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## G = C42.4D4order 128 = 27

### 4th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.4D4, 2- 1+4.C4, C2.11C2≀C4, (C2×Q8).2D4, C4.10D4.C4, C4⋊Q8.1C22, C4.6Q161C2, C42.3C44C2, D4.10D4.1C2, C22.4(C23⋊C4), (C2×Q8).2(C2×C4), (C2×C4).8(C22⋊C4), SmallGroup(128,137)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×Q8 — C42.4D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — D4.10D4 — C42.4D4
 Lower central C1 — C2 — C22 — C2×C4 — C2×Q8 — C42.4D4
 Upper central C1 — C2 — C22 — C2×C4 — C4⋊Q8 — C42.4D4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4⋊Q8 — C42.4D4

Generators and relations for C42.4D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=a-1b2, ab=ba, cac-1=a-1b, ad=da, cbc-1=a2b, dbd-1=a2b-1, dcd-1=a-1c3 >

Character table of C42.4D4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D 8E 8F 8G size 1 1 2 8 4 4 4 8 8 8 8 8 8 8 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ5 1 1 1 -1 -1 1 -1 -1 1 -1 -i i i -i i -i 1 linear of order 4 ρ6 1 1 1 1 -1 1 -1 1 1 -1 i -i -i i i -i -1 linear of order 4 ρ7 1 1 1 1 -1 1 -1 1 1 -1 -i i i -i -i i -1 linear of order 4 ρ8 1 1 1 -1 -1 1 -1 -1 1 -1 i -i -i i -i i 1 linear of order 4 ρ9 2 2 2 0 -2 2 -2 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 2 0 -2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 -4 -2 0 0 0 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ12 4 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ13 4 4 -4 2 0 0 0 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2≀C4 ρ14 4 -4 0 0 -2 0 2 0 0 0 √2 -√2 √2 -√2 0 0 0 symplectic faithful, Schur index 2 ρ15 4 -4 0 0 -2 0 2 0 0 0 -√2 √2 -√2 √2 0 0 0 symplectic faithful, Schur index 2 ρ16 4 -4 0 0 2 0 -2 0 0 0 -√-2 -√-2 √-2 √-2 0 0 0 complex faithful ρ17 4 -4 0 0 2 0 -2 0 0 0 √-2 √-2 -√-2 -√-2 0 0 0 complex faithful

Permutation representations of C42.4D4
On 16 points - transitive group 16T361
Generators in S16
(1 5)(2 13 6 9)(3 7)(4 11 8 15)(10 14)(12 16)
(1 16 5 12)(2 9 6 13)(3 14 7 10)(4 15 8 11)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 15 13 4 6 11 9 8)(3 14)(7 10)(12 16)

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

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

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

G:=TransitiveGroup(16,361);

Matrix representation of C42.4D4 in GL4(𝔽3) generated by

 2 0 0 0 0 0 2 0 0 1 0 0 0 0 0 2
,
 0 0 0 2 0 0 1 0 0 2 0 0 1 0 0 0
,
 0 2 1 0 0 0 0 1 1 0 0 0 0 2 2 0
,
 2 0 0 0 0 2 2 0 0 1 2 0 0 0 0 1
G:=sub<GL(4,GF(3))| [2,0,0,0,0,0,1,0,0,2,0,0,0,0,0,2],[0,0,0,1,0,0,2,0,0,1,0,0,2,0,0,0],[0,0,1,0,2,0,0,2,1,0,0,2,0,1,0,0],[2,0,0,0,0,2,1,0,0,2,2,0,0,0,0,1] >;

C42.4D4 in GAP, Magma, Sage, TeX

C_4^2._4D_4
% in TeX

G:=Group("C4^2.4D4");
// GroupNames label

G:=SmallGroup(128,137);
// by ID

G=gap.SmallGroup(128,137);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,723,352,346,745,248,1684,1411,718,375,172,4037,2028]);
// 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,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^-1*c^3>;
// generators/relations

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