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G = C4.4D4order 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, C425C2, C23.3C22, C22.14C23, (C2×Q8)⋊2C2, C2.8(C2×D4), C22⋊C45C2, (C2×D4).5C2, C2.7(C4○D4), (C2×C4).22C22, SmallGroup(32,31)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4.4D4
C1C2C22C2×C4C42 — C4.4D4
C1C22 — C4.4D4
C1C22 — C4.4D4
C1C22 — 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 >

4C2
4C2
2C22
2C22
2C22
2C4
2C4
2C22
2C4
2C22
2C22
2C4
2Q8
2Q8
2D4
2D4

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);

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

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