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G = C4.4D4  order 32 = 25

4th non-split extension by C4 of D4 acting via D4/C4=C2

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

Aliases: C4.4D4, C42⋊5C2, C23.3C22, C22.14C23, (C2×Q8)⋊2C2, C2.8(C2×D4), C22⋊C4⋊5C2, (C2×D4).5C2, C2.7(C4○D4), (C2×C4).22C22, SmallGroup(32,31)

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

Derived series
C1 — C22 — C4.4D4
Chief series
C1 — C2 — C22 — C2×C4 — C42 — C4.4D4
Lower central
C1 — C22 — C4.4D4
Upper central
C1 — C22 — C4.4D4
Jennings
C1 — C22 — C4.4D4

Generators and relations for C4.4D4
 G = < a,b,c | a4=b4=1, c2=a2, ab=ba, cac-1=a-1, cbc-1=a2b-1 >

C1
C2
C2
C2
4C2
4C2
C22
C4
C4
2C22
2C22
2C22
2C4
2C4
2C22
2C4
2C22
2C22
2C4
C2×C4
C23
C2×C4
C2×C4
C2×C4
C2×C4
C23
2Q8
2Q8
2D4
2D4
C2×Q8
C2×D4
C22⋊C4
C22⋊C4
C22⋊C4
C42
C22⋊C4
C4.4D4

Character table of C4.4D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H
 size 11114422222244
ρ111111111111111    trivial
ρ21111-11-1-1-111-11-1    linear of order 2
ρ31111-1-11-11-1-1-111    linear of order 2
ρ411111-1-11-1-1-111-1    linear of order 2
ρ511111-1-1-1-111-1-11    linear of order 2
ρ61111-1-1111111-1-1    linear of order 2
ρ71111-11-11-1-1-11-11    linear of order 2
ρ81111111-11-1-1-1-1-1    linear of order 2
ρ922-2-200-20200000    orthogonal lifted from D4
ρ1022-2-20020-200000    orthogonal lifted from D4
ρ112-22-200000-2i2i000    complex lifted from C4○D4
ρ122-22-2000002i-2i000    complex lifted from C4○D4
ρ132-2-220002i000-2i00    complex lifted from C4○D4
ρ142-2-22000-2i0002i00    complex lifted from C4○D4

Permutation representations of C4.4D4
►On 16 points - transitive group 16T30
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

(1 6 3 8)(2 5 4 7)(9 15 11 13)(10 14 12 16)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10,15,5)(2,11,16,6)(3,12,13,7)(4,9,14,8), (1,6,3,8)(2,5,4,7)(9,15,11,13)(10,14,12,16)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,10,15,5)(2,11,16,6)(3,12,13,7)(4,9,14,8), (1,6,3,8)(2,5,4,7)(9,15,11,13)(10,14,12,16) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,10,15,5),(2,11,16,6),(3,12,13,7),(4,9,14,8)], [(1,6,3,8),(2,5,4,7),(9,15,11,13),(10,14,12,16)]])
 
G:=TransitiveGroup(16,30);
 

C4.4D4 is a maximal subgroup of
 C42.C22  C42.C4  D4.8D4  D4.2D4  Q8.D4  C42.78C22  C42.28C22  C8⋊3D4  C23.36C23  C22.26C24  C22.29C24  C23.38C23  C22.32C24  C22.36C24  D4⋊5D4  Q8⋊5D4  C22.45C24  C22.49C24  C22.50C24  C22.53C24  C24⋊C22  C22.56C24  C22.57C24  C4.4S3≀C2
 C4p.D4: C8.12D4  C8.2D4  C42⋊7S3  C23.12D6  C12.23D4  C4.D20  C20.17D4  C20.23D4 ...
 C23.D2p: C42⋊3C4  D4.9D4  C23.11D6  Dic5.5D4  Dic7.D4  Dic11.D4  C23.6D26 ...
C4.4D4 is a maximal quotient of
 C24.3C22  C4.4S3≀C2
 C4.D4p: C4.4D8  C42⋊7S3  C4.D20  C4.D28  C4.D44  C4.D52 ...
 C23.D2p: C23.10D4  C23.11D4  C23.11D6  C23.12D6  Dic5.5D4  C20.17D4  Dic7.D4  C28.17D4 ...
 (C2×C4).D2p: C42⋊8C4  C24.C22  C23.67C23  C23⋊Q8  C23.83C23  C4.SD16  C42.78C22  C42.28C22 ...

Matrix representation of C4.4D4 ►in GL4(𝔽5) generated by

3400
0200
0021
0003
,
1300
0400
0030
0003
,
1300
1400
0030
0032
G:=sub<GL(4,GF(5))| [3,0,0,0,4,2,0,0,0,0,2,0,0,0,1,3],[1,0,0,0,3,4,0,0,0,0,3,0,0,0,0,3],[1,1,0,0,3,4,0,0,0,0,3,3,0,0,0,2] >;
 

C4.4D4 in GAP, Magma, Sage, TeX 

C_4._4D_4
 
% in TeX
 
G:=Group("C4.4D4");
 
// GroupNames label
 
G:=SmallGroup(32,31);
 
// by ID
 
G=gap.SmallGroup(32,31);
 
# by ID
 
G:=PCGroup([5,-2,2,2,-2,2,101,86,302,42]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^4=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^2*b^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C4.4D4 in TeX
Character table of C4.4D4 in TeX

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