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G = D₄○D₈  order 64 = 2⁶

Central product of D₄ and D₈

p-group, metabelian, nilpotent (class 3), monomial

Aliases: D₄○D₈, Q₈○Q₁₆, D₄.₁₁D₄, D₈⋊₇C₂², C₄.₈C₂⁴, C₈.₃C₂³, Q₈.₁₁D₄, Q₁₆⋊₇C₂², D₄.₅C₂³, Q₈.₅C₂³, SD₁₆⋊₄C₂², M₄(2)⋊₆C₂², 2₊¹⁺⁴⋊₃C₂, C₄○D₈⋊₄C₂, C₈○D₄⋊₃C₂, (C₂×D₈)⋊₁₂C₂, C₈⋊C₂²⋊₄C₂, (C₂×C₈)⋊₄C₂², C₄.₄₁(C₂×D₄), C₄○D₄⋊₁C₂², (C₂×D₄)⋊₈C₂², C₂².₅(C₂×D₄), (C₂×C₄).₄₃C₂³, C₂.₃₀(C₂²×D₄), SmallGroup(64,257)

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

Generators and relations for D₄○D₈
 G = < a,b,c,d | a⁴=b²=d²=1, c⁴=a², bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a²c³ >

Subgroups: 237 in 134 conjugacy classes, 79 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, C₄○D₄, C₈○D₄, C₂×D₈, C₄○D₈, C₈⋊C₂², 2₊¹⁺⁴, D₄○D₈
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, D₄○D₈

Character table of D₄○D₈

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

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

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

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

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

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

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

G:=TransitiveGroup(16,80);

D₄○D₈ is a maximal subgroup of
 D₈○SD₁₆  D₈⋊C₂³  Q₁₆.A₄  Q₈.₇S₄
 D₈⋊D_2p: D₈⋊₁₁D₄  D₈○D₈  D₈⋊₁₃D₆  D₈⋊₁₅D₆  D₈⋊₅D₆  D₈⋊₁₃D₁₀  D₈⋊₁₅D₁₀  D₈⋊₅D₁₀ ...
 D₄.D_4p: Q₁₆.₁₀D₄  D₈.₃D₄  D₄○D₁₆  D₄○SD₃₂  D₄.₁₂D₁₂  D₄.₁₂D₂₀  D₄.₁₂D₂₈ ...
 D_4p.C₂³: C₈.C₂⁴  D₁₂.₃₂C₂³  D₂₀.₃₂C₂³  D₂₈.₃₂C₂³ ...
D₄○D₈ is a maximal quotient of
 2₊¹⁺⁴⋊₅C₄  C₄○D₄.₈Q₈  C₄².₂₇₅C₂³  C₄².₂₇₇C₂³  C₄².₂₈₀C₂³  C₄².₁₄C₂³  (C₂×C₈)⋊₁₂D₄  (C₂×C₈)⋊₁₄D₄  C₄².₂₀C₂³  C₄².₂₂C₂³  (C₂×D₄).₃₀₁D₄  (C₂×D₄).₃₀₃D₄  C₄².₃₅₂C₂³  C₄².₃₅₃C₂³  C₄².₃₅₆C₂³  C₄².₃₅₈C₂³  C₄².₃₆₆C₂³  M₄(2)⋊₄Q₈  C₄².₃₈₇C₂³  C₄².₃₈₈C₂³  C₄².₃₉₁C₂³  C₄.2₊¹⁺⁴  C₄.₁₄2₊¹⁺⁴  C₄.₁₈2₊¹⁺⁴  C₄².₄₀₆C₂³  C₄².₄₁₀C₂³  C₄².₄₂₃C₂³  C₄².₄₂₅C₂³  C₄.2_-¹⁺⁴  C₄².₂₆C₂³  C₄².₂₉C₂³  SD₁₆⋊₇D₄  SD₁₆⋊₁D₄  D₄×D₈  Q₁₆⋊₁₃D₄  D₄⋊₇SD₁₆  C₄².₄₆₂C₂³  C₄².₄₇₀C₂³  C₄².₄₄C₂³  C₄².₄₉C₂³  C₄².₅₃C₂³  C₄².₅₄C₂³  C₄².₄₇₁C₂³  C₄².₄₇₄C₂³  C₄².₄₈₂C₂³  D₄⋊₆Q₁₆  C₄².₄₈₈C₂³  C₄².₄₉₀C₂³  C₄².₆₀C₂³  C₄².₆₁C₂³  C₄².₄₉₆C₂³  C₄².₄₉₈C₂³  C₄².₅₀₂C₂³  Q₈⋊₈SD₁₆  C₄².₅₀₇C₂³  C₄².₅₁₁C₂³  C₄².₅₁₆C₂³  D₈⋊₆Q₈  Q₈×Q₁₆  SD₁₆⋊Q₈  D₈⋊₅Q₈  Q₈⋊₅D₈  C₄².₅₃₀C₂³  C₄².₇₄C₂³  C₄².₅₃₃C₂³
 D₈⋊D_2p: D₈⋊₉D₄  D₈⋊₅D₄  D₈⋊₁₂D₄  D₈⋊₁₃D₆  D₈⋊₁₅D₆  D₈⋊₅D₆  D₈⋊₁₃D₁₀  D₈⋊₁₅D₁₀ ...
 M₄(2)⋊D_2p: M₄(2)⋊₁₆D₄  M₄(2)⋊₁₁D₄  D₄.₁₂D₁₂  D₄.₁₂D₂₀  D₄.₁₂D₂₈ ...
 C₄○D₄⋊D_2p: C₄○D₄⋊D₄  (C₂×D₄)⋊₂₁D₄  C₄².₁₈C₂³  D₁₂.₃₂C₂³  D₂₀.₃₂C₂³  D₂₈.₃₂C₂³ ...

Matrix representation of D₄○D₈ ►in GL₄(𝔽₇) generated by

G:=sub<GL(4,GF(7))| [0,5,1,5,6,0,1,2,2,2,5,1,6,1,5,2],[0,2,6,2,4,0,6,5,0,0,4,2,0,0,3,3],[2,1,1,5,0,2,6,5,5,2,6,1,1,1,5,5],[1,0,0,0,0,5,4,6,6,4,4,4,3,5,1,4] >;

D₄○D₈ in GAP, Magma, Sage, TeX

D_4\circ D_8

% in TeX

G:=Group("D4oD8");

// GroupNames label

G:=SmallGroup(64,257);

// by ID

G=gap.SmallGroup(64,257);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,255,1444,730,88]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^4=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^3>;

// generators/relations

Export

Character table of D₄○D₈ in TeX