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G = D42order 64 = 26

Direct product of D4 and D4

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

Aliases: D42, C243C22, C428C22, C22.39C24, C23.42C23, C2.122+ 1+4, C42(C2×D4), (C4×D4)⋊14C2, C41D47C2, C222(C2×D4), C22≀C25C2, C4⋊D410C2, C4⋊C416C22, (C2×D4)⋊5C22, (C22×D4)⋊8C2, C22⋊C46C22, (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

C1C22 — D42
C1C2C22C23C24C22×D4 — D42
C1C22 — D42
C1C22 — D42
C1C22 — 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 [×2], C2 [×12], C4 [×4], C4 [×5], C22, C22 [×8], C22 [×36], C2×C4, C2×C4 [×6], C2×C4 [×8], D4 [×8], D4 [×26], C23 [×8], C23 [×20], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×4], C2×D4 [×16], C2×D4 [×16], C24 [×4], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4 [×4], D42
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C22×D4 [×2], 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  D47SD16  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: D89D4  D84D4  D1219D4  D1211D4  D2019D4  D2011D4  D2819D4  D2811D4 ...
 C24⋊D2p: D4≀C2  C248D6  C244D10  C243D14 ...
 C8pD4⋊C2: D42SD16  SD16⋊D4  SD161D4  D44D8  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  C244Q8  C24.568C23  C24.269C23  C23.345C24  C23.349C24  C23.352C24  C24.276C23  C23.356C24  C24.282C23  C24.283C23  C23.364C24  C23.372C24  C23.391C24  C4218D4  C4220D4  C427Q8  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  Q164D4
 D4p⋊D4: D89D4  D810D4  D84D4  D85D4  D812D4  D813D4  D811D4  D86D4 ...
 C24⋊D2p: C247D4  C248D4  C248D6  C244D10  C243D14 ...
 C8pD4⋊C2: SD16⋊D4  SD167D4  Q1610D4  SD161D4  SD162D4  Q165D4  SD1611D4  Q1612D4 ...
 D4.pD4⋊C2: SD166D4  SD168D4  Q169D4  SD163D4  SD1610D4  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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