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G = D42  order 64 = 26

Direct product of D4 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: D42, C24⋊3C22, C42⋊8C22, C22.39C24, C23.42C23, C2.122+ 1+4, C4⋊2(C2×D4), (C4×D4)⋊14C2, C4⋊1D4⋊7C2, C22⋊2(C2×D4), C22≀C2⋊5C2, C4⋊D4⋊10C2, C4⋊C4⋊16C22, (C2×D4)⋊5C22, (C22×D4)⋊8C2, C22⋊C4⋊6C22, (C2×C4).26C23, C2.17(C22×D4), (C22×C4)⋊10C22, 2-Sylow(POmega+(4,7)), SmallGroup(64,226)

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

Derived series
C1 — C22 — D42
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — D42
Lower central
C1 — C22 — D42
Upper central
C1 — C22 — D42
Jennings
C1 — C22 — D42

Generators and relations for D42
 G = < a,b,c,d | a4=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 389 in 214 conjugacy classes, 91 normal (7 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, C4×D4, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, D42
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, D42

Character table of D42

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D4E4F4G4H4I
 size 1111222222224444222244444
ρ11111111111111111111111111    trivial
ρ21111-1-11-1-11-1-11-11-1-11-1111-11-1    linear of order 2
ρ31111-1-1-11-1-11-1-1-1111-11-1111-1-1    linear of order 2
ρ4111111-1-11-1-11-111-1-1-1-1-111-1-11    linear of order 2
ρ5111111-111-1111-11-1-11-11-1-11-1-1    linear of order 2
ρ61111-1-1-1-1-1-1-1-111111111-1-1-1-11    linear of order 2
ρ71111-1-111-111-1-111-1-1-1-1-1-1-1111    linear of order 2
ρ81111111-111-11-1-1111-11-1-1-1-11-1    linear of order 2
ρ911111-1-11-1-1111-1-11-1-1-1-1-11-111    linear of order 2
ρ101111-11-1-11-1-1-111-1-11-11-1-1111-1    linear of order 2
ρ111111-1111111-1-11-11-11-11-11-1-1-1    linear of order 2
ρ1211111-11-1-11-11-1-1-1-11111-111-11    linear of order 2
ρ1311111-111-111111-1-11-11-11-1-1-1-1    linear of order 2
ρ141111-111-111-1-11-1-11-1-1-1-11-11-11    linear of order 2
ρ151111-11-111-11-1-1-1-1-111111-1-111    linear of order 2
ρ1611111-1-1-1-1-1-11-11-11-11-111-111-1    linear of order 2
ρ1722-2-220200-20-2000020-2000000    orthogonal lifted from D4
ρ1822-2-2-20-200202000020-2000000    orthogonal lifted from D4
ρ192-22-20-20220-200000020-200000    orthogonal lifted from D4
ρ202-22-20202-20-2000000-20200000    orthogonal lifted from D4
ρ2122-2-2-20200-2020000-202000000    orthogonal lifted from D4
ρ2222-2-220-20020-20000-202000000    orthogonal lifted from D4
ρ232-22-20-20-2202000000-20200000    orthogonal lifted from D4
ρ242-22-2020-2-20200000020-200000    orthogonal lifted from D4
ρ254-4-44000000000000000000000    orthogonal lifted from 2+ 1+4

Permutation representations of D42
►On 16 points - transitive group 16T109
Generators in S16
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
 
G:=sub<Sym(16)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15), (1,10,6,13)(2,11,7,14)(3,12,8,15)(4,9,5,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15), (1,10,6,13)(2,11,7,14)(3,12,8,15)(4,9,5,16), (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,9)(8,10) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15)], [(1,10,6,13),(2,11,7,14),(3,12,8,15),(4,9,5,16)], [(1,15),(2,16),(3,13),(4,14),(5,11),(6,12),(7,9),(8,10)]])
 
G:=TransitiveGroup(16,109);
 

D42 is a maximal subgroup of
 D4⋊D8  D4.D8  D4⋊7SD16  C42.45C23  C42.473C23  C22.70C25  C22.73C25  C22.77C25  C4⋊2+ 1+4  C22.87C25  C22.102C25  C22.103C25  C22.108C25  C42⋊C23  C22.123C25  C22.126C25  C22.131C25  C22.132C25  C22.135C25  C22.138C25  C22.147C25
 D4p⋊D4: D8⋊9D4  D8⋊4D4  D12⋊19D4  D12⋊11D4  D20⋊19D4  D20⋊11D4  D28⋊19D4  D28⋊11D4 ...
 C24⋊D2p: D4≀C2  C24⋊8D6  C24⋊4D10  C24⋊3D14 ...
 C8⋊pD4⋊C2: D4⋊2SD16  SD16⋊D4  SD16⋊1D4  D4⋊4D8  C42.53C23  C42.474C23 ...
D42 is a maximal quotient of
 C23.240C24  C24.215C23  C24.219C23  C23.308C24  C23.316C24  C23.318C24  C23.322C24  C23.324C24  C23.328C24  C23.333C24  C24⋊4Q8  C24.568C23  C24.269C23  C23.345C24  C23.349C24  C23.352C24  C24.276C23  C23.356C24  C24.282C23  C24.283C23  C23.364C24  C23.372C24  C23.391C24  C42⋊18D4  C42⋊20D4  C42⋊7Q8  C23.455C24  C23.568C24  C23.569C24  C23.570C24  C23.571C24  C23.572C24  C23.573C24  C23.574C24  C24.384C23  C23.576C24  C24.385C23  C23.578C24  C25⋊C22  C23.580C24  C23.581C24  C24.389C23  C23.583C24  Q16⋊4D4
 D4p⋊D4: D8⋊9D4  D8⋊10D4  D8⋊4D4  D8⋊5D4  D8⋊12D4  D8⋊13D4  D8⋊11D4  D8⋊6D4 ...
 C24⋊D2p: C24⋊7D4  C24⋊8D4  C24⋊8D6  C24⋊4D10  C24⋊3D14 ...
 C8⋊pD4⋊C2: SD16⋊D4  SD16⋊7D4  Q16⋊10D4  SD16⋊1D4  SD16⋊2D4  Q16⋊5D4  SD16⋊11D4  Q16⋊12D4 ...
 D4.pD4⋊C2: SD16⋊6D4  SD16⋊8D4  Q16⋊9D4  SD16⋊3D4  SD16⋊10D4  D8.13D4  D8○SD16  D8○Q16 ...

Matrix representation of D42 ►in GL4(ℤ) generated by

-1000
0-100
000-1
0010
,
1000
0100
000-1
00-10
,
0-100
1000
0010
0001
,
0-100
-1000
00-10
000-1
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,0,1,0,0,-1,0],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,-1,0],[0,1,0,0,-1,0,0,0,0,0,1,0,0,0,0,1],[0,-1,0,0,-1,0,0,0,0,0,-1,0,0,0,0,-1] >;
 

D42 in GAP, Magma, Sage, TeX 

D_4^2
 
% in TeX
 
G:=Group("D4^2");
 
// GroupNames label
 
G:=SmallGroup(64,226);
 
// by ID
 
G=gap.SmallGroup(64,226);
 
# by ID
 
G:=PCGroup([6,-2,2,2,2,-2,2,217,650,297]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of D42 in TeX

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