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## G = C2×D4⋊4D4order 128 = 27

### Direct product of C2 and D4⋊4D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4⋊4D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×2+ 1+4 — C2×D4⋊4D4
 Lower central C1 — C2 — C2×C4 — C2×D4⋊4D4
 Upper central C1 — C22 — C22×C4 — C2×D4⋊4D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4⋊4D4

Generators and relations for C2×D44D4
G = < a,b,c,d,e | a2=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 980 in 422 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×4], C4 [×8], C22 [×3], C22 [×42], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×24], D4 [×4], D4 [×56], Q8 [×4], Q8 [×2], C23, C23 [×4], C23 [×30], C42 [×2], C42, C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×5], C2×D4 [×8], C2×D4 [×63], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×20], C24 [×2], C24 [×3], C4.D4 [×4], C4≀C2 [×8], C2×C42, C41D4 [×4], C41D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4 [×2], C22×D4 [×6], C2×C4○D4 [×2], C2×C4○D4 [×2], 2+ 1+4 [×4], 2+ 1+4 [×6], C2×C4.D4, C2×C4≀C2 [×2], D44D4 [×8], C2×C41D4, C2×C8⋊C22 [×2], C2×2+ 1+4, C2×D44D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], D44D4 [×2], C2×C22≀C2, C2×D44D4

Permutation representations of C2×D44D4
On 16 points - transitive group 16T265
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 10)(6 9)(7 12)(8 11)
(9 12 11 10)(13 16 15 14)
(1 7)(2 6)(3 5)(4 8)(9 14)(10 13)(11 16)(12 15)

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

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

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

G:=TransitiveGroup(16,265);

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 2N 2O 4A 4B 4C 4D 4E ··· 4L 8A 8B 8C 8D order 1 2 2 2 2 2 2 ··· 2 2 2 4 4 4 4 4 ··· 4 8 8 8 8 size 1 1 1 1 2 2 4 ··· 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4⋊4D4 kernel C2×D4⋊4D4 C2×C4.D4 C2×C4≀C2 D4⋊4D4 C2×C4⋊1D4 C2×C8⋊C22 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C24 C2 # reps 1 1 2 8 1 2 1 4 2 4 2 4

Matrix representation of C2×D44D4 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1 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 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 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 -1 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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0] >;

C2×D44D4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_4D_4
% in TeX

G:=Group("C2xD4:4D4");
// GroupNames label

G:=SmallGroup(128,1746);
// by ID

G=gap.SmallGroup(128,1746);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2804,1411,718,172,2028]);
// Polycyclic

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

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