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G = C8○D4  order 32 = 25

Central product of C8 and D4

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8○D4, C8○Q8, D4.C4, Q8.C4, C8○M4(2), C8.7C22, M4(2)⋊5C2, C4.12C23, (C2×C8)⋊7C2, C8○(C4○D4), C4.5(C2×C4), C4○D4.3C2, C22.1(C2×C4), C2.7(C22×C4), (C2×C4).25C22, SmallGroup(32,38)

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

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

Generators and relations for C8○D4
 G = < a,b,c | a8=c2=1, b2=a4, ab=ba, ac=ca, cbc=a4b >

C1
C2
2C2
2C2
2C2
C22
C4
C4
C22
C4
C4
C22
C8
C8
C2×C4
C8
C2×C4
C8
C2×C4
D4
D4
D4
Q8
C2×C8
M4(2)
C2×C8
M4(2)
C4○D4
C2×C8
M4(2)
C8○D4

Character table of C8○D4

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H8I8J
 size 11222112221111222222
ρ111111111111111111111    trivial
ρ2111-1-1111-1-11111-11-1-1-11    linear of order 2
ρ3111-1-1111-1-1-1-1-1-11-1111-1    linear of order 2
ρ41111111111-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-1111-11-1-1-1-1-1-11-1111    linear of order 2
ρ611-11-111-1-11-1-1-1-1111-1-11    linear of order 2
ρ711-11-111-1-111111-1-1-111-1    linear of order 2
ρ811-1-1111-11-111111-11-1-1-1    linear of order 2
ρ9111-11-1-1-1-11-i-iiiii-ii-i-i    linear of order 4
ρ1011-111-1-11-1-1ii-i-i-iiii-i-i    linear of order 4
ρ111111-1-1-1-11-1-i-iii-iii-ii-i    linear of order 4
ρ1211-1-1-1-1-1111ii-i-iii-i-ii-i    linear of order 4
ρ131111-1-1-1-11-1ii-i-ii-i-ii-ii    linear of order 4
ρ1411-1-1-1-1-1111-i-iii-i-iii-ii    linear of order 4
ρ15111-11-1-1-1-11ii-i-i-i-ii-iii    linear of order 4
ρ1611-111-1-11-1-1-i-iiii-i-i-iii    linear of order 4
ρ172-2000-2i2i0002ζ852ζ82ζ832ζ87000000    complex faithful
ρ182-20002i-2i0002ζ832ζ872ζ852ζ8000000    complex faithful
ρ192-2000-2i2i0002ζ82ζ852ζ872ζ83000000    complex faithful
ρ202-20002i-2i0002ζ872ζ832ζ82ζ85000000    complex faithful

Permutation representations of C8○D4
►On 16 points - transitive group 16T16
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

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

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

C8○D4 is a maximal subgroup of
 C8.A4  D4.F5  Q8.F5  C62.(C2×C4)  C12⋊S3.C4  Dic26.C4  D52.C4
 C8.D2p: D4.C8  C8○D8  C8.26D4  D4.3D4  D4.4D4  D4.5D4  C8○D12  D12.C4 ...
 C4p.C23: D4○C16  Q8○M4(2)  D4○D8  D4○SD16  Q8○D8  D4.Dic3  D4.Dic5  Q8.Dic7 ...
C8○D4 is a maximal quotient of
 (C22×C8)⋊C2  C42.6C22  C42.7C22  C8×D4  C8⋊9D4  C8⋊6D4  C8×Q8  C8⋊4Q8  C62.(C2×C4)  C12⋊S3.C4
 C4p.(C2×C4): C8○2M4(2)  C8○D12  D12.C4  D4.Dic3  D20.3C4  D20.2C4  D4.Dic5  D4.F5 ...

Matrix representation of C8○D4 ►in GL2(𝔽17) generated by

80
08
,
01
160
,
01
10
G:=sub<GL(2,GF(17))| [8,0,0,8],[0,16,1,0],[0,1,1,0] >;
 

C8○D4 in GAP, Magma, Sage, TeX 

C_8\circ D_4
 
% in TeX
 
G:=Group("C8oD4");
 
// GroupNames label
 
G:=SmallGroup(32,38);
 
// by ID
 
G=gap.SmallGroup(32,38);
 
# by ID
 
G:=PCGroup([5,-2,2,2,-2,-2,40,157,58]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=c^2=1,b^2=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b>;
 
// generators/relations
 

Export

Subgroup lattice of C8○D4 in TeX
Character table of C8○D4 in TeX

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