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G = C2.D16order 64 = 26

1st central extension by C2 of D16

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

Aliases: D81C4, C8.14D4, C2.1D16, C4.1SD16, C2.1SD32, C22.8D8, (C2×C16)⋊3C2, C8.7(C2×C4), C2.D81C2, (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

C1C8 — C2.D16
C1C2C4C2×C4C2×C8C2×D8 — C2.D16
C1C2C4C8 — C2.D16
C1C22C2×C4C2×C8 — C2.D16
C1C2C2C2C2C4C4C2×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 >

8C2
8C2
4C22
4C22
8C22
8C22
8C4
2D4
2D4
4C23
4D4
4C2×C4
2C16
2D8
2C2×D4
2C4⋊C4

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
ρ11222200-2-2000000-22-22-22-22    orthogonal lifted from D8
ρ122-22-200000022-2-21651631615169ζ165163ζ1615169ζ1651631615169165163ζ1615169    orthogonal lifted from D16
ρ13222200-2-20000002-22-22-22-2    orthogonal lifted from D8
ρ142-22-2000000-2-222ζ16151691651631615169ζ1651631615169165163ζ1615169ζ165163    orthogonal lifted from D16
ρ152-22-2000000-2-2221615169ζ165163ζ1615169165163ζ1615169ζ1651631615169165163    orthogonal lifted from D16
ρ162-22-200000022-2-2ζ165163ζ16151691651631615169165163ζ1615169ζ1651631615169    orthogonal lifted from D16
ρ1722-2-2000000-222-2ζ165163ζ1615169ζ165163ζ1615169ζ16131611ζ16716ζ16131611ζ16716    complex lifted from SD32
ρ182-2-22002-2000000-2-2--2--2-2--2--2-2    complex lifted from SD16
ρ1922-2-2000000-222-2ζ16131611ζ16716ζ16131611ζ16716ζ165163ζ1615169ζ165163ζ1615169    complex lifted from SD32
ρ202-2-22002-2000000--2--2-2-2--2-2-2--2    complex lifted from SD16
ρ2122-2-20000002-2-22ζ16716ζ165163ζ16716ζ165163ζ1615169ζ16131611ζ1615169ζ16131611    complex lifted from SD32
ρ2222-2-20000002-2-22ζ1615169ζ16131611ζ1615169ζ16131611ζ16716ζ165163ζ16716ζ165163    complex lifted from SD32

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

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

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

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

C2.D16 is a maximal subgroup of
C23.24D8  C23.39D8  C23.40D8  C4×D16  C4×SD32  SD323C4  D164C4  D87D4  D88D4  D8.9D4  D8.10D4  D82D4  Q162D4  D8.5D4  Q16.5D4  C167D4  C168D4  C16⋊D4  C162D4  D8⋊Q8  D8.Q8  C22.D16  C23.49D8  C23.19D8  C8.22SD16  C8.12SD16  C8.13SD16  D5.D16
 C2p.D16: D81Q8  C4.4D16  C6.D16  C2.D48  D81Dic3  C40.5D4  D407C4  C10.D16 ...
C2.D16 is a maximal quotient of
C22.SD32  C8.7C42  D162C4  Q322C4  D163C4  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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