C4^2:6C4 - GroupNames

G = C₄²:₆C₄ order 64 = 2⁶

3^rd semidirect product of C₄² and C₄ acting via C₄/C₂=C₂

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

Aliases: C₄²:₆C₄, C₄.₁C₄², M₄(2):₂C₄, C₂³.₃₂D₄, C₄:C₄:₃C₄, C₂.₃C₄wrC₂, C₄.₂(C₄:C₄), (C₂xC₄).₁₁Q₈, (C₂xC₄).₁₄₁D₄, (C₂xC₄²).₆C₂, C₂².₃(C₄:C₄), C₄.₂₇(C₂²:C₄), C₄²:C₂.₂C₂, (C₂xM₄(2)).₆C₂, C₂².₂₄(C₂²:C₄), C₂.₄(C₂.C₄²), (C₂²xC₄).₁₀₁C₂², (C₂xC₄).₆₃(C₂xC₄), SmallGroup(64,20)

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

Derived series

C₁ — C₄ — C₄²:₆C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂xC₄² — C₄²:₆C₄

Lower central

C₁ — C₂ — C₄ — C₄²:₆C₄

Upper central

C₁ — C₂xC₄ — C₂²xC₄ — C₄²:₆C₄

Jennings

C₁ — C₂ — C₂ — C₂²xC₄ — C₄²:₆C₄

Generators and relations for C₄²:₆C₄
G = < a,b,c | a⁴=b⁴=c⁴=1, cac^-1=ab=ba, cbc^-1=b^-1 >

Subgroups: 85 in 55 conjugacy classes, 29 normal (21 characteristic)
Quotients: C₁, C₂, C₄, C₂², C₂xC₄, D₄, Q₈, C₄², C₂²:C₄, C₄:C₄, C₂.C₄², C₄wrC₂, C₄²:₆C₄

C₁

C₂

₂C₂

C₂²

C₄

C₂²

C₄

₂C₂²

₂C₄

₂C₂²

₄C₄

C₂xC₄

C₂³

C₂xC₄

₂C₂xC₄

₂C₈

₂C₂xC₄

₂C₈

₂C₂xC₄

M₄(2)

C₄²

M₄(2)

C₄:C₄

C₂²xC₄

₂C₂²xC₄

₂C₄²

₂C₂²:C₄

₂C₂xC₈

₂C₄²

₂M₄(2)

C₂xC₄²

C₄²:C₂

C₂xM₄(2)

C₄²:₆C₄

Character table of C₄²:₆C₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	4Q	4R	8A	8B	8C	8D
size	1	1	1	1	2	2	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	1	1	1	1	linear of order 2
ρ₅	1	-1	1	-1	-1	1	1	1	-1	-1	-i	-i	-i	-i	-1	1	i	i	i	i	i	-i	-i	i	1	-1	-1	1	linear of order 4
ρ₆	1	-1	1	-1	1	-1	-1	-1	1	1	1	1	-1	-1	-1	1	-1	-1	1	1	i	i	-i	-i	-i	-i	i	i	linear of order 4
ρ₇	1	-1	1	-1	-1	1	1	1	-1	-1	i	i	i	i	-1	1	-i	-i	-i	-i	i	-i	-i	i	-1	1	1	-1	linear of order 4
ρ₈	1	-1	1	-1	-1	1	1	1	-1	-1	-i	-i	-i	-i	-1	1	i	i	i	i	-i	i	i	-i	-1	1	1	-1	linear of order 4
ρ₉	1	-1	1	-1	1	-1	-1	-1	1	1	1	1	-1	-1	-1	1	-1	-1	1	1	-i	-i	i	i	i	i	-i	-i	linear of order 4
ρ₁₀	1	-1	1	-1	-1	1	1	1	-1	-1	i	i	i	i	-1	1	-i	-i	-i	-i	-i	i	i	-i	1	-1	-1	1	linear of order 4
ρ₁₁	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	1	1	i	i	-i	-i	1	-1	1	-1	-i	i	-i	i	linear of order 4
ρ₁₂	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	1	1	-i	-i	i	i	1	-1	1	-1	i	-i	i	-i	linear of order 4
ρ₁₃	1	-1	1	-1	1	-1	-1	-1	1	1	-1	-1	1	1	-1	1	1	1	-1	-1	i	i	-i	-i	i	i	-i	-i	linear of order 4
ρ₁₄	1	1	1	1	-1	-1	-1	-1	-1	-1	-i	-i	i	i	1	1	-i	-i	i	i	-1	1	-1	1	-i	i	-i	i	linear of order 4
ρ₁₅	1	1	1	1	-1	-1	-1	-1	-1	-1	i	i	-i	-i	1	1	i	i	-i	-i	-1	1	-1	1	i	-i	i	-i	linear of order 4
ρ₁₆	1	-1	1	-1	1	-1	-1	-1	1	1	-1	-1	1	1	-1	1	1	1	-1	-1	-i	-i	i	i	-i	-i	i	i	linear of order 4
ρ₁₇	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	2	2	-2	-2	2	2	2	2	0	0	0	0	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	2	-2	2	-2	2	2	-2	-2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	2	-2	-2	2	-2	-2	2	2	0	0	0	0	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₁	2	-2	-2	2	0	0	-2i	2i	-2i	2i	1-i	-1+i	-1-i	1+i	0	0	-1+i	1-i	1+i	-1-i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₂	2	2	-2	-2	0	0	2i	-2i	-2i	2i	-1+i	1-i	-1-i	1+i	0	0	-1+i	1-i	-1-i	1+i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₃	2	-2	-2	2	0	0	2i	-2i	2i	-2i	1+i	-1-i	-1+i	1-i	0	0	-1-i	1+i	1-i	-1+i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₄	2	2	-2	-2	0	0	-2i	2i	2i	-2i	-1-i	1+i	-1+i	1-i	0	0	-1-i	1+i	-1+i	1-i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₅	2	-2	-2	2	0	0	2i	-2i	2i	-2i	-1-i	1+i	1-i	-1+i	0	0	1+i	-1-i	-1+i	1-i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₆	2	2	-2	-2	0	0	-2i	2i	2i	-2i	1+i	-1-i	1-i	-1+i	0	0	1+i	-1-i	1-i	-1+i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₇	2	2	-2	-2	0	0	2i	-2i	-2i	2i	1-i	-1+i	1+i	-1-i	0	0	1-i	-1+i	1+i	-1-i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂
ρ₂₈	2	-2	-2	2	0	0	-2i	2i	-2i	2i	-1+i	1-i	1+i	-1-i	0	0	1-i	-1+i	-1-i	1+i	0	0	0	0	0	0	0	0	complex lifted from C₄wrC₂

