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## G = C2×SD16order 32 = 25

### Direct product of C2 and SD16

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×SD16, C4.7D4, C83C22, C4.2C23, Q81C22, D4.1C22, C22.15D4, (C2×C8)⋊5C2, (C2×Q8)⋊3C2, (C2×D4).6C2, C2.12(C2×D4), (C2×C4).27C22, 2-Sylow(GL(3,3)), SmallGroup(32,40)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×SD16
 Chief series C1 — C2 — C4 — C2×C4 — C2×D4 — C2×SD16
 Lower central C1 — C2 — C4 — C2×SD16
 Upper central C1 — C22 — C2×C4 — C2×SD16
 Jennings C1 — C2 — C2 — C4 — C2×SD16

Generators and relations for C2×SD16
G = < a,b,c | a2=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Character table of C2×SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A 8B 8C 8D size 1 1 1 1 4 4 2 2 4 4 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 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 linear of order 2 ρ7 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 linear of order 2 ρ9 2 -2 2 -2 0 0 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ12 2 -2 -2 2 0 0 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ13 2 2 -2 -2 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ14 2 -2 -2 2 0 0 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16

Permutation representations of C2×SD16
On 16 points - transitive group 16T48
Generators in S16
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)

G:=sub<Sym(16)| (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)>;

G:=Group( (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15) );

G=PermutationGroup([(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15)])

G:=TransitiveGroup(16,48);

Matrix representation of C2×SD16 in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 5 12 0 5 5
,
 1 0 0 0 1 0 0 0 16
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,5,5,0,12,5],[1,0,0,0,1,0,0,0,16] >;

C2×SD16 in GAP, Magma, Sage, TeX

C_2\times {\rm SD}_{16}
% in TeX

G:=Group("C2xSD16");
// GroupNames label

G:=SmallGroup(32,40);
// by ID

G=gap.SmallGroup(32,40);
# by ID

G:=PCGroup([5,-2,2,2,-2,-2,80,101,483,248,58]);
// Polycyclic

G:=Group<a,b,c|a^2=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations

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