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G = C16:C22  order 64 = 26

The semidirect product of C16 and C22 acting faithfully

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

Aliases: C16:C22, C8.3D4, D16:2C2, C4.14D8, SD32:1C2, D8:2C22, C22.5D8, M5(2):1C2, C8.10C23, Q16:2C22, C4oD8:2C2, (C2xD8):10C2, (C2xC4).47D4, C4.11(C2xD4), C2.16(C2xD8), (C2xC8).23C22, 2-Sylow(PGammaL(2,49)), SmallGroup(64,190)

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

Derived series
C1 — C8 — C16:C22
Chief series
C1 — C2 — C4 — C8 — C2xC8 — C2xD8 — C16:C22
Lower central
C1 — C2 — C4 — C8 — C16:C22
Upper central
C1 — C2 — C2xC4 — C2xC8 — C16:C22
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C8 — C16:C22

Generators and relations for C16:C22
 G = < a,b,c | a16=b2=c2=1, bab=a7, cac=a9, bc=cb >

Subgroups: 113 in 45 conjugacy classes, 23 normal (15 characteristic)
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, C2xD8, C16:C22
C1
C2
2C2
8C2
8C2
8C2
C4
C22
C4
4C4
4C22
4C22
4C22
8C22
8C22
C8
C2xC4
C8
2D4
2D4
2D4
2Q8
4D4
4C23
4D4
4C2xC4
C16
C16
D8
C2xC8
D8
Q16
D8
2C4oD4
2SD16
2D8
2C2xD4
M5(2)
C2xD8
SD32
D16
SD32
D16
C4oD8
C16:C22

Character table of C16:C22

 class 12A2B2C2D2E4A4B4C8A8B8C16A16B16C16D
 size 1128882282244444
ρ11111111111111111    trivial
ρ211-11-1-11-1111-1-11-11    linear of order 2
ρ3111-11-1111111-1-1-1-1    linear of order 2
ρ411-1-1-111-1111-11-11-1    linear of order 2
ρ511-111-11-1-111-11-11-1    linear of order 2
ρ61111-1111-1111-1-1-1-1    linear of order 2
ρ711-1-1111-1-111-1-11-11    linear of order 2
ρ8111-1-1-111-11111111    linear of order 2
ρ9222000220-2-2-20000    orthogonal lifted from D4
ρ1022-20002-20-2-220000    orthogonal lifted from D4
ρ1122-2000-220000-√2√2√2-√2    orthogonal lifted from D8
ρ1222-2000-220000√2-√2-√2√2    orthogonal lifted from D8
ρ13222000-2-20000√2√2-√2-√2    orthogonal lifted from D8
ρ14222000-2-20000-√2-√2√2√2    orthogonal lifted from D8
ρ154-40000000-2√22√200000    orthogonal faithful
ρ164-400000002√2-2√200000    orthogonal faithful

Permutation representations of C16:C22
►On 16 points - transitive group 16T134
Generators in S16
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)

(1 9)(3 11)(5 13)(7 15)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14), (1,9)(3,11)(5,13)(7,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14), (1,9)(3,11)(5,13)(7,15) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14)], [(1,9),(3,11),(5,13),(7,15)]])
 
G:=TransitiveGroup(16,134);
 

C16:C22 is a maximal subgroup of
 D16:C4
 D8:D2p: D8:D4  D8:3D4  D8:D6  Q16:D6  D16:D5  D8:D10  D8:D14  Q16:D14 ...
 D8.D2p: Q16.10D4  Q16.D4  C8.3D8  D48:C2  D8.D6  C16:D10  D8.D10  D112:C2 ...
 D8p:C22: D16:C22  D4oD16  D4oSD32  C16:D6  D80:C2  C16:D14 ...
C16:C22 is a maximal quotient of
 C23.39D8  C23.40D8  M5(2):1C4  SD32:3C4  D16:4C4  D8:7D4  D8:2D4  D8:Q8  C4.Q32  D8.Q8  C23.49D8  C23.19D8  C23.51D8  C8.12SD16  C8.13SD16  C16:Q8
 C16:D2p: C16:D4  C16:2D4  C16:3D4  C16:D6  D8:D6  D48:C2  D80:C2  D16:D5 ...
 D8.D2p: D8.9D4  D8.5D4  D8.D6  D8.D10  D8.D14 ...
 Q16:D2p: D8:8D4  Q16:2D4  Q16:D6  D8:D10  Q16:D14 ...

Matrix representation of C16:C22 ►in GL4(F7) generated by

0536
5250
1314
5524
,
0401
4104
4416
6305
,
1063
0232
0322
0112
G:=sub<GL(4,GF(7))| [0,5,1,5,5,2,3,5,3,5,1,2,6,0,4,4],[0,4,4,6,4,1,4,3,0,0,1,0,1,4,6,5],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;
 

C16:C22 in GAP, Magma, Sage, TeX 

C_{16}\rtimes C_2^2
 
% in TeX
 
G:=Group("C16:C2^2");
 
// GroupNames label
 
G:=SmallGroup(64,190);
 
// by ID
 
G=gap.SmallGroup(64,190);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,-2,-2,121,650,579,297,165,1444,730,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^2=c^2=1,b*a*b=a^7,c*a*c=a^9,b*c=c*b>;
 
// generators/relations
 

Export

Subgroup lattice of C16:C22 in TeX
Character table of C16:C22 in TeX

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