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G = SD64  order 64 = 26

Semidihedral group

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

Aliases: SD64, C32⋊2C2, D16.C2, C8.6D4, C4.2D8, Q32⋊1C2, 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

Derived series
C1 — C16 — SD64
Chief series
C1 — C2 — C4 — C8 — C16 — D16 — SD64
Lower central
C1 — C2 — C4 — C8 — C16 — SD64
Upper central
C1 — C2 — C4 — C8 — C16 — SD64
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — SD64

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

C1
C2
16C2
C4
8C22
8C4
C8
4D4
4Q8
C16
2Q16
2D8
Q32
C32
D16
SD64

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-√2√2√2-√2√2√2    orthogonal lifted from D8
ρ722020-2-20000√2√2√2-√2-√2√2-√2-√2    orthogonal lifted from D8
ρ8220-2000-√2√2√2-√2-ζ167+ζ16-ζ167+ζ16ζ167-ζ16ζ165-ζ163ζ165-ζ163ζ167-ζ16-ζ165+ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ9220-2000√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163ζ165-ζ163-ζ167+ζ16-ζ167+ζ16ζ165-ζ163ζ167-ζ16ζ167-ζ16    orthogonal lifted from D16
ρ10220-2000√2-√2-√2√2ζ165-ζ163ζ165-ζ163-ζ165+ζ163ζ167-ζ16ζ167-ζ16-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16    orthogonal lifted from D16
ρ11220-2000-√2√2√2-√2ζ167-ζ16ζ167-ζ16-ζ167+ζ16-ζ165+ζ163-ζ165+ζ163-ζ167+ζ16ζ165-ζ163ζ165-ζ163    orthogonal lifted from D16
ρ122-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322ζ3211+ζ325ζ3227+ζ3221ζ3229+ζ3219ζ3225+ζ3223ζ329+ζ327ζ3213+ζ323ζ3231+ζ3217ζ3215+ζ32    complex faithful
ρ132-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322ζ3229+ζ3219ζ3213+ζ323ζ3227+ζ3221ζ3231+ζ3217ζ3215+ζ32ζ3211+ζ325ζ329+ζ327ζ3225+ζ3223    complex faithful
ρ142-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326ζ3225+ζ3223ζ329+ζ327ζ3231+ζ3217ζ3229+ζ3219ζ3213+ζ323ζ3215+ζ32ζ3227+ζ3221ζ3211+ζ325    complex faithful
ρ152-2000-√2√2ζ3214-ζ322-ζ3210+ζ326ζ3210-ζ326-ζ3214+ζ322ζ3213+ζ323ζ3229+ζ3219ζ3211+ζ325ζ3215+ζ32ζ3231+ζ3217ζ3227+ζ3221ζ3225+ζ3223ζ329+ζ327    complex faithful
ρ162-2000-√2√2-ζ3214+ζ322ζ3210-ζ326-ζ3210+ζ326ζ3214-ζ322ζ3227+ζ3221ζ3211+ζ325ζ3213+ζ323ζ329+ζ327ζ3225+ζ3223ζ3229+ζ3219ζ3215+ζ32ζ3231+ζ3217    complex faithful
ρ172-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326ζ3215+ζ32ζ3231+ζ3217ζ3225+ζ3223ζ3211+ζ325ζ3227+ζ3221ζ329+ζ327ζ3229+ζ3219ζ3213+ζ323    complex faithful
ρ182-2000√2-√2ζ3210-ζ326ζ3214-ζ322-ζ3214+ζ322-ζ3210+ζ326ζ329+ζ327ζ3225+ζ3223ζ3215+ζ32ζ3213+ζ323ζ3229+ζ3219ζ3231+ζ3217ζ3211+ζ325ζ3227+ζ3221    complex faithful
ρ192-2000√2-√2-ζ3210+ζ326-ζ3214+ζ322ζ3214-ζ322ζ3210-ζ326ζ3231+ζ3217ζ3215+ζ32ζ329+ζ327ζ3227+ζ3221ζ3211+ζ325ζ3225+ζ3223ζ3213+ζ323ζ3229+ζ3219    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
 C32⋊4C4
 C16.D2p: D16⋊2C4  Q32⋊2C4  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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