Copied to
clipboard

G = C8○D8  order 64 = 26

Central product of C8 and D8

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C8○D8, C8○Q16, D8⋊5C4, C8○SD16, Q16⋊5C4, C8.30D4, SD16⋊3C4, C42.74C22, M4(2).11C22, C8○C4≀C2, C4≀C2⋊7C2, (C4×C8)⋊10C2, C8○(C4○D8), C8○D4⋊6C2, C8.11(C2×C4), C4○D8.5C2, D4.3(C2×C4), C2.18(C4×D4), C4.79(C2×D4), Q8.3(C2×C4), C8○(C8.C4), C8.C4⋊8C2, C4.15(C22×C4), (C2×C4).79C23, C4○D4.7C22, (C2×C8).102C22, C22.1(C4○D4), 2-Sylow(CU(2,3)), SmallGroup(64,124)

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

Derived series
C1 — C4 — C8○D8
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C8○D4 — C8○D8
Lower central
C1 — C2 — C4 — C8○D8
Upper central
C1 — C8 — C2×C8 — C8○D8
Jennings
C1 — C2 — C2 — C2×C4 — C8○D8

Generators and relations for C8○D8
 G = < a,b,c | a8=c2=1, b4=a4, ab=ba, ac=ca, cbc=a4b3 >

C1
C2
2C2
4C2
4C2
C4
C22
C4
2C4
2C4
2C4
2C4
2C22
2C22
Q8
D4
C2×C4
C8
Q8
C8
C8
C8
D4
2C2×C4
2D4
2C2×C4
2D4
2C8
2C8
2C2×C4
SD16
C42
D8
Q16
SD16
C2×C8
C2×C8
M4(2)
M4(2)
C4○D4
C4○D4
2M4(2)
2M4(2)
2C2×C8
2C2×C8
C4≀C2
C4≀C2
C8.C4
C8○D4
C4×C8
C8○D4
C4○D8
C8○D8

Character table of C8○D8

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C8D8E8F8G8H8I8J8K8L8M8N
 size 1124411222224411112222224444
ρ11111111111111111111111111111    trivial
ρ21111-111111111-1-1-1-1-1-1-1-1-1-1-1-11-11    linear of order 2
ρ3111-1-11111111-1-11111111111-1-1-1-1    linear of order 2
ρ4111-111111111-11-1-1-1-1-1-1-1-1-1-11-11-1    linear of order 2
ρ5111-1111-1-11-1-1-111111-1-11-11-1-11-11    linear of order 2
ρ6111-1-111-1-11-1-1-1-1-1-1-1-111-11-111111    linear of order 2
ρ71111111-1-11-1-111-1-1-1-111-11-11-1-1-1-1    linear of order 2
ρ81111-111-1-11-1-11-11111-1-11-11-11-11-1    linear of order 2
ρ911-1-1-1-1-1-ii1-ii11i-i-ii-1-1-i1i1ii-i-i    linear of order 4
ρ1011-1-11-1-1-ii1-ii1-1-iii-i11i-1-i-1-iii-i    linear of order 4
ρ1111-11-1-1-1-ii1-ii-11-iii-i11i-1-i-1i-i-ii    linear of order 4
ρ1211-111-1-1-ii1-ii-1-1i-i-ii-1-1-i1i1-i-iii    linear of order 4
ρ1311-111-1-1i-i1i-i-1-1-iii-i-1-1i1-i1ii-i-i    linear of order 4
ρ1411-11-1-1-1i-i1i-i-11i-i-ii11-i-1i-1-iii-i    linear of order 4
ρ1511-1-11-1-1i-i1i-i1-1i-i-ii11-i-1i-1i-i-ii    linear of order 4
ρ1611-1-1-1-1-1i-i1i-i11-iii-i-1-1i1-i1-i-iii    linear of order 4
ρ1722-2002200-20000222200-20-200000    orthogonal lifted from D4
ρ1822-2002200-20000-2-2-2-20020200000    orthogonal lifted from D4
ρ1922200-2-200-200002i-2i-2i2i002i0-2i00000    complex lifted from C4○D4
ρ2022200-2-200-20000-2i2i2i-2i00-2i02i00000    complex lifted from C4○D4
ρ212-2000-2i2i-1-i1-i01+i-1+i002ζ832ζ852ζ82ζ87√-2-√-20-√20√20000    complex faithful
ρ222-2000-2i2i1+i-1+i0-1-i1-i002ζ872ζ82ζ852ζ83√-2-√-20-√20√20000    complex faithful
ρ232-20002i-2i-1+i1+i01-i-1-i002ζ852ζ832ζ872ζ8-√-2√-20-√20√20000    complex faithful
ρ242-20002i-2i1-i-1-i0-1+i1+i002ζ852ζ832ζ872ζ8√-2-√-20√20-√20000    complex faithful
ρ252-20002i-2i1-i-1-i0-1+i1+i002ζ82ζ872ζ832ζ85-√-2√-20-√20√20000    complex faithful
ρ262-20002i-2i-1+i1+i01-i-1-i002ζ82ζ872ζ832ζ85√-2-√-20√20-√20000    complex faithful
ρ272-2000-2i2i1+i-1+i0-1-i1-i002ζ832ζ852ζ82ζ87-√-2√-20√20-√20000    complex faithful
ρ282-2000-2i2i-1-i1-i01+i-1+i002ζ872ζ82ζ852ζ83-√-2√-20√20-√20000    complex faithful

