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G = C42.2D4order 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.C44C2, D4.9D4.1C2, C22.2(C23⋊C4), C4.4D4.1C22, C42.C225C2, (C2×D4).2(C2×C4), (C2×C4).6(C22⋊C4), SmallGroup(128,135)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×D4 — C42.2D4
C1C2C22C2×C4C2×Q8C4.4D4D4.9D4 — C42.2D4
C1C2C22C2×C4C2×D4 — C42.2D4
C1C2C22C2×C4C4.4D4 — C42.2D4
C1C2C2C22C2×C4C4.4D4 — 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 >

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

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

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

G:=Group( (1,5)(2,10,6,14)(4,12,8,16)(9,13), (1,9,5,13)(2,14,6,10)(3,15,7,11)(4,12,8,16), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(2,12,10,8,6,16,14,4)(3,15,7,11)(9,13) );

G=PermutationGroup([(1,5),(2,10,6,14),(4,12,8,16),(9,13)], [(1,9,5,13),(2,14,6,10),(3,15,7,11),(4,12,8,16)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(2,12,10,8,6,16,14,4),(3,15,7,11),(9,13)])

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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