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## G = C2×C4.D4order 64 = 26

### Direct product of C2 and C4.D4

direct product, p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C4.D4
G = < a,b,c,d | a2=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 185 in 93 conjugacy classes, 41 normal (11 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C24, C4.D4, C2×M4(2), C22×D4, C2×C4.D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4.D4, C2×C22⋊C4, C2×C4.D4

Character table of C2×C4.D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 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 -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 -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 1 -1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ7 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 1 linear of order 2 ρ9 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 i -i i i -i -i i -i linear of order 4 ρ10 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 -i i i -i -i i i -i linear of order 4 ρ11 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -i i -i -i i i -i i linear of order 4 ρ12 1 -1 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 i -i -i i i -i -i i linear of order 4 ρ13 1 -1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 -i -i -i i -i i i i linear of order 4 ρ14 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 i i -i -i -i -i i i linear of order 4 ρ15 1 -1 1 -1 -1 1 -1 -1 1 1 1 1 -1 -1 i i i -i i -i -i -i linear of order 4 ρ16 1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -i -i i i i i -i -i linear of order 4 ρ17 2 -2 2 -2 2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 -2 -2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4

Permutation representations of C2×C4.D4
On 16 points - transitive group 16T72
Generators in S16
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 13 5 9)(2 10 6 14)(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 10 13 6 5 14 9 2)(3 16 15 4 7 12 11 8)

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

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

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

G:=TransitiveGroup(16,72);

On 16 points - transitive group 16T99
Generators in S16
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 3 5 7)(2 8 6 4)(9 11 13 15)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10 3 16 5 14 7 12)(2 11 8 13 6 15 4 9)

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

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

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

G:=TransitiveGroup(16,99);

Matrix representation of C2×C4.D4 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0
,
 -1 -1 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 -1 0 0
,
 1 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 1 0 0 0

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0],[-1,2,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,-1,0,0] >;

C2×C4.D4 in GAP, Magma, Sage, TeX

C_2\times C_4.D_4
% in TeX

G:=Group("C2xC4.D4");
// GroupNames label

G:=SmallGroup(64,92);
// by ID

G=gap.SmallGroup(64,92);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,730,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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