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## G = Dic5order 20 = 22·5

### Dicyclic group

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5
 Chief series C1 — C5 — C10 — Dic5
 Lower central C5 — Dic5
 Upper central C1 — C2

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

Character table of Dic5

 class 1 2 4A 4B 5A 5B 10A 10B size 1 1 5 5 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 -1 -i i 1 1 -1 -1 linear of order 4 ρ4 1 -1 i -i 1 1 -1 -1 linear of order 4 ρ5 2 2 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 -2 0 0 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 symplectic faithful, Schur index 2 ρ8 2 -2 0 0 -1-√5/2 -1+√5/2 1-√5/2 1+√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);

Dic5 is a maximal subgroup of
C5⋊C8  C4×D5  C5⋊D4  C526C4  D5.D5  SL2(𝔽5)  C32⋊Dic5  C65⋊C4  2- 1+4.D5  C25.D5  C85⋊C4
Dic5p: Dic10  Dic15  Dic25  Dic35  Dic55  Dic65  Dic85  Dic95 ...
Dic5 is a maximal quotient of
D5.D5  C32⋊Dic5  C65⋊C4  C25.D5  C85⋊C4
C2p.D5: C52C8  Dic15  Dic25  C526C4  Dic35  Dic55  Dic65  Dic85 ...

Matrix representation of Dic5 in GL2(𝔽11) generated by

 9 3 3 6
,
 0 10 1 0
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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