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G = D27order 54 = 2·33

Dihedral group

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

Aliases: D27, C27⋊C2, C9.S3, C3.D9, sometimes denoted D54 or Dih27 or Dih54, SmallGroup(54,1)

Series: Derived Chief Lower central Upper central

C1C27 — D27
C1C3C9C27 — D27
C27 — D27
C1

Generators and relations for D27
 G = < a,b | a27=b2=1, bab=a-1 >

27C2
9S3
3D9

Character table of D27

 class 1239A9B9C27A27B27C27D27E27F27G27H27I
 size 1272222222222222
ρ1111111111111111    trivial
ρ21-11111111111111    linear of order 2
ρ3202222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ4202-1-1-1ζ9792ζ9792ζ9594ζ9594ζ9594ζ989ζ989ζ989ζ9792    orthogonal lifted from D9
ρ5202-1-1-1ζ9594ζ9594ζ989ζ989ζ989ζ9792ζ9792ζ9792ζ9594    orthogonal lifted from D9
ρ6202-1-1-1ζ989ζ989ζ9792ζ9792ζ9792ζ9594ζ9594ζ9594ζ989    orthogonal lifted from D9
ρ720-1ζ27152712ζ2721276ζ2724273ζ2723274ζ27142713ζ27172710ζ2719278ζ272627ζ2725272ζ2720277ζ27162711ζ2722275    orthogonal faithful
ρ820-1ζ2724273ζ27152712ζ2721276ζ2719278ζ272627ζ2720277ζ27162711ζ2725272ζ2723274ζ27142713ζ2722275ζ27172710    orthogonal faithful
ρ920-1ζ2724273ζ27152712ζ2721276ζ27172710ζ2719278ζ2725272ζ2720277ζ27162711ζ2722275ζ2723274ζ27142713ζ272627    orthogonal faithful
ρ1020-1ζ2721276ζ2724273ζ27152712ζ2720277ζ27162711ζ2723274ζ27142713ζ2722275ζ27172710ζ2719278ζ272627ζ2725272    orthogonal faithful
ρ1120-1ζ2721276ζ2724273ζ27152712ζ27162711ζ2725272ζ27142713ζ2722275ζ2723274ζ2719278ζ272627ζ27172710ζ2720277    orthogonal faithful
ρ1220-1ζ2721276ζ2724273ζ27152712ζ2725272ζ2720277ζ2722275ζ2723274ζ27142713ζ272627ζ27172710ζ2719278ζ27162711    orthogonal faithful
ρ1320-1ζ27152712ζ2721276ζ2724273ζ2722275ζ2723274ζ272627ζ27172710ζ2719278ζ27162711ζ2725272ζ2720277ζ27142713    orthogonal faithful
ρ1420-1ζ2724273ζ27152712ζ2721276ζ272627ζ27172710ζ27162711ζ2725272ζ2720277ζ27142713ζ2722275ζ2723274ζ2719278    orthogonal faithful
ρ1520-1ζ27152712ζ2721276ζ2724273ζ27142713ζ2722275ζ2719278ζ272627ζ27172710ζ2720277ζ27162711ζ2725272ζ2723274    orthogonal faithful

Permutation representations of D27
On 27 points - transitive group 27T8
Generators in S27
(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)
(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 17)(13 16)(14 15)

G:=sub<Sym(27)| (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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)>;

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), (2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15) );

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)], [(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,17),(13,16),(14,15)])

G:=TransitiveGroup(27,8);

Matrix representation of D27 in GL2(𝔽109) generated by

8758
5129
,
10
108108
G:=sub<GL(2,GF(109))| [87,51,58,29],[1,108,0,108] >;

D27 in GAP, Magma, Sage, TeX

D_{27}
% in TeX

G:=Group("D27");
// GroupNames label

G:=SmallGroup(54,1);
// by ID

G=gap.SmallGroup(54,1);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,81,125,362,82,579]);
// Polycyclic

G:=Group<a,b|a^27=b^2=1,b*a*b=a^-1>;
// generators/relations

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