Permutation representations of C₄²:₆C₄
►On 16 points - transitive group 16T121

Generators in S₁₆

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

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

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

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

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

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

G:=TransitiveGroup(16,121);

C₄²:₆C₄ is a maximal subgroup of
C₂⁴.₆₃D₄ C₄xC₄wrC₂ D₄.C₄² Q₈.C₄² D₄.₃C₄² C₄².₁₀₂D₄ C₂⁴.₇₀D₄ (C₂xC₄²):C₄ C₈:C₄:₁₇C₄ C₂⁴.₂₁D₄ C₄.₁₀D₄:₂C₄ C₄wrC₂:C₄ C₄²:₉(C₂xC₄) M₄(2).₄₁D₄ M₄(2).₄₂D₄ C₂⁴.₇₂D₄ C₈.C₂²:C₄ C₈:C₂²:C₄ M₄(2):₁₉D₄ C₂⁴.₂₃D₄ C₄:Q₈:₁₅C₄ C₂⁴.₂₄D₄ C₄.₄D₄:₁₃C₄ (C₂xC₄)wrC₂ C₄²:₇D₄ C₄².₄₂₆D₄ M₄(2).₃Q₈ M₄(2).₂₄D₄ C₄².₄₂₇D₄ C₄².₄₂₈D₄ C₄².₁₀₇D₄ C₄².₆₂Q₈ C₄².₂₈Q₈ M₄(2):₇Q₈ C₄²:₁₆Q₈ C₄²:Q₈ C₄²:₉D₄ C₄².₁₂₉D₄ C₄²:₁₀D₄ C₄².₁₃₀D₄ (C₂xD₄):₂Q₈ (C₂xQ₈):₂Q₈ C₄².₈D₄ M₄(2):₆D₄ M₄(2).₇D₄ M₄(2):Q₈ C₄²:₃Q₈ C₄².₃₂Q₈ C₂³.₉S₄
C_4p.C₄²: C₈.₁₄C₄² C₈.₅C₄² C₁₂.₈C₄² C₁₂.₂C₄² C₁₂.₃C₄² C₄²:₆Dic₅ C₂₀.₃₂C₄² C₂₀.₃₃C₄² ...
C₄²:₆C₄ is a maximal quotient of
C₄².₄₆Q₈ C₂⁴.₄₆D₄ C₄².₆Q₈ C₄².₇Q₈ C₂⁴.₄₈D₄ C₄²:₆F₅ M₄(2):₃F₅
C₂³.D_4p: C₂³.₃₀D₈ C₁₂.₃C₄² C₂₀.₃₃C₄² C₂₈.₃C₄² ...
C_2p.C₄wrC₂: C₄²:₆C₈ M₄(2):C₈ C₄².₄Q₈ C₄².₂₆D₄ C₄².₃₈₈D₄ C₄².₉Q₈ C₄².₃₇₀D₄ C₄².₁₀Q₈ ...

Matrix representation of C₄²:₆C₄ ►in GL₃(F₁₇) generated by

1	0	0
0	1	0
0	0	4

1	0	0
0	4	0
0	0	13

13	0	0
0	0	1
0	1	0

G:=sub<GL(3,GF(17))| [1,0,0,0,1,0,0,0,4],[1,0,0,0,4,0,0,0,13],[13,0,0,0,0,1,0,1,0] >;

C₄²:₆C₄ in GAP, Magma, Sage, TeX

C_4^2\rtimes_6C_4

% in TeX

G:=Group("C4^2:6C4");

// GroupNames label

G:=SmallGroup(64,20);

// by ID

G=gap.SmallGroup(64,20);

# by ID

G:=PCGroup([6,-2,2,-2,2,2,-2,48,73,103,650,158,1444]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₄²:₆C₄ in TeX
Character table of C₄²:₆C₄ in TeX

G = C42:6C4 order 64 = 26

3rd semidirect product of C42 and C4 acting via C4/C2=C2

G = C₄²:₆C₄ order 64 = 2⁶

3^rd semidirect product of C₄² and C₄ acting via C₄/C₂=C₂