Copied to
clipboard

G = Dic10order 40 = 23·5

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10, C5⋊Q8, C4.D5, C20.1C2, C2.3D10, C10.1C22, Dic5.1C2, SmallGroup(40,4)

Series: Derived Chief Lower central Upper central

C1C10 — Dic10
C1C5C10Dic5 — Dic10
C5C10 — Dic10
C1C2C4

Generators and relations for Dic10
 G = < a,b | a20=1, b2=a10, bab-1=a-1 >

5C4
5C4
5Q8

Character table of Dic10

 class 124A4B4C5A5B10A10B20A20B20C20D
 size 112101022222222
ρ11111111111111    trivial
ρ2111-1-111111111    linear of order 2
ρ311-11-11111-1-1-1-1    linear of order 2
ρ411-1-111111-1-1-1-1    linear of order 2
ρ522200-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ622-200-1-5/2-1+5/2-1-5/2-1+5/21+5/21+5/21-5/21-5/2    orthogonal lifted from D10
ρ722-200-1+5/2-1-5/2-1+5/2-1-5/21-5/21-5/21+5/21+5/2    orthogonal lifted from D10
ρ822200-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ92-200022-2-20000    symplectic lifted from Q8, Schur index 2
ρ102-2000-1-5/2-1+5/21+5/21-5/2ζ4ζ534ζ524ζ534ζ5243ζ5443ζ5ζ43ζ5443ζ5    symplectic faithful, Schur index 2
ρ112-2000-1+5/2-1-5/21-5/21+5/2ζ43ζ5443ζ543ζ5443ζ5ζ4ζ534ζ524ζ534ζ52    symplectic faithful, Schur index 2
ρ122-2000-1-5/2-1+5/21+5/21-5/24ζ534ζ52ζ4ζ534ζ52ζ43ζ5443ζ543ζ5443ζ5    symplectic faithful, Schur index 2
ρ132-2000-1+5/2-1-5/21-5/21+5/243ζ5443ζ5ζ43ζ5443ζ54ζ534ζ52ζ4ζ534ζ52    symplectic faithful, Schur index 2

Smallest permutation representation of Dic10
Regular action on 40 points
Generators in S40
(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 33 34 35 36 37 38 39 40)
(1 22 11 32)(2 21 12 31)(3 40 13 30)(4 39 14 29)(5 38 15 28)(6 37 16 27)(7 36 17 26)(8 35 18 25)(9 34 19 24)(10 33 20 23)

G:=sub<Sym(40)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,22,11,32)(2,21,12,31)(3,40,13,30)(4,39,14,29)(5,38,15,28)(6,37,16,27)(7,36,17,26)(8,35,18,25)(9,34,19,24)(10,33,20,23)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,22,11,32)(2,21,12,31)(3,40,13,30)(4,39,14,29)(5,38,15,28)(6,37,16,27)(7,36,17,26)(8,35,18,25)(9,34,19,24)(10,33,20,23) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,22,11,32),(2,21,12,31),(3,40,13,30),(4,39,14,29),(5,38,15,28),(6,37,16,27),(7,36,17,26),(8,35,18,25),(9,34,19,24),(10,33,20,23)])

Matrix representation of Dic10 in GL2(𝔽19) generated by

817
173
,
018
10
G:=sub<GL(2,GF(19))| [8,17,17,3],[0,1,18,0] >;

Dic10 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}
% in TeX

G:=Group("Dic10");
// GroupNames label

G:=SmallGroup(40,4);
// by ID

G=gap.SmallGroup(40,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-5,16,49,21,515]);
// Polycyclic

G:=Group<a,b|a^20=1,b^2=a^10,b*a*b^-1=a^-1>;
// generators/relations

׿
×
𝔽