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## G = (C2×C8)⋊4D4order 128 = 27

### 4th semidirect product of C2×C8 and D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — (C2×C8)⋊4D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — Q8○M4(2) — (C2×C8)⋊4D4
 Lower central C1 — C2 — C2×C4 — (C2×C8)⋊4D4
 Upper central C1 — C2 — C22×C4 — (C2×C8)⋊4D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8)⋊4D4

Generators and relations for (C2×C8)⋊4D4
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, cac-1=ab4, ad=da, cbc-1=dbd=ab-1, dcd=c-1 >

Subgroups: 412 in 172 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, D4⋊C4, C42⋊C2, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C41D4, C2×M4(2), C2×M4(2), C8○D4, C22×D4, C2×C4○D4, C4.9C42, C23.C23, C23.37D4, C22.29C24, Q8○M4(2), (C2×C8)⋊4D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C23.23D4, (C2×C8)⋊4D4

Character table of (C2×C8)⋊4D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 4 4 8 8 2 2 2 2 4 4 8 8 8 8 8 8 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 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 -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 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 -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 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 -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 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 -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 1 -1 1 i -i i -i 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 1 -1 1 -i i -i i -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 1 1 -1 -i i -i i 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 1 1 -1 i -i i -i -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 -1 1 -1 i -i -i i 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 -1 1 -1 -i i i -i -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 -1 -1 1 -i i i -i 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 -1 -1 1 i -i -i i -i -i i i -i i i -i linear of order 4 ρ17 2 2 -2 2 -2 2 -2 0 0 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 -2 2 orthogonal lifted from D4 ρ19 2 2 2 2 2 -2 2 0 0 -2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 -2 0 0 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 2 -2 -2 2 0 0 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 2 -2 orthogonal lifted from D4 ρ23 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 2 0 0 2 -2 -2 0 0 orthogonal lifted from D4 ρ24 2 2 2 -2 -2 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 0 0 -2 2 2 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 2i 2i 0 0 0 -2i -2i complex lifted from C4○D4 ρ26 2 2 2 -2 -2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 -2i -2i 0 0 0 2i 2i complex lifted from C4○D4 ρ27 2 2 -2 -2 2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 -2i 0 0 2i 2i -2i 0 0 complex lifted from C4○D4 ρ28 2 2 -2 -2 2 0 0 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 2i 0 0 -2i -2i 2i 0 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 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)⋊4D4
On 16 points - transitive group 16T238
Generators in S16
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 7)(2 11 6 15)(3 5)(4 9 8 13)(10 12)(14 16)
(1 10)(2 8)(3 16)(4 6)(5 14)(7 12)(9 11)(13 15)

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

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

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

G:=TransitiveGroup(16,238);

On 16 points - transitive group 16T298
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 14 5 10)(2 13)(3 12 7 16)(4 11)(6 9)(8 15)
(2 8)(3 7)(4 6)(9 11)(10 14)(13 15)

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

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

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

G:=TransitiveGroup(16,298);

Matrix representation of (C2×C8)⋊4D4 in GL8(ℤ)

 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 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
,
 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 -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
,
 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 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
,
 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 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

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

(C2×C8)⋊4D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_4D_4
% in TeX

G:=Group("(C2xC8):4D4");
// GroupNames label

G:=SmallGroup(128,642);
// by ID

G=gap.SmallGroup(128,642);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,2019,521,248,2804,1411,1027]);
// Polycyclic

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

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