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G = C₄×D₄  order 32 = 2⁵

Direct product of C₄ and D₄

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

Aliases: C₄×D₄, C₄²⋊₄C₂, C₂².₇C₂³, C₂³.₈C₂², C₄⋊C₄⋊₇C₂, C₄⋊₁(C₂×C₄), C₄○₂(C₄⋊C₄), C₂.₃(C₂×D₄), C₂²⋊C₄⋊₆C₂, (C₂²×C₄)⋊₂C₂, C₂²⋊₁(C₂×C₄), (C₂×D₄).₇C₂, C₄○₂(C₂²⋊C₄), C₂.₂(C₄○D₄), C₂.₄(C₂²×C₄), (C₂×C₄).₁₁C₂², (C₂×C₄)○(C₂×D₄), (C₂×C₄)○(C₄⋊C₄), (C₂×C₄)○(C₂²⋊C₄), 2-Sylow(CO(3,5)), SmallGroup(32,25)

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

Generators and relations for C₄×D₄
 G = < a,b,c | a⁴=b⁴=c²=1, ab=ba, ac=ca, cbc=b^-1 >

Character table of C₄×D₄

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

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

(5 16)(6 13)(7 14)(8 15)

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

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14,11,7)(2,15,12,8)(3,16,9,5)(4,13,10,6), (5,16)(6,13)(7,14)(8,15) );

G=PermutationGroup([(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14,11,7),(2,15,12,8),(3,16,9,5),(4,13,10,6)], [(5,16),(6,13),(7,14),(8,15)])

G:=TransitiveGroup(16,19);

C₄×D₄ is a maximal subgroup of
 D₄⋊C₈  C₈⋊₉D₄  C₈⋊₆D₄  SD₁₆⋊C₄  C₄⋊D₈  D₄.D₄  D₄.₂D₄  D₄⋊Q₈  D₄⋊₂Q₈  D₄.Q₈  C₂².₁₉C₂⁴  C₂³.₃₆C₂³  C₂².₂₆C₂⁴  C₂².₃₂C₂⁴  C₂².₃₃C₂⁴  C₂².₃₄C₂⁴  C₂².₃₆C₂⁴  D₄⋊₅D₄  D₄⋊₆D₄  Q₈⋊₅D₄  Q₈⋊₆D₄  C₂².₄₅C₂⁴  C₂².₄₆C₂⁴  C₂².₄₇C₂⁴  D₄⋊₃Q₈  C₂².₄₉C₂⁴  C₂².₅₀C₂⁴  C₂².₅₃C₂⁴
 D_4p⋊C₄: D₈⋊C₄  Dic₃⋊₅D₄  D₂₀⋊₈C₄  D₂₈⋊C₄  D₄₄⋊C₄  D₅₂⋊₈C₄ ...
 D_2p⋊(C₂×C₄): C₂².₁₁C₂⁴  C₂³.₃₃C₂³  Dic₃⋊₄D₄  Dic₅⋊₄D₄  Dic₇⋊₄D₄  Dic₁₁⋊₄D₄  Dic₁₃⋊₄D₄ ...
C₄×D₄ is a maximal quotient of
 C₂³.₆₃C₂³  C₂⁴.C₂²  C₂³.₆₅C₂³  C₂⁴.₃C₂²  C₈⋊₉D₄  C₈⋊₆D₄  SD₁₆⋊C₄  Q₁₆⋊C₄
 D_4p⋊C₄: D₈⋊C₄  C₈○D₈  C₈.₂₆D₄  Dic₃⋊₅D₄  D₂₀⋊₈C₄  D₂₈⋊C₄  D₄₄⋊C₄  D₅₂⋊₈C₄ ...
 C₂³.D_2p: C₂³.₈Q₈  C₂³.₂₃D₄  Dic₃⋊₄D₄  Dic₅⋊₄D₄  Dic₇⋊₄D₄  Dic₁₁⋊₄D₄  Dic₁₃⋊₄D₄ ...

Matrix representation of C₄×D₄ ►in GL₃(𝔽₅) generated by

G:=sub<GL(3,GF(5))| [2,0,0,0,4,0,0,0,4],[1,0,0,0,0,1,0,4,0],[1,0,0,0,1,0,0,0,4] >;

C₄×D₄ in GAP, Magma, Sage, TeX

C_4\times D_4

% in TeX

G:=Group("C4xD4");

// GroupNames label

G:=SmallGroup(32,25);

// by ID

G=gap.SmallGroup(32,25);

# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,72]);

// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₄×D₄ in TeX
Character table of C₄×D₄ in TeX