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G = C42.2D4  order 128 = 27

2nd non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 5), monomial

Aliases: C42.2D4, 2+ 1+4.C4, C2.9C2≀C4, C4.D4.C4, (C2×Q8).1D4, C42.C4⋊4C2, D4.9D4.1C2, C22.2(C23⋊C4), C4.4D4.1C22, C42.C22⋊5C2, (C2×D4).2(C2×C4), (C2×C4).6(C22⋊C4), SmallGroup(128,135)

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

Derived series
C1 — C2 — C2×D4 — C42.2D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — D4.9D4 — C42.2D4
Lower central
C1 — C2 — C22 — C2×C4 — C2×D4 — C42.2D4
Upper central
C1 — C2 — C22 — C2×C4 — C4.4D4 — C42.2D4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — C42.2D4

Generators and relations for C42.2D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a-1b2, ab=ba, cac-1=a-1b-1, ad=da, cbc-1=a2b-1, dbd-1=a2b, dcd-1=a-1c3 >

C1
C2
2C2
8C2
8C2
C22
2C4
4C22
4C22
4C4
4C4
4C4
8C22
16C22
C2×C4
2Q8
2C23
2C2×C4
2D4
2C2×C4
4C8
4C23
4D4
4C8
4C2×C4
4D4
4Q8
4D4
4C8
4D4
4C2×C4
8C8
C2×Q8
C2×D4
C42
2M4(2)
2C4○D4
4C4○D4
4C2×D4
4C2×D4
4SD16
4Q16
4C2×C8
4C22⋊C4
4M4(2)
C4.4D4
C4.D4
2+ 1+4
2C8⋊C4
2C4≀C2
2C8.C22
2C4.10D4
C42.C4
C42.C22
D4.9D4
C42.2D4

Character table of C42.2D4

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G
 size 11288444888888161616
ρ111111111111111111    trivial
ρ2111-111111-1-1-1-1-111-1    linear of order 2
ρ3111-111111-11111-1-1-1    linear of order 2
ρ41111111111-1-1-1-1-1-11    linear of order 2
ρ5111-11-11-1-1-1i-i-ii-ii1    linear of order 4
ρ611111-11-1-11-iii-i-ii-1    linear of order 4
ρ711111-11-1-11i-i-iii-i-1    linear of order 4
ρ8111-11-11-1-1-1-iii-ii-i1    linear of order 4
ρ92220-2222-200000000    orthogonal lifted from D4
ρ102220-2-22-2200000000    orthogonal lifted from D4
ρ1144-4200000-20000000    orthogonal lifted from C2≀C4
ρ12444000-40000000000    orthogonal lifted from C23⋊C4
ρ1344-4-20000020000000    orthogonal lifted from C2≀C4
ρ144-40002i0-2i001+i-1+i1-i-1-i000    complex faithful
ρ154-4000-2i02i00-1+i1+i-1-i1-i000    complex faithful
ρ164-40002i0-2i00-1-i1-i-1+i1+i000    complex faithful
ρ174-4000-2i02i001-i-1-i1+i-1+i000    complex faithful

Permutation representations of C42.2D4
►On 16 points - transitive group 16T398
Generators in S16
(1 5)(2 12 6 16)(4 14 8 10)(11 15)

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

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

(1 5)(2 14 12 8 6 10 16 4)(3 9 7 13)(11 15)
 
G:=sub<Sym(16)| (1,5)(2,12,6,16)(4,14,8,10)(11,15), (1,11,5,15)(2,16,6,12)(3,9,7,13)(4,14,8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(2,14,12,8,6,10,16,4)(3,9,7,13)(11,15)>;
 
G:=Group( (1,5)(2,12,6,16)(4,14,8,10)(11,15), (1,11,5,15)(2,16,6,12)(3,9,7,13)(4,14,8,10), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(2,14,12,8,6,10,16,4)(3,9,7,13)(11,15) );
 
G=PermutationGroup([[(1,5),(2,12,6,16),(4,14,8,10),(11,15)], [(1,11,5,15),(2,16,6,12),(3,9,7,13),(4,14,8,10)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(2,14,12,8,6,10,16,4),(3,9,7,13),(11,15)]])
 
G:=TransitiveGroup(16,398);
 

Matrix representation of C42.2D4 ►in GL4(𝔽5) generated by

1000
0200
0020
0004
,
3000
0200
0030
0002
,
0010
3000
0001
0300
,
3000
0010
0200
0004
G:=sub<GL(4,GF(5))| [1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,4],[3,0,0,0,0,2,0,0,0,0,3,0,0,0,0,2],[0,3,0,0,0,0,0,3,1,0,0,0,0,0,1,0],[3,0,0,0,0,0,2,0,0,1,0,0,0,0,0,4] >;
 

C42.2D4 in GAP, Magma, Sage, TeX 

C_4^2._2D_4
 
% in TeX
 
G:=Group("C4^2.2D4");
 
// GroupNames label
 
G:=SmallGroup(128,135);
 
// by ID
 
G=gap.SmallGroup(128,135);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1242,745,248,1684,1411,718,375,172,4037,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,a*d=d*a,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=a^-1*c^3>;
 
// generators/relations
 

Export

Subgroup lattice of C42.2D4 in TeX
Character table of C42.2D4 in TeX

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