Copied to
clipboard

## G = SD32order 32 = 25

### Semidihedral group

p-group, metacyclic, nilpotent (class 4), monomial

Aliases: SD32, D8.C2, C162C2, C4.2D4, C2.4D8, Q161C2, C8.3C22, 2-Sylow(GL(2,7)), also known as the quasi-dihedral group QD32, SmallGroup(32,19)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — SD32
 Chief series C1 — C2 — C4 — C8 — D8 — SD32
 Lower central C1 — C2 — C4 — C8 — SD32
 Upper central C1 — C2 — C4 — C8 — SD32
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — SD32

Generators and relations for SD32
G = < a,b | a16=b2=1, bab=a7 >

Character table of SD32

 class 1 2A 2B 4A 4B 8A 8B 16A 16B 16C 16D size 1 1 8 2 8 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 -1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 2 2 0 2 0 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ6 2 2 0 -2 0 0 0 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ7 2 2 0 -2 0 0 0 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ8 2 -2 0 0 0 -√2 √2 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 complex faithful ρ9 2 -2 0 0 0 √2 -√2 ζ167+ζ16 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 complex faithful ρ10 2 -2 0 0 0 -√2 √2 ζ1613+ζ1611 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 complex faithful ρ11 2 -2 0 0 0 √2 -√2 ζ1615+ζ169 ζ165+ζ163 ζ167+ζ16 ζ1613+ζ1611 complex faithful

Permutation representations of SD32
On 16 points - transitive group 16T55
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14)])

G:=TransitiveGroup(16,55);

Matrix representation of SD32 in GL2(𝔽7) generated by

 0 1 1 6
,
 1 6 0 6
G:=sub<GL(2,GF(7))| [0,1,1,6],[1,0,6,6] >;

SD32 in GAP, Magma, Sage, TeX

{\rm SD}_{32}
% in TeX

G:=Group("SD32");
// GroupNames label

G:=SmallGroup(32,19);
// by ID

G=gap.SmallGroup(32,19);
# by ID

G:=PCGroup([5,-2,2,-2,-2,-2,80,61,182,97,102,483,248,58]);
// Polycyclic

G:=Group<a,b|a^16=b^2=1,b*a*b=a^7>;
// generators/relations

׿
×
𝔽