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G = SD64order 64 = 26

Semidihedral group

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

Aliases: SD64, C322C2, D16.C2, C8.6D4, C4.2D8, Q321C2, C2.4D16, C16.3C22, 2-Sylow(GL(2,47)), also known as the quasi-dihedral group QD64, SmallGroup(64,53)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — SD64
C1C2C4C8C16D16 — SD64
C1C2C4C8C16 — SD64
C1C2C4C8C16 — SD64
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — SD64

Generators and relations for SD64
 G = < a,b | a32=b2=1, bab=a15 >

16C2
8C22
8C4
4D4
4Q8
2Q16
2D8

Character table of SD64

 class 12A2B4A4B8A8B16A16B16C16D32A32B32C32D32E32F32G32H
 size 111621622222222222222
ρ11111111111111111111    trivial
ρ211-11-111111111111111    linear of order 2
ρ31111-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ411-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ52202022-2-2-2-200000000    orthogonal lifted from D4
ρ622020-2-20000-2-2-222-222    orthogonal lifted from D8
ρ722020-2-20000222-2-22-2-2    orthogonal lifted from D8
ρ8220-2000-222-21671616716ζ16716ζ165163ζ165163ζ16716165163165163    orthogonal lifted from D16
ρ9220-20002-2-22165163165163ζ1651631671616716ζ165163ζ16716ζ16716    orthogonal lifted from D16
ρ10220-20002-2-22ζ165163ζ165163165163ζ16716ζ167161651631671616716    orthogonal lifted from D16
ρ11220-2000-222-2ζ16716ζ167161671616516316516316716ζ165163ζ165163    orthogonal lifted from D16
ρ122-2000-223214322ζ32103263210326ζ3214322ζ3211325ζ32273221ζ32293219ζ32253223ζ329327ζ3213323ζ32313217ζ321532    complex faithful
ρ132-2000-22ζ32143223210326ζ32103263214322ζ32293219ζ3213323ζ32273221ζ32313217ζ321532ζ3211325ζ329327ζ32253223    complex faithful
ρ142-20002-2ζ3210326ζ321432232143223210326ζ32253223ζ329327ζ32313217ζ32293219ζ3213323ζ321532ζ32273221ζ3211325    complex faithful
ρ152-2000-22ζ32143223210326ζ32103263214322ζ3213323ζ32293219ζ3211325ζ321532ζ32313217ζ32273221ζ32253223ζ329327    complex faithful
ρ162-2000-223214322ζ32103263210326ζ3214322ζ32273221ζ3211325ζ3213323ζ329327ζ32253223ζ32293219ζ321532ζ32313217    complex faithful
ρ172-20002-232103263214322ζ3214322ζ3210326ζ321532ζ32313217ζ32253223ζ3211325ζ32273221ζ329327ζ32293219ζ3213323    complex faithful
ρ182-20002-2ζ3210326ζ321432232143223210326ζ329327ζ32253223ζ321532ζ3213323ζ32293219ζ32313217ζ3211325ζ32273221    complex faithful
ρ192-20002-232103263214322ζ3214322ζ3210326ζ32313217ζ321532ζ329327ζ32273221ζ3211325ζ32253223ζ3213323ζ32293219    complex faithful

Smallest permutation representation of SD64
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 16)(3 31)(4 14)(5 29)(6 12)(7 27)(8 10)(9 25)(11 23)(13 21)(15 19)(18 32)(20 30)(22 28)(24 26)

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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26)>;

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), (2,16)(3,31)(4,14)(5,29)(6,12)(7,27)(8,10)(9,25)(11,23)(13,21)(15,19)(18,32)(20,30)(22,28)(24,26) );

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)], [(2,16),(3,31),(4,14),(5,29),(6,12),(7,27),(8,10),(9,25),(11,23),(13,21),(15,19),(18,32),(20,30),(22,28),(24,26)])

SD64 is a maximal subgroup of
 C4p.D8: C4○D32  C32⋊C22  Q64⋊C2  C32⋊S3  D16.S3  C3⋊SD64  C160⋊C2  D16.D5 ...
SD64 is a maximal quotient of
C324C4
 C16.D2p: D162C4  Q322C4  C32⋊S3  D16.S3  C3⋊SD64  C160⋊C2  D16.D5  C5⋊SD64 ...

Matrix representation of SD64 in GL2(𝔽47) generated by

01
143
,
143
046
G:=sub<GL(2,GF(47))| [0,1,1,43],[1,0,43,46] >;

SD64 in GAP, Magma, Sage, TeX

{\rm SD}_{64}
% in TeX

G:=Group("SD64");
// GroupNames label

G:=SmallGroup(64,53);
// by ID

G=gap.SmallGroup(64,53);
# by ID

G:=PCGroup([6,-2,2,-2,-2,-2,-2,192,73,218,116,122,579,297,165,1444,730,88]);
// Polycyclic

G:=Group<a,b|a^32=b^2=1,b*a*b=a^15>;
// generators/relations

Export

Subgroup lattice of SD64 in TeX
Character table of SD64 in TeX

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