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G = C2×SD32  order 64 = 26

Direct product of C2 and SD32

direct product, p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2×SD32, C4.7D8, C8.10D4, C16⋊3C22, C8.7C23, Q16⋊1C22, D8.1C22, C22.15D8, (C2×C16)⋊7C2, C4.8(C2×D4), (C2×Q16)⋊6C2, (C2×D8).4C2, C2.13(C2×D8), (C2×C4).82D4, (C2×C8).83C22, 2-Sylow(GL(3,7)), SmallGroup(64,187)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — C2×SD32
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×D8 — C2×SD32
Lower central
C1 — C2 — C4 — C8 — C2×SD32
Upper central
C1 — C22 — C2×C4 — C2×C8 — C2×SD32
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×SD32

Generators and relations for C2×SD32
 G = < a,b,c | a2=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

C1
C2
C2
C2
8C2
8C2
C4
C22
C4
4C4
4C22
4C4
4C22
8C22
8C22
C8
C2×C4
C8
2Q8
2D4
2D4
2Q8
4D4
4C23
4C2×C4
4Q8
C16
C16
D8
C2×C8
Q16
Q16
D8
2C2×Q8
2Q16
2D8
2C2×D4
C2×C16
C2×D8
SD32
SD32
SD32
SD32
C2×Q16
C2×SD32

Character table of C2×SD32

 class 12A2B2C2D2E4A4B4C4D8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111882288222222222222
ρ11111111111111111111111    trivial
ρ21-1-11-11-11-111-11-111-11-11-1-1    linear of order 2
ρ31-1-111-1-111-11-11-111-11-11-1-1    linear of order 2
ρ41111-1-111-1-1111111111111    linear of order 2
ρ511111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ61-1-11-11-111-11-11-1-1-11-11-111    linear of order 2
ρ71-1-111-1-11-111-11-1-1-11-11-111    linear of order 2
ρ81111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ92222002200-2-2-2-200000000    orthogonal lifted from D4
ρ102-2-2200-2200-22-2200000000    orthogonal lifted from D4
ρ112-2-22002-2000000-√2√2-√2√2√2-√2√2-√2    orthogonal lifted from D8
ρ12222200-2-2000000-√2√2√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ13222200-2-2000000√2-√2-√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ142-2-22002-2000000√2-√2√2-√2-√2√2-√2√2    orthogonal lifted from D8
ρ152-22-2000000-√2-√2√2√2ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169ζ1615+ζ169ζ165+ζ163ζ165+ζ163ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ1622-2-2000000-√2√2√2-√2ζ165+ζ163ζ1615+ζ169ζ1615+ζ169ζ167+ζ16ζ165+ζ163ζ1613+ζ1611ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ1722-2-2000000-√2√2√2-√2ζ1613+ζ1611ζ167+ζ16ζ167+ζ16ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ182-22-2000000-√2-√2√2√2ζ165+ζ163ζ1615+ζ169ζ167+ζ16ζ167+ζ16ζ1613+ζ1611ζ1613+ζ1611ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ1922-2-2000000√2-√2-√2√2ζ167+ζ16ζ165+ζ163ζ165+ζ163ζ1613+ζ1611ζ167+ζ16ζ1615+ζ169ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2022-2-2000000√2-√2-√2√2ζ1615+ζ169ζ1613+ζ1611ζ1613+ζ1611ζ165+ζ163ζ1615+ζ169ζ167+ζ16ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ212-22-2000000√2√2-√2-√2ζ167+ζ16ζ165+ζ163ζ1613+ζ1611ζ1613+ζ1611ζ1615+ζ169ζ1615+ζ169ζ167+ζ16ζ165+ζ163    complex lifted from SD32
ρ222-22-2000000√2√2-√2-√2ζ1615+ζ169ζ1613+ζ1611ζ165+ζ163ζ165+ζ163ζ167+ζ16ζ167+ζ16ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32

Smallest permutation representation of C2×SD32
►On 32 points
Generators in S32
(1 32)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)

(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)

(1 24)(2 31)(3 22)(4 29)(5 20)(6 27)(7 18)(8 25)(9 32)(10 23)(11 30)(12 21)(13 28)(14 19)(15 26)(16 17)
 
G:=sub<Sym(32)| (1,32)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,24)(2,31)(3,22)(4,29)(5,20)(6,27)(7,18)(8,25)(9,32)(10,23)(11,30)(12,21)(13,28)(14,19)(15,26)(16,17)>;
 
G:=Group( (1,32)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (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), (1,24)(2,31)(3,22)(4,29)(5,20)(6,27)(7,18)(8,25)(9,32)(10,23)(11,30)(12,21)(13,28)(14,19)(15,26)(16,17) );
 
G=PermutationGroup([[(1,32),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31)], [(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)], [(1,24),(2,31),(3,22),(4,29),(5,20),(6,27),(7,18),(8,25),(9,32),(10,23),(11,30),(12,21),(13,28),(14,19),(15,26),(16,17)]])
 

C2×SD32 is a maximal subgroup of
 SD32⋊3C4  Q16⋊7D4  D8⋊8D4  D8.9D4  D8.10D4  Q16.D4  D8.3D4  Q16⋊2D4  D8.4D4  D8.5D4  Q16.5D4  C16⋊8D4  C16⋊2D4  D4.5D8  C16⋊5D4  C8.21D8  C16⋊3D4  C8.7D8  D4○SD32
C2×SD32 is a maximal quotient of
 Q16⋊7D4  D8.9D4  Q16⋊2D4  D8.4D4  C16⋊8D4  Q16⋊Q8  D8⋊Q8  C23.49D8  C23.50D8  C4.4D16  C4.SD32  C16⋊5D4  C16⋊3Q8

Matrix representation of C2×SD32 ►in GL4(𝔽17) generated by

16000
01600
0010
0001
,
0100
16000
001114
00108
,
16000
0100
00160
00161
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[0,16,0,0,1,0,0,0,0,0,11,10,0,0,14,8],[16,0,0,0,0,1,0,0,0,0,16,16,0,0,0,1] >;
 

C2×SD32 in GAP, Magma, Sage, TeX 

C_2\times {\rm SD}_{32}
 
% in TeX
 
G:=Group("C2xSD32");
 
// GroupNames label
 
G:=SmallGroup(64,187);
 
// by ID
 
G=gap.SmallGroup(64,187);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,-2,-2,192,121,579,297,165,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
 
// generators/relations
 

Export

Subgroup lattice of C2×SD32 in TeX
Character table of C2×SD32 in TeX

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