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## G = C22×D4order 32 = 25

### Direct product of C22 and D4

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

Aliases: C22×D4, C4⋊C23, C242C2, C22⋊C23, C2.1C24, C233C22, (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 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ3 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ8 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ10 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ11 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ12 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ13 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ14 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ16 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ17 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 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  C232D4  C23.10D4  C23.37D4  C22⋊D8  C22⋊SD16  C22.11C24  C233D4  C22.29C24  D45D4
C22×D4 is a maximal quotient of
C22.19C24  C22.26C24  C233D4  C22.29C24  C23.38C23  C22.31C24  D45D4  D46D4  Q85D4  Q86D4  D8⋊C22  D4○D8  D4○SD16  Q8○D8

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

 -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
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

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