Permutation representations of C8○D8
►On 16 points - transitive group 16T114
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 6 3 8 5 2 7 4)(9 12 15 10 13 16 11 14)

(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6,3,8,5,2,7,4)(9,12,15,10,13,16,11,14), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,6,3,8,5,2,7,4)(9,12,15,10,13,16,11,14), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,6,3,8,5,2,7,4),(9,12,15,10,13,16,11,14)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14)]])
 
G:=TransitiveGroup(16,114);
 

C8○D8 is a maximal subgroup of
 C42.283C23  M4(2).51D4  D8○SD16  D8⋊6D4  D8○D8  D8○Q16  CU2(𝔽3)  SD16⋊3F5
 D8p⋊C4: C8○D16  D16⋊5C4  D24⋊11C4  D24⋊7C4  D40⋊17C4  D40⋊13C4  Q16⋊5F5  D56⋊11C4 ...
 C42.D2p: C8≀C2  C8.32D8  C8.3D8  D8⋊3D4  C8.5D8  D8⋊3Q8  D8.2Q8  C42.196D6 ...
 C8p.D4: C16○D8  D8.C8  C24.100D4  C40.93D4  C56.93D4 ...
 (Cp×D8)⋊C4: M4(2)○D8  D8⋊5Dic3  D8⋊5Dic5  D8⋊5F5  D8⋊5Dic7 ...
C8○D8 is a maximal quotient of
 C8×D8  C8×SD16  C8×Q16  D4.M4(2)  D4⋊2M4(2)  Q8.M4(2)  Q8⋊2M4(2)  C8⋊6D8  C8⋊9SD16  C8⋊6Q16  M4(2).3Q8  C42.62Q8
 C42.D2p: C42.428D4  C42.326D4  D24⋊11C4  C42.196D6  D40⋊17C4  C42.196D10  D56⋊11C4  C42.196D14 ...
 M4(2).D2p: M4(2).42D4  M4(2).43D4  M4(2).24D4  D24⋊7C4  C24.100D4  D40⋊13C4  C40.93D4  D56⋊7C4 ...
 C2p.(C4×D4): Q8.C42  C8.14C42  D8⋊5Dic3  D8⋊5Dic5  D8⋊5F5  SD16⋊3F5  Q16⋊5F5  D8⋊5Dic7 ...

Matrix representation of C8○D8 ►in GL2(𝔽17) generated by

20
02
,
20
09
,
01
10
G:=sub<GL(2,GF(17))| [2,0,0,2],[2,0,0,9],[0,1,1,0] >;
 

C8○D8 in GAP, Magma, Sage, TeX 

C_8\circ D_8
 
% in TeX
 
G:=Group("C8oD8");
 
// GroupNames label
 
G:=SmallGroup(64,124);
 
// by ID
 
G=gap.SmallGroup(64,124);
 
# by ID
 
G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,86,963,489,117,88]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=c^2=1,b^4=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C8○D8 in TeX
Character table of C8○D8 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