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G = C2×SD16  order 32 = 25

Direct product of C2 and SD16

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

Aliases: C2×SD16, C4.7D4, C8⋊3C22, C4.2C23, Q8⋊1C22, 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 >

C1
C2
C2
C2
4C2
4C2
C4
C4
C22
2C4
2C4
2C22
2C22
4C22
4C22
D4
Q8
C8
Q8
D4
C8
C2×C4
2C2×C4
2D4
2Q8
2C23
SD16
SD16
SD16
SD16
C2×C8
C2×D4
C2×Q8
C2×SD16

Character table of C2×SD16

 class 12A2B2C2D2E4A4B4C4D8A8B8C8D
 size 11114422442222
ρ111111111111111    trivial
ρ21-11-11-1-111-1-1-111    linear of order 2
ρ31111-1-111-1-11111    linear of order 2
ρ41-11-1-11-11-11-1-111    linear of order 2
ρ51111-1-11111-1-1-1-1    linear of order 2
ρ61-11-1-11-111-111-1-1    linear of order 2
ρ711111111-1-1-1-1-1-1    linear of order 2
ρ81-11-11-1-11-1111-1-1    linear of order 2
ρ92-22-2002-2000000    orthogonal lifted from D4
ρ10222200-2-2000000    orthogonal lifted from D4
ρ1122-2-2000000-√-2√-2√-2-√-2    complex lifted from SD16
ρ122-2-22000000-√-2√-2-√-2√-2    complex lifted from SD16
ρ1322-2-2000000√-2-√-2-√-2√-2    complex lifted from SD16
ρ142-2-22000000√-2-√-2√-2-√-2    complex lifted from SD16

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

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 4)(3 7)(6 8)(9 15)(11 13)(12 16)
 
G:=sub<Sym(16)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)>;
 
G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,9), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16) );
 
G=PermutationGroup([(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,9)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16)])
 
G:=TransitiveGroup(16,48);
 

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

1600
0160
0016
,
100
0512
055
,
100
010
0016
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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