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G = C4○D16order 64 = 26

Central product of C4 and D16

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

Aliases: C4D16, C4Q32, C4SD32, D163C2, Q323C2, C8.13D4, C4.20D8, SD323C2, C8.9C23, C22.1D8, C16.6C22, D8.2C22, Q16.2C22, (C2×C16)⋊6C2, C4○D81C2, C2.15(C2×D8), (C2×C4).84D4, C4.10(C2×D4), (C2×C8).89C22, SmallGroup(64,189)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C4○D16
C1C2C4C8C2×C8C4○D8 — C4○D16
C1C2C4C8 — C4○D16
C1C4C2×C4C2×C8 — C4○D16
C1C2C2C2C2C4C4C8 — C4○D16

Generators and relations for C4○D16
 G = < a,b,c | a4=c2=1, b8=a2, ab=ba, ac=ca, cbc=a2b7 >

2C2
8C2
8C2
4C4
4C22
4C22
4C4
2D4
2Q8
2D4
2Q8
4D4
4C2×C4
4D4
4C2×C4
2C4○D4
2C4○D4
2SD16
2SD16

Character table of C4○D16

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D16A16B16C16D16E16F16G16H
 size 1128811288222222222222
ρ11111111111111111111111    trivial
ρ2111-1-1111-1-1111111111111    linear of order 2
ρ311-1-1-1-1-11111-11-1-1-11-11-111    linear of order 2
ρ411-111-1-11-1-11-11-1-1-11-11-111    linear of order 2
ρ511-1-11-1-11-111-11-111-11-11-1-1    linear of order 2
ρ611-11-1-1-111-11-11-111-11-11-1-1    linear of order 2
ρ71111-1111-111111-1-1-1-1-1-1-1-1    linear of order 2
ρ8111-111111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ922-200-2-2200-22-2200000000    orthogonal lifted from D4
ρ102220022200-2-2-2-200000000    orthogonal lifted from D4
ρ1122-20022-2000000-2222-2-2-22    orthogonal lifted from D8
ρ1222200-2-2-20000002-22-2-22-22    orthogonal lifted from D8
ρ1322200-2-2-2000000-22-222-22-2    orthogonal lifted from D8
ρ1422-20022-20000002-2-2-2222-2    orthogonal lifted from D8
ρ152-2000-2i2i0002--2-2-2ζ16716ζ16516316716ζ16131611165163ζ1615169ζ165163ζ16716    complex faithful
ρ162-2000-2i2i000-2-22--2ζ165163ζ1615169ζ165163ζ1671616716ζ16131611ζ16716165163    complex faithful
ρ172-20002i-2i0002-2-2--2ζ16716ζ165163ζ16716ζ16131611ζ165163ζ161516916516316716    complex faithful
ρ182-20002i-2i000-2--22-2ζ165163ζ1615169165163ζ16716ζ16716ζ1613161116716ζ165163    complex faithful
ρ192-2000-2i2i0002--2-2-2ζ1615169ζ16131611ζ16716ζ165163ζ165163ζ1671616516316716    complex faithful
ρ202-2000-2i2i000-2-22--2ζ16131611ζ16716165163ζ1615169ζ16716ζ16516316716ζ165163    complex faithful
ρ212-20002i-2i000-2--22-2ζ16131611ζ16716ζ165163ζ161516916716ζ165163ζ16716165163    complex faithful
ρ222-20002i-2i0002-2-2--2ζ1615169ζ1613161116716ζ165163165163ζ16716ζ165163ζ16716    complex faithful

Smallest permutation representation of C4○D16
On 32 points
Generators in S32
(1 19 9 27)(2 20 10 28)(3 21 11 29)(4 22 12 30)(5 23 13 31)(6 24 14 32)(7 25 15 17)(8 26 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)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)

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

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

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

C4○D16 is a maximal subgroup of
D16.C4  D163C4  D16⋊C22
 D16p⋊C2: C4○D32  C32⋊C22  D487C2  D485C2  D807C2  D805C2  D1127C2  Q323D7 ...
 D2p.D8: D4○D16  D4○SD32  Q8○D16  D163S3  D6.2D8  D163D5  SD323D5  D163D7 ...
 C4p.D8: C8○D16  D165C4  Q64⋊C2  Q16.D6  C40.30C23  C56.30C23 ...
C4○D16 is a maximal quotient of
C23.24D8  C23.25D8  C4×D16  C4×SD32  C4×Q32  D88D4  Q16.5D4  C167D4  C168D4  D8.Q8  Q16.Q8  C23.19D8  C23.20D8  C8.22SD16  C16.5Q8
 D8.D2p: D8.10D4  D8.5D4  D163S3  D6.2D8  Q16.D6  D163D5  SD323D5  C40.30C23 ...
 C8.D4p: C8.21D8  D487C2  D807C2  D1127C2 ...
 C16.D2p: C16.19D4  D485C2  D805C2  Q323D7 ...

Matrix representation of C4○D16 in GL2(𝔽17) generated by

130
013
,
1311
613
,
1311
114
G:=sub<GL(2,GF(17))| [13,0,0,13],[13,6,11,13],[13,11,11,4] >;

C4○D16 in GAP, Magma, Sage, TeX

C_4\circ D_{16}
% in TeX

G:=Group("C4oD16");
// GroupNames label

G:=SmallGroup(64,189);
// by ID

G=gap.SmallGroup(64,189);
# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,121,158,579,297,165,1444,730,88]);
// Polycyclic

G:=Group<a,b,c|a^4=c^2=1,b^8=a^2,a*b=b*a,a*c=c*a,c*b*c=a^2*b^7>;
// generators/relations

Export

Subgroup lattice of C4○D16 in TeX
Character table of C4○D16 in TeX

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