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## G = C2×SD32order 64 = 26

### Direct product of C2 and SD32

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

Aliases: C2×SD32, C4.7D8, C8.10D4, C163C22, C8.7C23, Q161C22, 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 >

Character table of C2×SD32

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 1 1 8 8 2 2 8 8 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ7 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 -1 1 -1 1 1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 2 0 0 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 0 -2 2 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 -2 2 0 0 2 -2 0 0 0 0 0 0 -√2 √2 -√2 √2 √2 -√2 √2 -√2 orthogonal lifted from D8 ρ12 2 2 2 2 0 0 -2 -2 0 0 0 0 0 0 -√2 √2 √2 √2 -√2 -√2 -√2 √2 orthogonal lifted from D8 ρ13 2 2 2 2 0 0 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 -√2 √2 √2 √2 -√2 orthogonal lifted from D8 ρ14 2 -2 -2 2 0 0 2 -2 0 0 0 0 0 0 √2 -√2 √2 -√2 -√2 √2 -√2 √2 orthogonal lifted from D8 ρ15 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ1615+ζ169 ζ165+ζ163 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ16 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ165+ζ163 ζ1615+ζ169 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1613+ζ1611 ζ167+ζ16 complex lifted from SD32 ρ17 2 2 -2 -2 0 0 0 0 0 0 -√2 √2 √2 -√2 ζ1613+ζ1611 ζ167+ζ16 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ18 2 -2 2 -2 0 0 0 0 0 0 -√2 -√2 √2 √2 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ167+ζ16 ζ1613+ζ1611 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 complex lifted from SD32 ρ19 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ167+ζ16 ζ165+ζ163 ζ165+ζ163 ζ1613+ζ1611 ζ167+ζ16 ζ1615+ζ169 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32 ρ20 2 2 -2 -2 0 0 0 0 0 0 √2 -√2 -√2 √2 ζ1615+ζ169 ζ1613+ζ1611 ζ1613+ζ1611 ζ165+ζ163 ζ1615+ζ169 ζ167+ζ16 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32 ρ21 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ167+ζ16 ζ165+ζ163 ζ1613+ζ1611 ζ1613+ζ1611 ζ1615+ζ169 ζ1615+ζ169 ζ167+ζ16 ζ165+ζ163 complex lifted from SD32 ρ22 2 -2 2 -2 0 0 0 0 0 0 √2 √2 -√2 -√2 ζ1615+ζ169 ζ1613+ζ1611 ζ165+ζ163 ζ165+ζ163 ζ167+ζ16 ζ167+ζ16 ζ1615+ζ169 ζ1613+ζ1611 complex lifted from SD32

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

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

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

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

C2×SD32 is a maximal subgroup of
SD323C4  Q167D4  D88D4  D8.9D4  D8.10D4  Q16.D4  D8.3D4  Q162D4  D8.4D4  D8.5D4  Q16.5D4  C168D4  C162D4  D4.5D8  C165D4  C8.21D8  C163D4  C8.7D8  D4○SD32
C2×SD32 is a maximal quotient of
Q167D4  D8.9D4  Q162D4  D8.4D4  C168D4  Q16⋊Q8  D8⋊Q8  C23.49D8  C23.50D8  C4.4D16  C4.SD32  C165D4  C163Q8

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

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 16 0 0 0 0 0 11 14 0 0 10 8
,
 16 0 0 0 0 1 0 0 0 0 16 0 0 0 16 1
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

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