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## G = C2×Q16order 32 = 25

### Direct product of C2 and Q16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×Q16, C4.8D4, C4.3C23, C8.5C22, C22.16D4, Q8.1C22, (C2×C8).4C2, C2.13(C2×D4), (C2×Q8).4C2, (C2×C4).28C22, SmallGroup(32,41)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×Q16
 Chief series C1 — C2 — C4 — C2×C4 — C2×Q8 — C2×Q16
 Lower central C1 — C2 — C4 — C2×Q16
 Upper central C1 — C22 — C2×C4 — C2×Q16
 Jennings C1 — C2 — C2 — C4 — C2×Q16

Generators and relations for C2×Q16
G = < a,b,c | a2=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Q16

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 -1 1 -1 -1 1 1 -1 1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ9 2 2 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 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ12 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ13 2 -2 -2 2 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ14 2 -2 -2 2 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2

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

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

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

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

Matrix representation of C2×Q16 in GL3(𝔽17) generated by

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

C2×Q16 in GAP, Magma, Sage, TeX

C_2\times Q_{16}
% in TeX

G:=Group("C2xQ16");
// GroupNames label

G:=SmallGroup(32,41);
// by ID

G=gap.SmallGroup(32,41);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,80,101,86,483,248,58]);
// Polycyclic

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

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