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G = C2×D16  order 64 = 26

Direct product of C2 and D16

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

Aliases: C2×D16, C8.9D4, C4.6D8, C16⋊2C22, D8⋊1C22, C8.6C23, C22.14D8, (C2×C16)⋊5C2, (C2×D8)⋊6C2, C4.7(C2×D4), (C2×C4).81D4, C2.12(C2×D8), (C2×C8).82C22, 2-Sylow(SO-(4,17)), SmallGroup(64,186)

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

Derived series
C1 — C8 — C2×D16
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×D8 — C2×D16
Lower central
C1 — C2 — C4 — C8 — C2×D16
Upper central
C1 — C22 — C2×C4 — C2×C8 — C2×D16
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C2×D16

Generators and relations for C2×D16
 G = < a,b,c | a2=b16=c2=1, ab=ba, ac=ca, cbc=b-1 >

C1
C2
C2
C2
8C2
8C2
8C2
8C2
C4
C4
C22
4C22
4C22
4C22
4C22
8C22
8C22
8C22
8C22
C2×C4
C8
C8
2D4
2D4
2D4
2D4
4D4
4C23
4C23
4D4
C16
D8
D8
C2×C8
D8
C16
D8
2C2×D4
2C2×D4
2D8
2D8
D16
D16
D16
D16
C2×C16
C2×D8
C2×D8
C2×D16

Character table of C2×D16

 class 12A2B2C2D2E2F2G4A4B8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111888822222222222222
ρ11111111111111111111111    trivial
ρ21-1-11-11-11-11-11-11111-1-1-11-1    linear of order 2
ρ31-1-11-111-1-11-11-11-1-1-1111-11    linear of order 2
ρ4111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-111-1-11-11-11-11-1-1-1111-11    linear of order 2
ρ61111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-1-1-1-111111111111111    linear of order 2
ρ81-1-111-11-1-11-11-11111-1-1-11-1    linear of order 2
ρ92-2-220000-222-22-200000000    orthogonal lifted from D4
ρ102222000022-2-2-2-200000000    orthogonal lifted from D4
ρ1122220000-2-20000-√2√2-√2-√2√2-√2√2√2    orthogonal lifted from D8
ρ1222220000-2-20000√2-√2√2√2-√2√2-√2-√2    orthogonal lifted from D8
ρ132-2-2200002-20000-√2√2-√2√2-√2√2√2-√2    orthogonal lifted from D8
ρ142-2-2200002-20000√2-√2√2-√2√2-√2-√2√2    orthogonal lifted from D8
ρ1522-2-2000000√2√2-√2-√2-ζ167+ζ16ζ165-ζ163ζ167-ζ16-ζ167+ζ16-ζ165+ζ163ζ167-ζ16-ζ165+ζ163ζ165-ζ163    orthogonal lifted from D16
ρ162-22-2000000√2-√2-√2√2-ζ165+ζ163-ζ167+ζ16ζ165-ζ163ζ165-ζ163-ζ167+ζ16-ζ165+ζ163ζ167-ζ16ζ167-ζ16    orthogonal lifted from D16
ρ1722-2-2000000-√2-√2√2√2ζ165-ζ163ζ167-ζ16-ζ165+ζ163ζ165-ζ163-ζ167+ζ16-ζ165+ζ163-ζ167+ζ16ζ167-ζ16    orthogonal lifted from D16
ρ182-22-2000000-√2√2√2-√2ζ167-ζ16-ζ165+ζ163-ζ167+ζ16-ζ167+ζ16-ζ165+ζ163ζ167-ζ16ζ165-ζ163ζ165-ζ163    orthogonal lifted from D16
ρ192-22-2000000√2-√2-√2√2ζ165-ζ163ζ167-ζ16-ζ165+ζ163-ζ165+ζ163ζ167-ζ16ζ165-ζ163-ζ167+ζ16-ζ167+ζ16    orthogonal lifted from D16
ρ2022-2-2000000-√2-√2√2√2-ζ165+ζ163-ζ167+ζ16ζ165-ζ163-ζ165+ζ163ζ167-ζ16ζ165-ζ163ζ167-ζ16-ζ167+ζ16    orthogonal lifted from D16
ρ2122-2-2000000√2√2-√2-√2ζ167-ζ16-ζ165+ζ163-ζ167+ζ16ζ167-ζ16ζ165-ζ163-ζ167+ζ16ζ165-ζ163-ζ165+ζ163    orthogonal lifted from D16
ρ222-22-2000000-√2√2√2-√2-ζ167+ζ16ζ165-ζ163ζ167-ζ16ζ167-ζ16ζ165-ζ163-ζ167+ζ16-ζ165+ζ163-ζ165+ζ163    orthogonal lifted from D16

Smallest permutation representation of C2×D16
►On 32 points
Generators in S32
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 17)(16 18)

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

C2×D16 is a maximal subgroup of
 D16⋊2C4  M6(2)⋊C2  D16⋊4C4  D8⋊7D4  D8⋊8D4  Q16.10D4  D8⋊2D4  D8.5D4  C16⋊7D4  C16⋊D4  D4.3D8  C4⋊D16  C8.21D8  C16⋊3D4  C32⋊C22  D4○D16
C2×D16 is a maximal quotient of
 D8⋊7D4  D8⋊2D4  C16⋊7D4  D8⋊1Q8  C22.D16  C4.4D16  C4⋊D16  C16⋊2Q8  C4○D32  C32⋊C22  Q64⋊C2

Matrix representation of C2×D16 ►in GL4(𝔽17) generated by

16000
01600
0010
0001
,
14300
141400
001311
00613
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[14,14,0,0,3,14,0,0,0,0,13,6,0,0,11,13],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;
 

C2×D16 in GAP, Magma, Sage, TeX 

C_2\times D_{16}
 
% in TeX
 
G:=Group("C2xD16");
 
// GroupNames label
 
G:=SmallGroup(64,186);
 
// by ID
 
G=gap.SmallGroup(64,186);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,-2,-2,121,579,297,165,1444,730,88]);
 
// 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^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C2×D16 in TeX
Character table of C2×D16 in TeX

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