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## G = SD16order 16 = 24

### Semidihedral group

p-group, metacyclic, nilpotent (class 3), monomial

Aliases: SD16, Q8⋊C2, C82C2, D4.C2, C2.4D4, C4.2C22, 2-Sylow(GL(2,3)), also known as the quasi-dihedral group QD16, SmallGroup(16,8)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — SD16
 Chief series C1 — C2 — C4 — D4 — SD16
 Lower central C1 — C2 — C4 — SD16
 Upper central C1 — C2 — C4 — SD16
 Jennings C1 — C2 — C2 — C4 — SD16

Generators and relations for SD16
G = < a,b | a8=b2=1, bab=a3 >

Character table of SD16

 class 1 2A 2B 4A 4B 8A 8B size 1 1 4 2 4 2 2 ρ1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 linear of order 2 ρ5 2 2 0 -2 0 0 0 orthogonal lifted from D4 ρ6 2 -2 0 0 0 -√-2 √-2 complex faithful ρ7 2 -2 0 0 0 √-2 -√-2 complex faithful

Permutation representations of SD16
On 8 points - transitive group 8T8
Generators in S8
(1 2 3 4 5 6 7 8)
(2 4)(3 7)(6 8)

G:=sub<Sym(8)| (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8)>;

G:=Group( (1,2,3,4,5,6,7,8), (2,4)(3,7)(6,8) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8)], [(2,4),(3,7),(6,8)]])

G:=TransitiveGroup(8,8);

Regular action on 16 points - transitive group 16T12
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)

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

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

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

G:=TransitiveGroup(16,12);

SD16 is a maximal subgroup of
GL2(𝔽3)  C322SD16  AΓL1(𝔽9)  C52⋊SD16
C2p.D4: C4○D8  C8⋊C22  C8.C22  C24⋊C2  D4.S3  Q82S3  C40⋊C2  D4.D5 ...
SD16 is a maximal quotient of
C4.Q8  C322SD16  AΓL1(𝔽9)  C52⋊SD16
C2p.D4: D4⋊C4  Q8⋊C4  C24⋊C2  D4.S3  Q82S3  C40⋊C2  D4.D5  Q8⋊D5 ...

Polynomial with Galois group SD16 over ℚ
actionf(x)Disc(f)
8T8x8-14x6-10x5+31x4+15x3-14x2-5x+156·712·1013

Matrix representation of SD16 in GL2(𝔽3) generated by

 0 1 1 1
,
 1 1 0 2
G:=sub<GL(2,GF(3))| [0,1,1,1],[1,0,1,2] >;

SD16 in GAP, Magma, Sage, TeX

{\rm SD}_{16}
% in TeX

G:=Group("SD16");
// GroupNames label

G:=SmallGroup(16,8);
// by ID

G=gap.SmallGroup(16,8);
# by ID

G:=PCGroup([4,-2,2,-2,-2,32,49,146,78,34]);
// Polycyclic

G:=Group<a,b|a^8=b^2=1,b*a*b=a^3>;
// generators/relations

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