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G = SD16  order 16 = 24

Semidihedral group

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

Aliases: SD16, Q8⋊C2, C8⋊2C2, 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 >

C1
C2
4C2
C4
2C4
2C22
Q8
C8
D4
SD16

Character table of SD16

 class 12A2B4A4B8A8B
 size 1142422
ρ11111111    trivial
ρ21111-1-1-1    linear of order 2
ρ311-111-1-1    linear of order 2
ρ411-11-111    linear of order 2
ρ5220-2000    orthogonal lifted from D4
ρ62-2000-√-2√-2    complex faithful
ρ72-2000√-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)  C32⋊2SD16  AΓL1(𝔽9)  C52⋊SD16
 C2p.D4: C4○D8  C8⋊C22  C8.C22  C24⋊C2  D4.S3  Q8⋊2S3  C40⋊C2  D4.D5 ...
SD16 is a maximal quotient of
 C4.Q8  C32⋊2SD16  AΓL1(𝔽9)  C52⋊SD16
 C2p.D4: D4⋊C4  Q8⋊C4  C24⋊C2  D4.S3  Q8⋊2S3  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

01
11
,
11
02
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
 

Export

Subgroup lattice of SD16 in TeX
Character table of SD16 in TeX

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