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G = Dic9order 36 = 22·32

Dicyclic group

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

Aliases: Dic9, C9⋊C4, C2.D9, C18.C2, C6.1S3, C3.Dic3, SmallGroup(36,1)

Series: Derived Chief Lower central Upper central

C1C9 — Dic9
C1C3C9C18 — Dic9
C9 — Dic9
C1C2

Generators and relations for Dic9
 G = < a,b | a18=1, b2=a9, bab-1=a-1 >

9C4
3Dic3

Character table of Dic9

 class 1234A4B69A9B9C18A18B18C
 size 112992222222
ρ1111111111111    trivial
ρ2111-1-11111111    linear of order 2
ρ31-11i-i-1111-1-1-1    linear of order 4
ρ41-11-ii-1111-1-1-1    linear of order 4
ρ5222002-1-1-1-1-1-1    orthogonal lifted from S3
ρ622-100-1ζ9792ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ722-100-1ζ9594ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ822-100-1ζ989ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ92-2200-2-1-1-1111    symplectic lifted from Dic3, Schur index 2
ρ102-2-1001ζ9594ζ989ζ979297929594989    symplectic faithful, Schur index 2
ρ112-2-1001ζ989ζ9792ζ959495949899792    symplectic faithful, Schur index 2
ρ122-2-1001ζ9792ζ9594ζ98998997929594    symplectic faithful, Schur index 2

Smallest permutation representation of Dic9
Regular action on 36 points
Generators in S36
(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)
(1 20 10 29)(2 19 11 28)(3 36 12 27)(4 35 13 26)(5 34 14 25)(6 33 15 24)(7 32 16 23)(8 31 17 22)(9 30 18 21)

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

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

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

Matrix representation of Dic9 in GL2(𝔽17) generated by

210
108
,
131
04
G:=sub<GL(2,GF(17))| [2,10,10,8],[13,0,1,4] >;

Dic9 in GAP, Magma, Sage, TeX

{\rm Dic}_9
% in TeX

G:=Group("Dic9");
// GroupNames label

G:=SmallGroup(36,1);
// by ID

G=gap.SmallGroup(36,1);
# by ID

G:=PCGroup([4,-2,-2,-3,-3,8,242,82,387]);
// Polycyclic

G:=Group<a,b|a^18=1,b^2=a^9,b*a*b^-1=a^-1>;
// generators/relations

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