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G = C16⋊C22order 64 = 26

The semidirect product of C16 and C22 acting faithfully

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

Aliases: C16⋊C22, C8.3D4, D162C2, C4.14D8, SD321C2, D82C22, C22.5D8, M5(2)⋊1C2, C8.10C23, Q162C22, C4○D82C2, (C2×D8)⋊10C2, (C2×C4).47D4, C4.11(C2×D4), C2.16(C2×D8), (C2×C8).23C22, 2-Sylow(PGammaL(2,49)), SmallGroup(64,190)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C16⋊C22
C1C2C4C8C2×C8C2×D8 — C16⋊C22
C1C2C4C8 — C16⋊C22
C1C2C2×C4C2×C8 — C16⋊C22
C1C2C2C2C2C4C4C8 — C16⋊C22

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

2C2
8C2
8C2
8C2
4C4
4C22
4C22
4C22
8C22
8C22
2D4
2D4
2D4
2Q8
4D4
4C23
4D4
4C2×C4
2C4○D4
2SD16
2D8
2C2×D4

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
ρ1122-2000-220000-222-2    orthogonal lifted from D8
ρ1222-2000-2200002-2-22    orthogonal lifted from D8
ρ13222000-2-2000022-2-2    orthogonal lifted from D8
ρ14222000-2-20000-2-222    orthogonal lifted from D8
ρ154-40000000-222200000    orthogonal faithful
ρ164-4000000022-2200000    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  D83D4  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  D4○D16  D4○SD32  C16⋊D6  D80⋊C2  C16⋊D14 ...
C16⋊C22 is a maximal quotient of
C23.39D8  C23.40D8  M5(2)⋊1C4  SD323C4  D164C4  D87D4  D82D4  D8⋊Q8  C4.Q32  D8.Q8  C23.49D8  C23.19D8  C23.51D8  C8.12SD16  C8.13SD16  C16⋊Q8
 C16⋊D2p: C16⋊D4  C162D4  C163D4  C16⋊D6  D8⋊D6  D48⋊C2  D80⋊C2  D16⋊D5 ...
 D8.D2p: D8.9D4  D8.5D4  D8.D6  D8.D10  D8.D14 ...
 Q16⋊D2p: D88D4  Q162D4  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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