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G = C2×D16order 64 = 26

Direct product of C2 and D16

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

Aliases: C2×D16, C8.9D4, C4.6D8, C162C22, D81C22, 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

C1C8 — C2×D16
C1C2C4C8C2×C8C2×D8 — C2×D16
C1C2C4C8 — C2×D16
C1C22C2×C4C2×C8 — C2×D16
C1C2C2C2C2C4C4C8 — C2×D16

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

8C2
8C2
8C2
8C2
4C22
4C22
4C22
4C22
8C22
8C22
8C22
8C22
2D4
2D4
2D4
2D4
4D4
4C23
4C23
4D4
2C2×D4
2C2×D4
2D8
2D8

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
ρ1122220000-2-20000-22-2-22-222    orthogonal lifted from D8
ρ1222220000-2-200002-222-22-2-2    orthogonal lifted from D8
ρ132-2-2200002-20000-22-22-222-2    orthogonal lifted from D8
ρ142-2-2200002-200002-22-22-2-22    orthogonal lifted from D8
ρ1522-2-200000022-2-216716ζ165163ζ1671616716165163ζ16716165163ζ165163    orthogonal lifted from D16
ρ162-22-20000002-2-2216516316716ζ165163ζ16516316716165163ζ16716ζ16716    orthogonal lifted from D16
ρ1722-2-2000000-2-222ζ165163ζ16716165163ζ1651631671616516316716ζ16716    orthogonal lifted from D16
ρ182-22-2000000-222-2ζ167161651631671616716165163ζ16716ζ165163ζ165163    orthogonal lifted from D16
ρ192-22-20000002-2-22ζ165163ζ16716165163165163ζ16716ζ1651631671616716    orthogonal lifted from D16
ρ2022-2-2000000-2-22216516316716ζ165163165163ζ16716ζ165163ζ1671616716    orthogonal lifted from D16
ρ2122-2-200000022-2-2ζ1671616516316716ζ16716ζ16516316716ζ165163165163    orthogonal lifted from D16
ρ222-22-2000000-222-216716ζ165163ζ16716ζ16716ζ16516316716165163165163    orthogonal lifted from D16

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

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

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

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

C2×D16 is a maximal subgroup of
D162C4  M6(2)⋊C2  D164C4  D87D4  D88D4  Q16.10D4  D82D4  D8.5D4  C167D4  C16⋊D4  D4.3D8  C4⋊D16  C8.21D8  C163D4  C32⋊C22  D4○D16
C2×D16 is a maximal quotient of
D87D4  D82D4  C167D4  D81Q8  C22.D16  C4.4D16  C4⋊D16  C162Q8  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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