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## G = (C4×C8)⋊C4order 128 = 27

### 3rd semidirect product of C4×C8 and C4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: (C4×C8)⋊3C4, (C2×D4).8D4, (C2×Q8).8D4, C42.C22C4, C42.19(C2×C4), C423C4.2C2, C2.8(C423C4), C42.C4.2C2, C4.4D4.5C22, C22.24(C23⋊C4), C42.78C22.1C2, (C2×C4).40(C22⋊C4), SmallGroup(128,146)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — (C4×C8)⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — C42.78C22 — (C4×C8)⋊C4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — (C4×C8)⋊C4
 Upper central C1 — C2 — C22 — C2×C4 — C4.4D4 — (C4×C8)⋊C4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — (C4×C8)⋊C4

Generators and relations for (C4×C8)⋊C4
G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=a-1b2, cbc-1=a-1b-1 >

Character table of (C4×C8)⋊C4

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 8A 8B 8C 8D 8E 8F size 1 1 2 8 4 4 4 8 16 16 16 4 4 4 4 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 -i 1 i -1 -1 -1 -1 -i i linear of order 4 ρ6 1 1 1 -1 1 1 1 -1 i -1 -i 1 1 1 1 -i i linear of order 4 ρ7 1 1 1 -1 1 1 1 -1 -i -1 i 1 1 1 1 i -i linear of order 4 ρ8 1 1 1 -1 1 1 1 -1 i 1 -i -1 -1 -1 -1 i -i linear of order 4 ρ9 2 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 4 0 0 -4 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ12 4 4 -4 0 0 0 0 0 0 0 0 2i 2i -2i -2i 0 0 complex lifted from C42⋊3C4 ρ13 4 4 -4 0 0 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 complex lifted from C42⋊3C4 ρ14 4 -4 0 0 2i 0 -2i 0 0 0 0 2ζ8 2ζ85 2ζ83 2ζ87 0 0 complex faithful ρ15 4 -4 0 0 2i 0 -2i 0 0 0 0 2ζ85 2ζ8 2ζ87 2ζ83 0 0 complex faithful ρ16 4 -4 0 0 -2i 0 2i 0 0 0 0 2ζ83 2ζ87 2ζ8 2ζ85 0 0 complex faithful ρ17 4 -4 0 0 -2i 0 2i 0 0 0 0 2ζ87 2ζ83 2ζ85 2ζ8 0 0 complex faithful

Smallest permutation representation of (C4×C8)⋊C4
On 32 points
Generators in S32
(1 12 27 20)(2 13 28 21)(3 14 29 22)(4 15 30 23)(5 16 31 24)(6 9 32 17)(7 10 25 18)(8 11 26 19)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(2 17 26 19)(3 29 7 25)(4 11 32 9)(6 21 30 23)(8 15 28 13)(10 20 18 16)(12 14 24 22)(27 31)

G:=sub<Sym(32)| (1,12,27,20)(2,13,28,21)(3,14,29,22)(4,15,30,23)(5,16,31,24)(6,9,32,17)(7,10,25,18)(8,11,26,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,17,26,19)(3,29,7,25)(4,11,32,9)(6,21,30,23)(8,15,28,13)(10,20,18,16)(12,14,24,22)(27,31)>;

G:=Group( (1,12,27,20)(2,13,28,21)(3,14,29,22)(4,15,30,23)(5,16,31,24)(6,9,32,17)(7,10,25,18)(8,11,26,19), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,17,26,19)(3,29,7,25)(4,11,32,9)(6,21,30,23)(8,15,28,13)(10,20,18,16)(12,14,24,22)(27,31) );

G=PermutationGroup([[(1,12,27,20),(2,13,28,21),(3,14,29,22),(4,15,30,23),(5,16,31,24),(6,9,32,17),(7,10,25,18),(8,11,26,19)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(2,17,26,19),(3,29,7,25),(4,11,32,9),(6,21,30,23),(8,15,28,13),(10,20,18,16),(12,14,24,22),(27,31)]])

Matrix representation of (C4×C8)⋊C4 in GL4(𝔽17) generated by

 2 15 15 15 15 2 15 15 2 2 2 15 2 2 15 2
,
 13 1 1 13 1 13 13 1 16 4 13 1 4 16 1 13
,
 1 0 0 0 0 16 0 0 0 0 0 16 0 0 1 0
G:=sub<GL(4,GF(17))| [2,15,2,2,15,2,2,2,15,15,2,15,15,15,15,2],[13,1,16,4,1,13,4,16,1,13,13,1,13,1,1,13],[1,0,0,0,0,16,0,0,0,0,0,1,0,0,16,0] >;

(C4×C8)⋊C4 in GAP, Magma, Sage, TeX

(C_4\times C_8)\rtimes C_4
% in TeX

G:=Group("(C4xC8):C4");
// GroupNames label

G:=SmallGroup(128,146);
// by ID

G=gap.SmallGroup(128,146);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,232,422,1059,184,1690,745,1684,1411,375,172,4037]);
// Polycyclic

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

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