Copied to
clipboard

G = D32order 64 = 26

Dihedral group

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

Aliases: D32, C321C2, C4.1D8, C8.5D4, D161C2, 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

C1C16 — D32
C1C2C4C8C16D16 — D32
C1C2C4C8C16 — D32
C1C2C4C8C16 — D32
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — D32

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

16C2
16C2
8C22
8C22
4D4
4D4
2D8
2D8

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
ρ622002-2-20000-2-2-222-222    orthogonal lifted from D8
ρ722002-2-20000222-2-22-2-2    orthogonal lifted from D8
ρ82200-2002-2-22165163165163ζ165163ζ1615169ζ1615169ζ16516316151691615169    orthogonal lifted from D16
ρ92200-2002-2-22ζ165163ζ16516316516316151691615169165163ζ1615169ζ1615169    orthogonal lifted from D16
ρ102200-200-222-216151691615169ζ1615169165163165163ζ1615169ζ165163ζ165163    orthogonal lifted from D16
ρ112200-200-222-2ζ1615169ζ16151691615169ζ165163ζ1651631615169165163165163    orthogonal lifted from D16
ρ122-2000-22ζ32143223210326ζ32103263214322ζ32113253211325ζ321332332253223ζ322532233213323321532ζ321532    orthogonal faithful
ρ132-20002-2ζ3210326ζ321432232143223210326321532ζ321532ζ322532233211325ζ3211325322532233213323ζ3213323    orthogonal faithful
ρ142-20002-232103263214322ζ3214322ζ321032632253223ζ32253223321532ζ32133233213323ζ3215323211325ζ3211325    orthogonal faithful
ρ152-2000-223214322ζ32103263210326ζ3214322ζ321332332133233211325321532ζ321532ζ3211325ζ3225322332253223    orthogonal faithful
ρ162-2000-22ζ32143223210326ζ321032632143223211325ζ32113253213323ζ3225322332253223ζ3213323ζ321532321532    orthogonal faithful
ρ172-20002-232103263214322ζ3214322ζ3210326ζ3225322332253223ζ3215323213323ζ3213323321532ζ32113253211325    orthogonal faithful
ρ182-2000-223214322ζ32103263210326ζ32143223213323ζ3213323ζ3211325ζ321532321532321132532253223ζ32253223    orthogonal faithful
ρ192-20002-2ζ3210326ζ321432232143223210326ζ32153232153232253223ζ32113253211325ζ32253223ζ32133233213323    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
C323C4
 D32p: D64  D96  D160  D224 ...
 C8p.D4: D162C4  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

׿
×
𝔽