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G = Q₈○M₄(2)  order 64 = 2⁶

Central product of Q₈ and M₄(2)

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

Aliases: Q₈○M₄(2), D₄○M₄(2), C₄.₁₇C₂⁴, C₈.₁₅C₂³, M₄(2)○M₄(2), M₄(2)⋊₁₂C₂², C₈○D₄⋊₈C₂, (C₂×C₈)⋊₉C₂², C₄○D₄.₂C₄, D₄.₉(C₂×C₄), (C₂×Q₈).₈C₄, (C₂×D₄).₁₀C₄, Q₈.₁₀(C₂×C₄), M₄(2)○(C₄○D₄), C₂.₁₂(C₂³×C₄), C₂³.₁₃(C₂×C₄), C₄.₂₃(C₂²×C₄), (C₂×M₄(2))⋊₁₆C₂, (C₂×C₄).₁₃₄C₂³, C₄○D₄.₁₅C₂², C₂².₅(C₂²×C₄), (C₂²×C₄).₇₈C₂², (C₂×C₄).₃₂(C₂×C₄), (C₂×C₄○D₄).₁₁C₂, SmallGroup(64,249)

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

Generators and relations for Q₈○M₄(2)
 G = < a,b,c,d | a⁴=d²=1, b²=c⁴=a², bab^-1=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a²c >

Subgroups: 145 in 129 conjugacy classes, 119 normal (9 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×M₄(2), C₈○D₄, C₂×C₄○D₄, Q₈○M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₂³×C₄, Q₈○M₄(2)

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

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

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

(2 6)(4 8)(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,70);

Q₈○M₄(2) is a maximal subgroup of
 M₄(2).₄₀D₄  M₄(2).₄₁D₄  (C₂×D₄).Q₈  M₄(2).₄₄D₄  M₄(2)⋊₁₉D₄  (C₂×C₈)⋊D₄  M₄(2).₄₆D₄  M₄(2).₄₇D₄  C₄.(C₄×D₄)  (C₂×C₈)⋊₄D₄  M₄(2)⋊₂₁D₄  M₄(2).₅₀D₄  M₄(2).₂₄C₂³  M₄(2).₂₅C₂³  2_-¹⁺⁴⋊₅C₄  M₄(2).₂₉C₂³  M₄(2).₅₁D₄  M₄(2)○D₈  M₄(2).₃₇D₄  M₄(2).₃₈D₄  M₄(2).A₄  Dic₅.C₂⁴  Dic₅.₂₀C₂⁴  Dic₅.₂₂C₂⁴
 C_4p.C₂⁴: C₄.₂₂C₂⁵  D₈⋊C₂³  C₄.C₂⁵  M₄(2)⋊₂₆D₆  M₄(2)⋊₂₈D₆  C₁₂.₇₆C₂⁴  C₄₀.₄₇C₂³  C₂₀.₇₂C₂⁴ ...
Q₈○M₄(2) is a maximal quotient of
 M₄(2)○₂M₄(2)  D₄.₅C₄²  C₂⁴.₇₃(C₂×C₄)  C₄².₂₅₇C₂³  C₄².₂₅₉C₂³  C₄².₂₆₁C₂³  C₄².₂₆₂C₂³  C₄².₆₇₈C₂³  C₄².₂₆₅C₂³  C₄².₂₆₆C₂³  D₄×M₄(2)  C₄².₂₈₇C₂³  M₄(2)⋊₉Q₈  Q₈×M₄(2)  C₄².₂₉₂C₂³  C₄².₂₉₃C₂³  C₄².₂₉₄C₂³  D₄⋊₆M₄(2)  Q₈⋊₆M₄(2)  C₄².₆₉₁C₂³  C₂³⋊₃M₄(2)  C₄².₆₉₃C₂³  C₄².₂₉₇C₂³  C₄².₂₉₉C₂³  C₄².₃₀₀C₂³  C₄².₃₀₁C₂³  C₄².₆₉₅C₂³  C₄².₃₀₂C₂³  C₄².₆₉₆C₂³  C₄².₃₀₅C₂³  C₄².₆₉₇C₂³  C₄².₆₉₈C₂³  C₄².₃₀₇C₂³  C₄².₃₀₈C₂³  C₄².₃₁₀C₂³  Dic₅.C₂⁴  Dic₅.₂₀C₂⁴  Dic₅.₂₂C₂⁴
 M₄(2)⋊D_2p: M₄(2)⋊₂₂D₄  M₄(2)⋊₂₃D₄  M₄(2)⋊₂₆D₆  M₄(2)⋊₂₈D₆  C₄₀.₄₇C₂³  C₂₀.₇₂C₂⁴  C₂₈.₇₀C₂⁴  C₅₆.₄₉C₂³ ...
 C₄○D₄.D_2p: D₄○(C₂²⋊C₈)  C₄².₆₇₄C₂³  C₁₂.₇₆C₂⁴  C₂₀.₇₆C₂⁴  C₂₈.₇₆C₂⁴ ...

34 conjugacy classes

34 irreducible representations

Matrix representation of Q₈○M₄(2) ►in GL₄(𝔽₅) generated by

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

Q₈○M₄(2) in GAP, Magma, Sage, TeX

Q_8\circ M_4(2)

% in TeX

G:=Group("Q8oM4(2)");

// GroupNames label

G:=SmallGroup(64,249);

// by ID

G=gap.SmallGroup(64,249);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,96,332,963,88]);

// Polycyclic

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

// generators/relations