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G = C2.D16  order 64 = 26

1st central extension by C2 of D16

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

Aliases: D8⋊1C4, C8.14D4, C2.1D16, C4.1SD16, C2.1SD32, C22.8D8, (C2×C16)⋊3C2, C8.7(C2×C4), C2.D8⋊1C2, (C2×D8).1C2, (C2×C4).60D4, C4.1(C22⋊C4), (C2×C8).69C22, C2.6(D4⋊C4), 2-Sylow(CSO-(4,7)), SmallGroup(64,38)

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

Derived series
C1 — C8 — C2.D16
Chief series
C1 — C2 — C4 — C2×C4 — 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 — C2×C8 — C2.D16

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

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

Character table of C2.D16

 class 12A2B2C2D2E4A4B4C4D8A8B8C8D16A16B16C16D16E16F16G16H
 size 1111882288222222222222
ρ11111111111111111111111    trivial
ρ21111-1-111-1-1111111111111    linear of order 2
ρ311111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-1-111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51-1-111-1-11i-i1-11-1i-i-iiii-i-i    linear of order 4
ρ61-1-11-11-11-ii1-11-1i-i-iiii-i-i    linear of order 4
ρ71-1-111-1-11-ii1-11-1-iii-i-i-iii    linear of order 4
ρ81-1-11-11-11i-i1-11-1-iii-i-i-iii    linear of order 4
ρ92222002200-2-2-2-200000000    orthogonal lifted from D4
ρ102-2-2200-2200-22-2200000000    orthogonal lifted from D4
ρ11222200-2-2000000-√2√2-√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ122-22-2000000√2√2-√2-√2-ζ165+ζ163-ζ1615+ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169    orthogonal lifted from D16
ρ13222200-2-2000000√2-√2√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ142-22-2000000-√2-√2√2√2ζ1615-ζ169-ζ165+ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163    orthogonal lifted from D16
ρ152-22-2000000-√2-√2√2√2-ζ1615+ζ169ζ165-ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163    orthogonal lifted from D16
ρ162-22-2000000√2√2-√2-√2ζ165-ζ163ζ1615-ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169    orthogonal lifted from D16
ρ1722-2-2000000-√2√2√2-√2ζ165+ζ163ζ1615+ζ169ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611ζ167+ζ16    complex lifted from SD32
ρ182-2-22002-2000000√-2√-2-√-2-√-2√-2-√-2-√-2√-2    complex lifted from SD16
ρ1922-2-2000000-√2√2√2-√2ζ1613+ζ1611ζ167+ζ16ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ165+ζ163ζ1615+ζ169    complex lifted from SD32
ρ202-2-22002-2000000-√-2-√-2√-2√-2-√-2√-2√-2-√-2    complex lifted from SD16
ρ2122-2-2000000√2-√2-√2√2ζ167+ζ16ζ165+ζ163ζ167+ζ16ζ165+ζ163ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611    complex lifted from SD32
ρ2222-2-2000000√2-√2-√2√2ζ1615+ζ169ζ1613+ζ1611ζ1615+ζ169ζ1613+ζ1611ζ167+ζ16ζ165+ζ163ζ167+ζ16ζ165+ζ163    complex lifted from SD32

Smallest permutation representation of C2.D16
►On 32 points
Generators in S32
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(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 26 27 16)(2 15 28 25)(3 24 29 14)(4 13 30 23)(5 22 31 12)(6 11 32 21)(7 20 17 10)(8 9 18 19)
 
G:=sub<Sym(32)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19)>;
 
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,17)(8,18)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(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), (1,26,27,16)(2,15,28,25)(3,24,29,14)(4,13,30,23)(5,22,31,12)(6,11,32,21)(7,20,17,10)(8,9,18,19) );
 
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,17),(8,18),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(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)], [(1,26,27,16),(2,15,28,25),(3,24,29,14),(4,13,30,23),(5,22,31,12),(6,11,32,21),(7,20,17,10),(8,9,18,19)]])
 

C2.D16 is a maximal subgroup of
 C23.24D8  C23.39D8  C23.40D8  C4×D16  C4×SD32  SD32⋊3C4  D16⋊4C4  D8⋊7D4  D8⋊8D4  D8.9D4  D8.10D4  D8⋊2D4  Q16⋊2D4  D8.5D4  Q16.5D4  C16⋊7D4  C16⋊8D4  C16⋊D4  C16⋊2D4  D8⋊Q8  D8.Q8  C22.D16  C23.49D8  C23.19D8  C8.22SD16  C8.12SD16  C8.13SD16  D5.D16
 C2p.D16: D8⋊1Q8  C4.4D16  C6.D16  C2.D48  D8⋊1Dic3  C40.5D4  D40⋊7C4  C10.D16 ...
C2.D16 is a maximal quotient of
 C22.SD32  C8.7C42  D16⋊2C4  Q32⋊2C4  D16⋊3C4  D5.D16
 C2p.D16: C4.16D16  C4.D16  C4.10D16  D16.C4  M6(2)⋊C2  C16.18D4  C6.D16  C2.D48 ...

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

16000
01600
00160
00016
,
7200
101000
0071
00167
,
7200
91000
0071
00110
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[7,10,0,0,2,10,0,0,0,0,7,16,0,0,1,7],[7,9,0,0,2,10,0,0,0,0,7,1,0,0,1,10] >;
 

C2.D16 in GAP, Magma, Sage, TeX 

C_2.D_{16}
 
% in TeX
 
G:=Group("C2.D16");
 
// GroupNames label
 
G:=SmallGroup(64,38);
 
// by ID
 
G=gap.SmallGroup(64,38);
 
# by ID
 
G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,362,188,230,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^16=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
 
// generators/relations
 

Export

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

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