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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

C1C8 — SD32
C1C2C4C8D8 — SD32
C1C2C4C8 — SD32
C1C2C4C8 — SD32
C1C2C2C2C2C4C4C8 — SD32

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

8C2
4C4
4C22
2Q8
2D4

Character table of SD32

 class 12A2B4A4B8A8B16A16B16C16D
 size 11828222222
ρ111111111111    trivial
ρ21111-111-1-1-1-1    linear of order 2
ρ311-11-1111111    linear of order 2
ρ411-11111-1-1-1-1    linear of order 2
ρ522020-2-20000    orthogonal lifted from D4
ρ6220-20002-22-2    orthogonal lifted from D8
ρ7220-2000-22-22    orthogonal lifted from D8
ρ82-2000-22ζ165163ζ16716ζ16131611ζ1615169    complex faithful
ρ92-20002-2ζ16716ζ16131611ζ1615169ζ165163    complex faithful
ρ102-2000-22ζ16131611ζ1615169ζ165163ζ16716    complex faithful
ρ112-20002-2ζ1615169ζ165163ζ16716ζ16131611    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

01
16
,
16
06
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

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