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G = C₄²⋊C₂  order 32 = 2⁵

1^st semidirect product of C₄² and C₂ acting faithfully

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

Aliases: C₄²⋊₁C₂, C₂².₆C₂³, C₂³.₇C₂², C₄○(C₄⋊C₄), C₄⋊C₄⋊₆C₂, (C₂×C₄)⋊₄C₄, C₄.₉(C₂×C₄), C₄○(C₂²⋊C₄), C₂.₁(C₄○D₄), C₂²⋊C₄.₃C₂, C₂².₅(C₂×C₄), C₂.₃(C₂²×C₄), (C₂²×C₄).₄C₂, (C₂×C₄).₁₀C₂², SmallGroup(32,24)

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

Generators and relations for C₄²⋊C₂
 G = < a,b,c | a⁴=b⁴=c²=1, ab=ba, cac=ab², bc=cb >

Character table of C₄²⋊C₂

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

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

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

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

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

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

G:=TransitiveGroup(16,17);

C₄²⋊C₂ is a maximal subgroup of
 C₄×C₄○D₄  C₂².₁₁C₂⁴  C₂³.₃₂C₂³  C₂³.₃₃C₂³  C₂².₁₉C₂⁴  C₂³.₃₆C₂³  C₂³.₃₇C₂³  C₂².₂₉C₂⁴  C₂².₃₄C₂⁴  C₂².₃₅C₂⁴  C₂².₃₆C₂⁴  C₂³.₄₁C₂³  C₂².₄₅C₂⁴  C₂².₄₆C₂⁴  C₂².₄₇C₂⁴  C₂².₄₉C₂⁴  C₂².₅₀C₂⁴  D₁₀.C₂³  (C₆×C₁₂)⋊₅C₄  D₂₆.C₂³
 C₂³.D_2p: C₄.₉C₄²  C₄²⋊₆C₄  C₂³.₂₄D₄  C₂³.₃₇D₄  C₂³.₃₈D₄  C₄²⋊C₂²  C₂³.₂₅D₄  M₄(2)⋊C₄ ...
 (C₄×C_4p)⋊C₂: C₈○₂M₄(2)  C₄².₇C₂²  C₄²⋊₂S₃  C₄²⋊D₅  C₄²⋊D₇  C₄²⋊D₁₁  C₄²⋊D₁₃ ...
 (C₂×C₄).D_2p: M₄(2)⋊₄C₄  C₂³.C₂³  C₄².₆C₂²  C₂³.₃₈C₂³  C₄⋊C₄⋊₇S₃  C₄⋊C₄⋊₇D₅  C₄⋊C₄⋊₇D₇  C₄⋊C₄⋊₇D₁₁ ...
C₄²⋊C₂ is a maximal quotient of
 C₄²⋊₄C₄  C₄²⋊₅C₄  C₄².₆C₄  D₁₀.C₂³  (C₆×C₁₂)⋊₅C₄  D₂₆.C₂³
 C₂³.D_2p: C₂³.₇Q₈  C₂³.₃₄D₄  C₂³.₁₆D₆  C₂³.₂₆D₆  C₂³.₁₁D₁₀  C₂³.₂₁D₁₀  C₂³.₁₁D₁₄  C₂³.₂₁D₁₄ ...
 (C₄×C_4p)⋊C₂: C₄².₁₂C₄  C₄².₇C₂²  C₄²⋊₂S₃  C₄²⋊D₅  C₄²⋊D₇  C₄²⋊D₁₁  C₄²⋊D₁₃ ...
 (C₂×C₄).D_2p: C₄×C₂²⋊C₄  C₄×C₄⋊C₄  C₄²⋊₈C₄  C₂³.₆₃C₂³  C₂⁴.C₂²  C₄⋊C₄⋊₇S₃  C₄⋊C₄⋊₇D₅  C₄⋊C₄⋊₇D₇ ...

Matrix representation of C₄²⋊C₂ ►in GL₃(𝔽₅) generated by

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

C₄²⋊C₂ in GAP, Magma, Sage, TeX

C_4^2\rtimes C_2

% in TeX

G:=Group("C4^2:C2");

// GroupNames label

G:=SmallGroup(32,24);

// by ID

G=gap.SmallGroup(32,24);

# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,42]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₄²⋊C₂ in TeX
Character table of C₄²⋊C₂ in TeX