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G = D27  order 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

Derived series
C1 — C27 — D27
Chief series
C1 — C3 — C9 — C27 — D27
Lower central
C27 — D27
Upper central
C1

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

C1
27C2
C3
9S3
C9
3D9
C27
D27

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
ρ4202-1-1-1ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ5202-1-1-1ζ95+ζ94ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ6202-1-1-1ζ98+ζ9ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ720-1ζ2715+ζ2712ζ2721+ζ276ζ2724+ζ273ζ2723+ζ274ζ2714+ζ2713ζ2717+ζ2710ζ2719+ζ278ζ2726+ζ27ζ2725+ζ272ζ2720+ζ277ζ2716+ζ2711ζ2722+ζ275    orthogonal faithful
ρ820-1ζ2724+ζ273ζ2715+ζ2712ζ2721+ζ276ζ2719+ζ278ζ2726+ζ27ζ2720+ζ277ζ2716+ζ2711ζ2725+ζ272ζ2723+ζ274ζ2714+ζ2713ζ2722+ζ275ζ2717+ζ2710    orthogonal faithful
ρ920-1ζ2724+ζ273ζ2715+ζ2712ζ2721+ζ276ζ2717+ζ2710ζ2719+ζ278ζ2725+ζ272ζ2720+ζ277ζ2716+ζ2711ζ2722+ζ275ζ2723+ζ274ζ2714+ζ2713ζ2726+ζ27    orthogonal faithful
ρ1020-1ζ2721+ζ276ζ2724+ζ273ζ2715+ζ2712ζ2720+ζ277ζ2716+ζ2711ζ2723+ζ274ζ2714+ζ2713ζ2722+ζ275ζ2717+ζ2710ζ2719+ζ278ζ2726+ζ27ζ2725+ζ272    orthogonal faithful
ρ1120-1ζ2721+ζ276ζ2724+ζ273ζ2715+ζ2712ζ2716+ζ2711ζ2725+ζ272ζ2714+ζ2713ζ2722+ζ275ζ2723+ζ274ζ2719+ζ278ζ2726+ζ27ζ2717+ζ2710ζ2720+ζ277    orthogonal faithful
ρ1220-1ζ2721+ζ276ζ2724+ζ273ζ2715+ζ2712ζ2725+ζ272ζ2720+ζ277ζ2722+ζ275ζ2723+ζ274ζ2714+ζ2713ζ2726+ζ27ζ2717+ζ2710ζ2719+ζ278ζ2716+ζ2711    orthogonal faithful
ρ1320-1ζ2715+ζ2712ζ2721+ζ276ζ2724+ζ273ζ2722+ζ275ζ2723+ζ274ζ2726+ζ27ζ2717+ζ2710ζ2719+ζ278ζ2716+ζ2711ζ2725+ζ272ζ2720+ζ277ζ2714+ζ2713    orthogonal faithful
ρ1420-1ζ2724+ζ273ζ2715+ζ2712ζ2721+ζ276ζ2726+ζ27ζ2717+ζ2710ζ2716+ζ2711ζ2725+ζ272ζ2720+ζ277ζ2714+ζ2713ζ2722+ζ275ζ2723+ζ274ζ2719+ζ278    orthogonal faithful
ρ1520-1ζ2715+ζ2712ζ2721+ζ276ζ2724+ζ273ζ2714+ζ2713ζ2722+ζ275ζ2719+ζ278ζ2726+ζ27ζ2717+ζ2710ζ2720+ζ277ζ2716+ζ2711ζ2725+ζ272ζ2723+ζ274    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);
 

D27 is a maximal subgroup of   D81  C27⋊C6  C27⋊S3  C9.S4  D135  D189
D27 is a maximal quotient of   Dic27  D81  C27⋊S3  C9.S4  D135  D189

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
 

Export

Subgroup lattice of D27 in TeX
Character table of D27 in TeX

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