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## G = C42.(C2×C4)  order 128 = 27

### 2nd non-split extension by C42 of C2×C4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C4.10C4≀C2, C4⋊Q8.1C4, C16⋊C4.C2, (C2×C8).24D4, C42.2(C2×C4), C42.C2.2C4, C8⋊C4.84C22, C22.11(C4.D4), C42.30C22.2C2, C2.5(C42.C22), (C2×C4).57(C22⋊C4), SmallGroup(128,88)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C42 — C42.(C2×C4)
 Chief series C1 — C2 — C22 — C2×C4 — C2×C8 — C8⋊C4 — C42.30C22 — C42.(C2×C4)
 Lower central C1 — C2 — C2×C4 — C42 — C42.(C2×C4)
 Upper central C1 — C2 — C2×C4 — C8⋊C4 — C42.(C2×C4)
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C8⋊C4 — C42.(C2×C4)

Generators and relations for C42.(C2×C4)
G = < a,b,c,d | a4=b4=1, c2=b2, d4=b, ab=ba, cac-1=a-1, dad-1=ab-1, cbc-1=b-1, bd=db, dcd-1=ab-1c >

Character table of C42.(C2×C4)

 class 1 2A 2B 4A 4B 4C 4D 4E 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 2 2 2 8 16 16 4 4 4 4 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 trivial ρ2 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 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 linear of order 2 ρ5 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 i -i -i i -i i i -i linear of order 4 ρ6 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 i -i i -i i i -i -i linear of order 4 ρ7 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -i i i -i i -i -i i linear of order 4 ρ8 1 1 1 1 1 1 -1 1 -1 -1 -1 -1 -i i -i i -i -i i i linear of order 4 ρ9 2 2 2 2 2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 0 0 0 0 0 -2i 2i 0 0 1-i 1+i -1+i 0 -1-i 0 complex lifted from C4≀C2 ρ12 2 2 -2 -2 2 0 0 0 2i -2i 0 0 1-i -1-i 0 0 0 -1+i 0 1+i complex lifted from C4≀C2 ρ13 2 2 -2 -2 2 0 0 0 2i -2i 0 0 -1+i 1+i 0 0 0 1-i 0 -1-i complex lifted from C4≀C2 ρ14 2 2 -2 2 -2 0 0 0 0 0 2i -2i 0 0 -1-i -1+i 1+i 0 1-i 0 complex lifted from C4≀C2 ρ15 2 2 -2 2 -2 0 0 0 0 0 2i -2i 0 0 1+i 1-i -1-i 0 -1+i 0 complex lifted from C4≀C2 ρ16 2 2 -2 2 -2 0 0 0 0 0 -2i 2i 0 0 -1+i -1-i 1-i 0 1+i 0 complex lifted from C4≀C2 ρ17 2 2 -2 -2 2 0 0 0 -2i 2i 0 0 -1-i 1-i 0 0 0 1+i 0 -1+i complex lifted from C4≀C2 ρ18 2 2 -2 -2 2 0 0 0 -2i 2i 0 0 1+i -1+i 0 0 0 -1-i 0 1-i complex lifted from C4≀C2 ρ19 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ20 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C42.(C2×C4)
On 32 points
Generators in S32
(1 22 9 30)(2 19)(3 32 11 24)(4 29)(5 26 13 18)(6 23)(7 20 15 28)(8 17)(10 27)(12 21)(14 31)(16 25)
(1 5 9 13)(2 6 10 14)(3 7 11 15)(4 8 12 16)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 15 9 7)(2 25 10 17)(3 5 11 13)(4 23 12 31)(6 21 14 29)(8 19 16 27)(18 24 26 32)(20 30 28 22)
(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)

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

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

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

Matrix representation of C42.(C2×C4) in GL8(𝔽17)

 7 0 6 11 0 0 0 0 0 7 3 0 0 0 0 0 0 6 10 0 0 0 0 0 14 6 0 10 0 0 0 0 0 0 0 0 7 3 0 11 0 0 0 0 7 10 3 11 0 0 0 0 6 11 10 14 0 0 0 0 3 0 10 7
,
 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 1 16
,
 0 0 16 0 0 0 0 0 0 0 16 1 0 0 0 0 1 0 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 0 0 6 10 0 0 0 0 0 3 0 10 7 0 0 0 0 10 0 11 6 0 0 0 0 10 7 14 6
,
 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0

G:=sub<GL(8,GF(17))| [7,0,0,14,0,0,0,0,0,7,6,6,0,0,0,0,6,3,10,0,0,0,0,0,11,0,0,10,0,0,0,0,0,0,0,0,7,7,6,3,0,0,0,0,3,10,11,0,0,0,0,0,0,3,10,10,0,0,0,0,11,11,14,7],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,10,10,0,0,0,0,6,0,0,7,0,0,0,0,10,10,11,14,0,0,0,0,0,7,6,6],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C42.(C2×C4) in GAP, Magma, Sage, TeX

C_4^2.(C_2\times C_4)
% in TeX

G:=Group("C4^2.(C2xC4)");
// GroupNames label

G:=SmallGroup(128,88);
// by ID

G=gap.SmallGroup(128,88);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,184,1690,521,80,1411,172,4037]);
// Polycyclic

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

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