(11,13)(12,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,6,14,11),(2,7,15,12),(3,8,16,9),(4,5,13,10)], [(1,5),(2,8),(3,7),(4,6),(9,15),(10,14),(11,13),(12,16)]])
 
G:=TransitiveGroup(16,51);
 

C4⋊1D4 is a maximal subgroup of
 C4.D8  C42⋊C4  D4⋊4D4  C4⋊D8  C4⋊SD16  C4.4D8  C42.29C22  C22.26C24  C22.29C24  C22.34C24  D42  Q8⋊6D4  C22.53C24  C22.54C24  C23.A4  C4⋊S3≀C2
 C4p⋊D4: C8⋊5D4  C8⋊4D4  C8⋊3D4  C4⋊D12  C12⋊3D4  C20⋊4D4  C20⋊D4  C28⋊4D4 ...
C4⋊1D4 is a maximal quotient of
 C42⋊9C4  C24.3C22  C23⋊2D4  C23.4Q8  C4⋊Q16  C8.12D4  C8.2D4  C4⋊S3≀C2
 C4p⋊D4: C8⋊5D4  C8⋊4D4  C8⋊3D4  C4⋊D12  C12⋊3D4  C20⋊4D4  C20⋊D4  C28⋊4D4 ...

Matrix representation of C4⋊1D4 ►in GL4(ℤ) generated by

0100
-1000
0010
0001
,
1000
0100
0001
00-10
,
1000
0-100
00-10
0001
G:=sub<GL(4,Integers())| [0,-1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1] >;
 

C4⋊1D4 in GAP, Magma, Sage, TeX 

C_4\rtimes_1D_4
 
% in TeX
 
G:=Group("C4:1D4");
 
// GroupNames label
 
G:=SmallGroup(32,34);
 
// by ID
 
G=gap.SmallGroup(32,34);
 
# by ID
 
G:=PCGroup([5,-2,2,2,-2,2,101,46,302,72]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
 
// generators/relations
 

Export

Character table of C4⋊1D4 in TeX

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