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## G = (C2×C8).D4order 128 = 27

### 5th non-split extension by C2×C8 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×M4(2) — C2×C4.D4 — (C2×C8).D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).D4
 Upper central C1 — C2 — C22×C4 — (C2×C8).D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).D4

Generators and relations for (C2×C8).D4
G = < a,b,c,d | a2=b8=d2=1, c4=b4, cbc-1=ab=ba, cac-1=ab4, ad=da, dbd=ab-1, dcd=ab6c3 >

Subgroups: 264 in 102 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C24, C4.D4, D4⋊C4, C8.C4, C42⋊C2, C2×M4(2), C22×D4, C4.9C42, M4(2)⋊4C4, C2×C4.D4, C23.37D4, M4(2).C4, (C2×C8).D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C4⋊D4, C22.D4, C4.4D4, C422C2, C23.11D4, (C2×C8).D4

Character table of (C2×C8).D4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 8 8 2 2 2 2 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 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 -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 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 -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 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 -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 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 -1 linear of order 2 ρ9 2 2 2 -2 -2 -2 2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 2 0 0 -2 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 -2 0 0 2 0 orthogonal lifted from D4 ρ12 2 2 2 -2 -2 2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 -2 2 -2 0 0 2 2 -2 -2 0 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ14 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 -2i 0 0 0 0 0 0 2i complex lifted from C4○D4 ρ15 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 -2i 0 0 2i 0 0 complex lifted from C4○D4 ρ16 2 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 2i 0 0 -2i 0 0 complex lifted from C4○D4 ρ17 2 2 2 -2 -2 0 0 -2 2 -2 2 0 0 0 0 2i 0 0 0 0 0 0 -2i complex lifted from C4○D4 ρ18 2 2 -2 2 -2 0 0 2 2 -2 -2 0 2i 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 -2 2 0 0 2 -2 -2 2 0 0 0 0 0 2i 0 0 -2i 0 0 0 complex lifted from C4○D4 ρ20 2 2 -2 -2 2 0 0 -2 2 2 -2 2i 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 2 -2 -2 2 0 0 2 -2 -2 2 0 0 0 0 0 -2i 0 0 2i 0 0 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 2 0 0 -2 2 2 -2 -2i 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Permutation representations of (C2×C8).D4
On 16 points - transitive group 16T412
Generators in S16
(9 13)(10 14)(11 15)(12 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13 7 11 5 9 3 15)(2 14 4 16 6 10 8 12)
(2 8)(3 7)(4 6)(9 13)(10 16)(12 14)

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

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

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

G:=TransitiveGroup(16,412);

Matrix representation of (C2×C8).D4 in GL8(ℤ)

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

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

(C2×C8).D4 in GAP, Magma, Sage, TeX

(C_2\times C_8).D_4
% in TeX

G:=Group("(C2xC8).D4");
// GroupNames label

G:=SmallGroup(128,813);
// by ID

G=gap.SmallGroup(128,813);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,1018,248,1411,718,172,4037,1027]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^4=b^4,c*b*c^-1=a*b=b*a,c*a*c^-1=a*b^4,a*d=d*a,d*b*d=a*b^-1,d*c*d=a*b^6*c^3>;
// generators/relations

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