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G = C22×D4  order 32 = 25

Direct product of C22 and D4

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

Aliases: C22×D4, C4⋊C23, C24⋊2C2, C22⋊C23, C2.1C24, C23⋊3C22, (C2×C4)⋊4C22, (C22×C4)⋊5C2, 2-Sylow(GO-(4,3)), SmallGroup(32,46)

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

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

Generators and relations for C22×D4
 G = < a,b,c,d | a2=b2=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 158 in 118 conjugacy classes, 78 normal (5 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C22×C4, C2×D4, C24, C22×D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4

Character table of C22×D4

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D
 size 11111111222222222222
ρ111111111111111111111    trivial
ρ211-111-1-1-11-1-11-111-1-111-1    linear of order 2
ρ31-1-1-111-11-1-11111-1-1-1-111    linear of order 2
ρ41-11-11-11-1-11-11-11-111-11-1    linear of order 2
ρ511111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ61-11-11-11-11-11-11-11-11-11-1    linear of order 2
ρ711-111-1-1-1-111-11-1-11-111-1    linear of order 2
ρ81-1-1-111-1111-1-1-1-111-1-111    linear of order 2
ρ9111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ101-11-11-11-1-11-111-11-1-11-11    linear of order 2
ρ1111-111-1-1-11-1-111-1-111-1-11    linear of order 2
ρ121-1-1-111-11-1-111-1-11111-1-1    linear of order 2
ρ1311-111-1-1-1-111-1-111-11-1-11    linear of order 2
ρ141-1-1-111-1111-1-111-1-111-1-1    linear of order 2
ρ1511111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ161-11-11-11-11-11-1-11-11-11-11    linear of order 2
ρ1722-2-2-2-222000000000000    orthogonal lifted from D4
ρ18222-2-22-2-2000000000000    orthogonal lifted from D4
ρ192-222-2-2-22000000000000    orthogonal lifted from D4
ρ202-2-22-222-2000000000000    orthogonal lifted from D4

Permutation representations of C22×D4
►On 16 points - transitive group 16T25
Generators in S16
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)

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

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

(1 14)(2 13)(3 16)(4 15)(5 9)(6 12)(7 11)(8 10)
 
G:=sub<Sym(16)| (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,9)(6,12)(7,11)(8,10)>;
 
G:=Group( (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,9)(6,12)(7,11)(8,10) );
 
G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,15),(10,16),(11,13),(12,14)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14),(2,13),(3,16),(4,15),(5,9),(6,12),(7,11),(8,10)]])
 
G:=TransitiveGroup(16,25);
 

C22×D4 is a maximal subgroup of
 C23.23D4  C24.3C22  C23⋊2D4  C23.10D4  C23.37D4  C22⋊D8  C22⋊SD16  C22.11C24  C23⋊3D4  C22.29C24  D4⋊5D4
C22×D4 is a maximal quotient of
 C22.19C24  C22.26C24  C23⋊3D4  C22.29C24  C23.38C23  C22.31C24  D4⋊5D4  D4⋊6D4  Q8⋊5D4  Q8⋊6D4  D8⋊C22  D4○D8  D4○SD16  Q8○D8

Matrix representation of C22×D4 ►in GL4(ℤ) generated by

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

C22×D4 in GAP, Magma, Sage, TeX 

C_2^2\times D_4
 
% in TeX
 
G:=Group("C2^2xD4");
 
// GroupNames label
 
G:=SmallGroup(32,46);
 
// by ID
 
G=gap.SmallGroup(32,46);
 
# by ID
 
G:=PCGroup([5,-2,2,2,2,-2,181]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^4=d^2=1,a*b=b*a,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 C22×D4 in TeX

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