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G = C4○D16  order 64 = 26

Central product of C4 and D16

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

Aliases: C4○D16, C4○Q32, C4○SD32, D16⋊3C2, Q32⋊3C2, C8.13D4, C4.20D8, SD32⋊3C2, C8.9C23, C22.1D8, C16.6C22, D8.2C22, Q16.2C22, (C2×C16)⋊6C2, C4○D8⋊1C2, 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

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

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

C1
C2
2C2
8C2
8C2
C4
C22
C4
4C4
4C22
4C22
4C4
C2×C4
C8
C8
2D4
2Q8
2D4
2Q8
4D4
4C2×C4
4D4
4C2×C4
C16
C2×C8
Q16
D8
C16
D8
Q16
2C4○D4
2C4○D4
2SD16
2SD16
SD32
C4○D8
C2×C16
C4○D8
SD32
D16
Q32
C4○D16

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-√2√2√2√2-√2-√2-√2√2    orthogonal lifted from D8
ρ1222200-2-2-2000000√2-√2√2-√2-√2√2-√2√2    orthogonal lifted from D8
ρ1322200-2-2-2000000-√2√2-√2√2√2-√2√2-√2    orthogonal lifted from D8
ρ1422-20022-2000000√2-√2-√2-√2√2√2√2-√2    orthogonal lifted from D8
ρ152-2000-2i2i000√2-√-2-√2√-2ζ167+ζ16ζ165+ζ163-ζ167+ζ16ζ1613+ζ1611-ζ165+ζ163ζ1615+ζ169ζ165-ζ163ζ167-ζ16    complex faithful
ρ162-2000-2i2i000-√2√-2√2-√-2ζ165+ζ163ζ1615+ζ169ζ165-ζ163ζ167+ζ16-ζ167+ζ16ζ1613+ζ1611ζ167-ζ16-ζ165+ζ163    complex faithful
ρ172-20002i-2i000√2√-2-√2-√-2ζ167+ζ16ζ165+ζ163ζ167-ζ16ζ1613+ζ1611ζ165-ζ163ζ1615+ζ169-ζ165+ζ163-ζ167+ζ16    complex faithful
ρ182-20002i-2i000-√2-√-2√2√-2ζ165+ζ163ζ1615+ζ169-ζ165+ζ163ζ167+ζ16ζ167-ζ16ζ1613+ζ1611-ζ167+ζ16ζ165-ζ163    complex faithful
ρ192-2000-2i2i000√2-√-2-√2√-2ζ1615+ζ169ζ1613+ζ1611ζ167-ζ16ζ165+ζ163ζ165-ζ163ζ167+ζ16-ζ165+ζ163-ζ167+ζ16    complex faithful
ρ202-2000-2i2i000-√2√-2√2-√-2ζ1613+ζ1611ζ167+ζ16-ζ165+ζ163ζ1615+ζ169ζ167-ζ16ζ165+ζ163-ζ167+ζ16ζ165-ζ163    complex faithful
ρ212-20002i-2i000-√2-√-2√2√-2ζ1613+ζ1611ζ167+ζ16ζ165-ζ163ζ1615+ζ169-ζ167+ζ16ζ165+ζ163ζ167-ζ16-ζ165+ζ163    complex faithful
ρ222-20002i-2i000√2√-2-√2-√-2ζ1615+ζ169ζ1613+ζ1611-ζ167+ζ16ζ165+ζ163-ζ165+ζ163ζ167+ζ16ζ165-ζ163ζ167-ζ16    complex faithful

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

(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)| (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (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( (1,27,9,19)(2,28,10,20)(3,29,11,21)(4,30,12,22)(5,31,13,23)(6,32,14,24)(7,17,15,25)(8,18,16,26), (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([[(1,27,9,19),(2,28,10,20),(3,29,11,21),(4,30,12,22),(5,31,13,23),(6,32,14,24),(7,17,15,25),(8,18,16,26)], [(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  D16⋊3C4  D16⋊C22
 D16p⋊C2: C4○D32  C32⋊C22  D48⋊7C2  D48⋊5C2  D80⋊7C2  D80⋊5C2  D112⋊7C2  Q32⋊3D7 ...
 D2p.D8: D4○D16  D4○SD32  Q8○D16  D16⋊3S3  D6.2D8  D16⋊3D5  SD32⋊3D5  D16⋊3D7 ...
 C4p.D8: C8○D16  D16⋊5C4  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  D8⋊8D4  Q16.5D4  C16⋊7D4  C16⋊8D4  D8.Q8  Q16.Q8  C23.19D8  C23.20D8  C8.22SD16  C16.5Q8
 D8.D2p: D8.10D4  D8.5D4  D16⋊3S3  D6.2D8  Q16.D6  D16⋊3D5  SD32⋊3D5  C40.30C23 ...
 C8.D4p: C8.21D8  D48⋊7C2  D80⋊7C2  D112⋊7C2 ...
 C16.D2p: C16.19D4  D48⋊5C2  D80⋊5C2  Q32⋊3D7 ...

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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