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G = D32  order 64 = 26

Dihedral group

p-group, metacyclic, nilpotent (class 5), monomial

Aliases: D32, C32⋊1C2, C4.1D8, C8.5D4, D16⋊1C2, C2.3D16, C16.2C22, 2-Sylow(PGL(2,31)), sometimes denoted D64 or Dih32 or Dih64, SmallGroup(64,52)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C16 — D32
Chief series
C1 — C2 — C4 — C8 — C16 — D16 — D32
Lower central
C1 — C2 — C4 — C8 — C16 — D32
Upper central
C1 — C2 — C4 — C8 — C16 — D32
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — D32

Generators and relations for D32
 G = < a,b | a32=b2=1, bab=a-1 >

C1
C2
16C2
16C2
C4
8C22
8C22
C8
4D4
4D4
C16
2D8
2D8
D16
C32
D16
D32

Character table of D32

 class 12A2B2C48A8B16A16B16C16D32A32B32C32D32E32F32G32H
 size 111616222222222222222
ρ11111111111111111111    trivial
ρ211-1-1111111111111111    linear of order 2
ρ3111-11111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ52200222-2-2-2-200000000    orthogonal lifted from D4
ρ622002-2-20000-√2-√2-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ722002-2-20000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ82200-200√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163ζ165-ζ163ζ1615-ζ169ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ1615+ζ169    orthogonal lifted from D16
ρ92200-200√2-√2-√2√2ζ165-ζ163ζ165-ζ163-ζ165+ζ163-ζ1615+ζ169-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ1615-ζ169    orthogonal lifted from D16
ρ102200-200-√2√2√2-√2-ζ1615+ζ169-ζ1615+ζ169ζ1615-ζ169-ζ165+ζ163-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ165-ζ163    orthogonal lifted from D16
ρ112200-200-√2√2√2-√2ζ1615-ζ169ζ1615-ζ169-ζ1615+ζ169ζ165-ζ163ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ122-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322ζ3211-ζ325-ζ3211+ζ325ζ3213-ζ323-ζ3225+ζ3223ζ3225-ζ3223-ζ3213+ζ323-ζ3215+ζ32ζ3215-ζ32    orthogonal faithful
ρ132-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326-ζ3215+ζ32ζ3215-ζ32ζ3225-ζ3223-ζ3211+ζ325ζ3211-ζ325-ζ3225+ζ3223-ζ3213+ζ323ζ3213-ζ323    orthogonal faithful
ρ142-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326-ζ3225+ζ3223ζ3225-ζ3223-ζ3215+ζ32ζ3213-ζ323-ζ3213+ζ323ζ3215-ζ32-ζ3211+ζ325ζ3211-ζ325    orthogonal faithful
ρ152-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322ζ3213-ζ323-ζ3213+ζ323-ζ3211+ζ325-ζ3215+ζ32ζ3215-ζ32ζ3211-ζ325ζ3225-ζ3223-ζ3225+ζ3223    orthogonal faithful
ρ162-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322-ζ3211+ζ325ζ3211-ζ325-ζ3213+ζ323ζ3225-ζ3223-ζ3225+ζ3223ζ3213-ζ323ζ3215-ζ32-ζ3215+ζ32    orthogonal faithful
ρ172-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326ζ3225-ζ3223-ζ3225+ζ3223ζ3215-ζ32-ζ3213+ζ323ζ3213-ζ323-ζ3215+ζ32ζ3211-ζ325-ζ3211+ζ325    orthogonal faithful
ρ182-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322-ζ3213+ζ323ζ3213-ζ323ζ3211-ζ325ζ3215-ζ32-ζ3215+ζ32-ζ3211+ζ325-ζ3225+ζ3223ζ3225-ζ3223    orthogonal faithful
ρ192-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326ζ3215-ζ32-ζ3215+ζ32-ζ3225+ζ3223ζ3211-ζ325-ζ3211+ζ325ζ3225-ζ3223ζ3213-ζ323-ζ3213+ζ323    orthogonal faithful

Smallest permutation representation of D32
►On 32 points
Generators in S32
(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)

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

D32 is a maximal subgroup of
 SD128
 D32p: D64  D96  D160  D224 ...
 D16p⋊C2: C4○D32  C32⋊C22  C3⋊D32  C5⋊D32  C7⋊D32 ...
D32 is a maximal quotient of
 C32⋊3C4
 D32p: D64  D96  D160  D224 ...
 C8p.D4: D16⋊2C4  SD128  Q128  C3⋊D32  C5⋊D32  C7⋊D32 ...

Matrix representation of D32 ►in GL2(𝔽31) generated by

030
19
,
918
3022
G:=sub<GL(2,GF(31))| [0,1,30,9],[9,30,18,22] >;
 

D32 in GAP, Magma, Sage, TeX 

D_{32}
 
% in TeX
 
G:=Group("D32");
 
// GroupNames label
 
G:=SmallGroup(64,52);
 
// by ID
 
G=gap.SmallGroup(64,52);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,-2,-2,-2,73,218,116,122,579,297,165,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b|a^32=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D32 in TeX
Character table of D32 in TeX

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