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G = Dic5order 20 = 22·5

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: Dic5, C52C4, C2.D5, C10.C2, SmallGroup(20,1)

Series: Derived Chief Lower central Upper central

C1C5 — Dic5
C1C5C10 — Dic5
C5 — Dic5
C1C2

Generators and relations for Dic5
 G = < a,b | a10=1, b2=a5, bab-1=a-1 >

5C4

Character table of Dic5

 class 124A4B5A5B10A10B
 size 11552222
ρ111111111    trivial
ρ211-1-11111    linear of order 2
ρ31-1-ii11-1-1    linear of order 4
ρ41-1i-i11-1-1    linear of order 4
ρ52200-1+5/2-1-5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ62200-1-5/2-1+5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ72-200-1+5/2-1-5/21+5/21-5/2    symplectic faithful, Schur index 2
ρ82-200-1-5/2-1+5/21-5/21+5/2    symplectic faithful, Schur index 2

Permutation representations of Dic5
Regular action on 20 points - transitive group 20T2
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,14,6,19),(2,13,7,18),(3,12,8,17),(4,11,9,16),(5,20,10,15)])

G:=TransitiveGroup(20,2);

Matrix representation of Dic5 in GL2(𝔽11) generated by

93
36
,
010
10
G:=sub<GL(2,GF(11))| [9,3,3,6],[0,1,10,0] >;

Dic5 in GAP, Magma, Sage, TeX

{\rm Dic}_5
% in TeX

G:=Group("Dic5");
// GroupNames label

G:=SmallGroup(20,1);
// by ID

G=gap.SmallGroup(20,1);
# by ID

G:=PCGroup([3,-2,-2,-5,6,146]);
// Polycyclic

G:=Group<a,b|a^10=1,b^2=a^5,b*a*b^-1=a^-1>;
// generators/relations

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