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G = Q16order 16 = 24

Generalised quaternion group

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: Q16, Dic4, C8.C2, Q8.C2, C2.5D4, C4.3C22, 2-Sylow(SL(2,7)), SmallGroup(16,9)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q16
C1C2C4Q8 — Q16
C1C2C4 — Q16
C1C2C4 — Q16
C1C2C2C4 — Q16

Generators and relations for Q16
 G = < a,b | a8=1, b2=a4, bab-1=a-1 >

2C4
2C4

Character table of Q16

 class 124A4B4C8A8B
 size 1124422
ρ11111111    trivial
ρ21111-1-1-1    linear of order 2
ρ3111-11-1-1    linear of order 2
ρ4111-1-111    linear of order 2
ρ522-20000    orthogonal lifted from D4
ρ62-20002-2    symplectic faithful, Schur index 2
ρ72-2000-22    symplectic faithful, Schur index 2

Permutation representations of Q16
Regular action on 16 points - transitive group 16T14
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16 5 12)(2 15 6 11)(3 14 7 10)(4 13 8 9)

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

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

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

G:=TransitiveGroup(16,14);

Matrix representation of Q16 in GL2(𝔽7) generated by

06
13
,
65
11
G:=sub<GL(2,GF(7))| [0,1,6,3],[6,1,5,1] >;

Q16 in GAP, Magma, Sage, TeX

Q_{16}
% in TeX

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

G:=SmallGroup(16,9);
// by ID

G=gap.SmallGroup(16,9);
# by ID

G:=PCGroup([4,-2,2,-2,-2,32,49,37,146,78,34]);
// Polycyclic

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

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