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## G = (C2×C4).Q16order 128 = 27

### 1st non-split extension by C2×C4 of Q16 acting via Q16/C2=D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C4).Q16
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4⋊C4 — C23.83C23 — (C2×C4).Q16
 Lower central C1 — C2 — C23 — C22×C4 — (C2×C4).Q16
 Upper central C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).Q16
 Jennings C1 — C22 — C23 — C2×C4⋊C4 — (C2×C4).Q16

Generators and relations for (C2×C4).Q16
G = < a,b,c,d | a2=b4=c8=1, d2=ab2c4, cbc-1=ab=ba, cac-1=ab2, ad=da, dbd-1=ab-1, dcd-1=b-1c-1 >

Character table of (C2×C4).Q16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 2 2 4 4 4 4 8 8 8 8 8 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 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 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 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 linear of order 2 ρ5 1 1 1 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 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 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 1 -1 1 -i -i -i i i i -i i linear of order 4 ρ9 2 2 2 2 2 2 2 -2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 2 -2 2 -2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 √2 0 0 √2 0 0 -√2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 -2 2 -2 2 -2 -2 0 2 0 0 0 0 0 0 -√2 0 0 -√2 0 0 √2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 1+i -1-i 0 1-i -1+i 0 0 complex lifted from C4≀C2 ρ14 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 1-i -1+i 0 1+i -1-i 0 0 complex lifted from C4≀C2 ρ15 2 -2 2 -2 -2 2 0 -2i 0 2i 0 0 0 0 0 0 -1+i 1-i 0 -1-i 1+i 0 0 complex lifted from C4≀C2 ρ16 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 -√-2 0 0 √-2 0 0 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 2 -2 2 -2 2 0 -2 0 0 0 0 0 0 √-2 0 0 -√-2 0 0 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 2 -2 -2 2 0 2i 0 -2i 0 0 0 0 0 0 -1-i 1+i 0 -1+i 1-i 0 0 complex lifted from C4≀C2 ρ19 4 4 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 -4 -4 0 0 0 0 0 0 -2i 0 0 0 2i 0 0 0 0 0 0 0 0 complex lifted from C23.D4 ρ21 4 -4 -4 4 0 0 0 0 0 0 0 2i 0 -2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 -2i 0 2i 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ23 4 4 -4 -4 0 0 0 0 0 0 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 complex lifted from C23.D4

Smallest permutation representation of (C2×C4).Q16
On 32 points
Generators in S32
(1 29)(2 6)(3 31)(4 8)(5 25)(7 27)(9 13)(10 23)(11 15)(12 17)(14 19)(16 21)(18 22)(20 24)(26 30)(28 32)
(1 10 25 19)(2 24 26 15)(3 21 27 12)(4 9 28 18)(5 14 29 23)(6 20 30 11)(7 17 31 16)(8 13 32 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)
(2 9 26 18)(3 27)(4 11 28 20)(6 13 30 22)(7 31)(8 15 32 24)(10 14)(12 17)(16 21)(19 23)

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

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

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

Matrix representation of (C2×C4).Q16 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 12 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 13 0 0 0 0 0 4 4
,
 7 13 0 0 0 0 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 12 1 2 0 0 0 4 0 0 0 0 15 11 11 5
,
 10 4 0 0 0 0 13 7 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 12 1 2 0 0 0 0 16 16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,12,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,4,0,0,0,0,0,4],[7,4,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,15,0,0,0,12,4,11,0,0,1,1,0,11,0,0,0,2,0,5],[10,13,0,0,0,0,4,7,0,0,0,0,0,0,1,0,0,0,0,0,0,16,12,0,0,0,0,0,1,16,0,0,0,0,2,16] >;

(C2×C4).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_4).Q_{16}
% in TeX

G:=Group("(C2xC4).Q16");
// GroupNames label

G:=SmallGroup(128,85);
// by ID

G=gap.SmallGroup(128,85);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,120,422,387,520,1690,521,248,2804]);
// Polycyclic

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

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