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G = C₈.₂₆D₄  order 64 = 2⁶

13^rd non-split extension by C₈ of D₄ acting via D₄/C₂²=C₂

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

Aliases: D₈⋊₄C₄, Q₁₆⋊₄C₄, C₈.₂₆D₄, SD₁₆⋊₂C₄, C₄².₁₂C₂², M₄(2).₁₂C₂², C₄≀C₂⋊₆C₂, C₈○D₄⋊₇C₂, C₈.₆(C₂×C₄), C₈⋊C₄⋊₃C₂, C₄○D₈.₃C₂, D₄.₄(C₂×C₄), C₂.₁₉(C₄×D₄), C₄.₈₀(C₂×D₄), Q₈.₄(C₂×C₄), C₈.C₄⋊₄C₂, (C₂×C₄).₈₀C₂³, (C₂×C₈).₅₁C₂², C₄.₁₆(C₂²×C₄), C₄○D₄.₈C₂², C₂².₂(C₄○D₄), SmallGroup(64,125)

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

Generators and relations for C₈.₂₆D₄
 G = < a,b,c | a⁸=b⁴=1, c²=a², bab^-1=cac^-1=a⁵, cbc^-1=a²b^-1 >

Character table of C₈.₂₆D₄

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

(2 6)(4 8)(9 11 13 15)(10 16 14 12)

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

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

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

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

G:=TransitiveGroup(16,113);

C₈.₂₆D₄ is a maximal subgroup of
 C₄².₂₈₃C₂³  D₈⋊₁₁D₄  D₈.₁₃D₄  C₈.₅S₄
 D_8p⋊C₄: D₁₆⋊C₄  D₂₄⋊₄C₄  D₂₄⋊₁₀C₄  D₄₀⋊₁₀C₄  D₄₀⋊₁₆C₄  Q₁₆⋊F₅  D₅₆⋊₄C₄  D₅₆⋊₁₀C₄ ...
 M₄(2).D_2p: M₄(2).₅₁D₄  M₄(2)○D₈  M₄(2).₂₂D₆  C₂₄.₅₄D₄  M₄(2).₂₂D₁₀  C₄₀.₅₀D₄  M₄(2).₂₂D₁₄  C₅₆.₅₀D₄ ...
 C_8p.(C₂×C₄): Q₃₂⋊C₄  D₈⋊₄Dic₃  D₈⋊₄Dic₅  D₈⋊F₅  SD₁₆⋊₂F₅  D₈⋊₄Dic₇ ...
C₈.₂₆D₄ is a maximal quotient of
 SD₁₆⋊C₈  Q₁₆⋊₅C₈  D₈⋊₅C₈  C₈⋊₉D₈  C₈⋊₁₂SD₁₆  C₈⋊₁₅SD₁₆  C₈⋊₉Q₁₆  C₈⋊M₄(2)  C₈.M₄(2)  C₈⋊₃M₄(2)  D₄.₃C₄²  C₈.₅C₄²  M₄(2).₃Q₈  C₄².₂₈Q₈  SD₁₆⋊₂F₅  Q₁₆⋊F₅
 M₄(2).D_2p: M₄(2).₄₂D₄  M₄(2).₄₃D₄  M₄(2).₂₄D₄  M₄(2).₂₂D₆  D₂₄⋊₁₀C₄  C₂₄.₅₄D₄  M₄(2).₂₂D₁₀  D₄₀⋊₁₆C₄ ...
 (C_p×D₈)⋊C₄: C₄².₁₁₆D₄  D₈⋊₄Dic₃  D₈⋊₄Dic₅  D₈⋊F₅  D₈⋊₄Dic₇ ...
 C₄².D_2p: C₄².₁₀₇D₄  D₂₄⋊₄C₄  D₄₀⋊₁₀C₄  D₅₆⋊₄C₄ ...

Matrix representation of C₈.₂₆D₄ ►in GL₄(𝔽₅) generated by

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

C₈.₂₆D₄ in GAP, Magma, Sage, TeX

C_8._{26}D_4

% in TeX

G:=Group("C8.26D4");

// GroupNames label

G:=SmallGroup(64,125);

// by ID

G=gap.SmallGroup(64,125);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,650,86,963,489,117,88]);

// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=a^2,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^2*b^-1>;

// generators/relations

Export

Subgroup lattice of C₈.₂₆D₄ in TeX
Character table of C₈.₂₆D₄ in TeX