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## G = (C4×C8)⋊6C4order 128 = 27

### 6th semidirect product of C4×C8 and C4 acting faithfully

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

Aliases: (C4×C8)⋊6C4, C4⋊Q82C4, (C2×D4).6D4, C85D4.6C2, C42.16(C2×C4), C42⋊C4.1C2, C41D4.3C22, C2.9(C42⋊C4), C22.19(C23⋊C4), (C2×C4).35(C22⋊C4), SmallGroup(128,141)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — (C4×C8)⋊6C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×D4 — C4⋊1D4 — C8⋊5D4 — (C4×C8)⋊6C4
 Lower central C1 — C2 — C22 — C2×C4 — C42 — (C4×C8)⋊6C4
 Upper central C1 — C2 — C22 — C2×C4 — C4⋊1D4 — (C4×C8)⋊6C4
 Jennings C1 — C2 — C2 — C22 — C2×C4 — C4⋊1D4 — (C4×C8)⋊6C4

Generators and relations for (C4×C8)⋊6C4
G = < a,b,c | a4=b8=c4=1, ab=ba, cac-1=ab2, cbc-1=ab3 >

Character table of (C4×C8)⋊6C4

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

Permutation representations of (C4×C8)⋊6C4
On 16 points - transitive group 16T363
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 11 13 15)(10 12 14 16)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 11)(2 10 4 16)(3 9 7 13)(5 15)(6 14 8 12)

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

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

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

G:=TransitiveGroup(16,363);

Matrix representation of (C4×C8)⋊6C4 in GL4(𝔽3) generated by

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

(C4×C8)⋊6C4 in GAP, Magma, Sage, TeX

(C_4\times C_8)\rtimes_6C_4
% in TeX

G:=Group("(C4xC8):6C4");
// GroupNames label

G:=SmallGroup(128,141);
// by ID

G=gap.SmallGroup(128,141);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,1059,520,794,745,1684,1411,375,172,4037]);
// Polycyclic

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

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