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## G = (C4×C8).C4order 128 = 27

### 6th non-split extension by C4×C8 of C4 acting faithfully

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

Aliases: (C4×C8).6C4, (C2×Q8).5D4, C41D4.4C4, C85D4.7C2, C4⋊Q8.2C22, C42.17(C2×C4), C42.3C45C2, C2.10(C42⋊C4), C22.20(C23⋊C4), (C2×C4).36(C22⋊C4), SmallGroup(128,142)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — (C4×C8).C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×Q8 — C4⋊Q8 — C8⋊5D4 — (C4×C8).C4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — (C4×C8).C4
 Upper central C1 — C2 — C22 — C2×C4 — C4⋊Q8 — (C4×C8).C4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4⋊Q8 — (C4×C8).C4

Generators and relations for (C4×C8).C4
G = < a,b,c | a4=b8=1, c4=b4, ab=ba, cac-1=ab2, cbc-1=ab3 >

Character table of (C4×C8).C4

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

Permutation representations of (C4×C8).C4
On 16 points - transitive group 16T378
Generators in S16
(9 15 13 11)(10 16 14 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 10 2 13 5 14 6 9)(3 16 8 15 7 12 4 11)

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

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

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

G:=TransitiveGroup(16,378);

Matrix representation of (C4×C8).C4 in GL4(𝔽3) generated by

 2 0 0 2 0 1 0 0 0 0 1 0 2 0 0 1
,
 0 0 0 2 0 1 1 0 0 2 1 0 2 0 0 2
,
 0 0 2 0 1 0 0 2 0 0 0 1 0 1 1 0
G:=sub<GL(4,GF(3))| [2,0,0,2,0,1,0,0,0,0,1,0,2,0,0,1],[0,0,0,2,0,1,2,0,0,1,1,0,2,0,0,2],[0,1,0,0,0,0,0,1,2,0,0,1,0,2,1,0] >;

(C4×C8).C4 in GAP, Magma, Sage, TeX

(C_4\times C_8).C_4
% in TeX

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

G:=SmallGroup(128,142);
// by ID

G=gap.SmallGroup(128,142);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,1059,520,794,745,248,1684,1411,375,172,4037]);
// Polycyclic

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

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