C4^2:C2^2 - GroupNames

G = C₄²⋊C₂² order 64 = 2⁶

1^st semidirect product of C₄² and C₂² acting faithfully

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

Aliases: C₄²⋊₁C₂², C₂³.₁₃D₄, M₄(2)⋊₁₀C₂², C₄≀C₂⋊₅C₂, C₄○D₄⋊₄C₄, (C₂×Q₈)⋊₈C₄, (C₂×D₄)⋊₁₀C₄, D₄.₇(C₂×C₄), (C₂×C₄).₂₁D₄, C₄.₇₀(C₂×D₄), Q₈.₇(C₂×C₄), C₄.₈(C₂²×C₄), C₄²⋊C₂⋊₄C₂, (C₂×C₄).₆₆C₂³, C₄○D₄.₆C₂², C₂².₁₁(C₂×D₄), C₄.₂₅(C₂²⋊C₄), (C₂×M₄(2))⋊₁₃C₂, C₂².₅(C₂²⋊C₄), (C₂²×C₄).₃₇C₂², (C₂×C₄).₂₄(C₂×C₄), (C₂×C₄○D₄).₇C₂, C₂.₂₄(C₂×C₂²⋊C₄), SmallGroup(64,102)

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

Derived series

C₁ — C₄ — C₄²⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₄²⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₄²⋊C₂²

Upper central

C₁ — C₄ — C₂²×C₄ — C₄²⋊C₂²

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄²⋊C₂²

Generators and relations for C₄²⋊C₂²
G = < a,b,c,d | a⁴=b⁴=c²=d²=1, ab=ba, cac=a^-1b, dad=ab², bc=cb, bd=db, cd=dc >

Subgroups: 129 in 77 conjugacy classes, 39 normal (23 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₄≀C₂, C₄²⋊C₂, C₂×M₄(2), C₂×C₄○D₄, C₄²⋊C₂²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×C₂²⋊C₄, C₄²⋊C₂²

Character table of C₄²⋊C₂²

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D
size	1	1	2	2	2	4	4	1	1	2	2	2	4	4	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	trivial
ρ₂	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	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	linear of order 2
ρ₅	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
ρ₆	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
ρ₇	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
ρ₈	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
ρ₉	1	1	-1	-1	1	-1	1	-1	-1	1	-1	1	1	i	-i	-i	i	-1	i	i	-i	-i	linear of order 4
ρ₁₀	1	1	-1	-1	1	1	-1	-1	-1	1	-1	1	-1	i	-i	-i	i	1	-i	-i	i	i	linear of order 4
ρ₁₁	1	1	-1	-1	1	-1	1	-1	-1	1	-1	1	1	-i	i	i	-i	-1	-i	-i	i	i	linear of order 4
ρ₁₂	1	1	-1	-1	1	1	-1	-1	-1	1	-1	1	-1	-i	i	i	-i	1	i	i	-i	-i	linear of order 4
ρ₁₃	1	1	-1	1	-1	1	1	-1	-1	1	1	-1	-1	-i	i	-i	i	-1	-i	i	i	-i	linear of order 4
ρ₁₄	1	1	-1	1	-1	-1	-1	-1	-1	1	1	-1	1	-i	i	-i	i	1	i	-i	-i	i	linear of order 4
ρ₁₅	1	1	-1	1	-1	1	1	-1	-1	1	1	-1	-1	i	-i	i	-i	-1	i	-i	-i	i	linear of order 4
ρ₁₆	1	1	-1	1	-1	-1	-1	-1	-1	1	1	-1	1	i	-i	i	-i	1	-i	i	i	-i	linear of order 4
ρ₁₇	2	2	-2	-2	2	0	0	2	2	-2	2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	2	-2	2	-2	0	0	2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	2	2	-2	-2	0	0	-2	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	2	2	2	2	0	0	-2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	4	-4	0	0	0	0	0	-4i	4i	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful
ρ₂₂	4	-4	0	0	0	0	0	4i	-4i	0	0	0	0	0	0	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₄²⋊C₂²
►On 16 points - transitive group 16T106

Generators in S₁₆

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

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

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

(2 3)(6 8)(9 11)(14 16)

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

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

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

G:=TransitiveGroup(16,106);

►On 16 points - transitive group 16T107

Generators in S₁₆

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

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

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

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

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

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

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

G:=TransitiveGroup(16,107);

►On 16 points - transitive group 16T122

Generators in S₁₆

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

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

(2 15)(3 9)(4 7)(5 12)(8 14)(10 13)

(1 11)(3 9)(6 16)(8 14)

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

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

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

G:=TransitiveGroup(16,122);

C₄²⋊C₂² is a maximal subgroup of
C₄².₅D₄ C₄².₆D₄ C₄².₇D₄ C₄².₄₂₇D₄ M₄(2).₃₀D₄ M₄(2).D₄ C₄².₈D₄ M₄(2).₈D₄ M₄(2).₉D₄ C₄².₉D₄ C₄².₁₀D₄ C₄².₂₈₃C₂³ M₄(2)○D₈ M₄(2)⋊C₂³ C₄².₁₂C₂³ (C₂×D₄)⋊₆F₅ (C₂×Q₈)⋊₆F₅ D₄⋊F₅⋊C₂
C₄²⋊D_2p: C₄²⋊D₄ C₄²⋊₂D₄ C₄²⋊₃D₆ C₄²⋊₆D₆ C₄²⋊D₁₀ C₄²⋊₄D₁₀ C₄²⋊D₁₄ C₄²⋊₄D₁₄ ...
M₄(2)⋊D_2p: M₄(2)⋊₁₉D₄ M₄(2)⋊₅D₄ M₄(2)⋊₂₄D₆ C₂³.₂₀D₂₀ C₂³.₂₀D₂₈ ...
C₄○D₄.D_2p: 2₊¹⁺⁴⋊₄C₄ C₄○D₄.D₄ (C₂²×Q₈)⋊C₄ C₈○D₄⋊C₄ M₄(2).₄₁D₄ M₄(2).₄₄D₄ M₄(2).₄₆D₄ M₄(2).₄₇D₄ ...
C₄²⋊C₂² is a maximal quotient of
C₄².₃₉₇D₄ C₄².₃₉₈D₄ C₄².₃₉₉D₄ C₄².₄₅D₄ C₄².₄₆D₄ C₄².₃₇₃D₄ D₄⋊M₄(2) Q₈⋊M₄(2) C₄².₃₇₄D₄ C₄².₅₂D₄ C₄².₅₃D₄ C₄².₅₄D₄ C₂⁴.₅₆D₄ C₂⁴.₅₇D₄ C₄².₅₈D₄ C₂⁴.₅₈D₄ C₄².₅₉D₄ C₄².₆₀D₄ C₂⁴.₅₉D₄ C₄².₆₁D₄ C₄².₆₂D₄ C₂⁴.₆₁D₄ C₄².₆₃D₄ C₄².₄₀₇D₄ C₄².₄₀₈D₄ C₄².₃₇₆D₄ C₄².₆₇D₄ C₄².₆₈D₄ C₄².₆₉D₄ C₄².₇₀D₄ C₄².₇₁D₄ C₄².₇₂D₄ C₄².₇₃D₄ C₄².₇₄D₄ C₂⁴.₆₃D₄ D₄.C₄² C₂⁴.₇₀D₄ C₂⁴.₇₂D₄ M₄(2)⋊₁₃D₄ M₄(2)⋊₇Q₈ C₄²⋊Q₈ (C₂×D₄)⋊₆F₅ (C₂×Q₈)⋊₆F₅ D₄⋊F₅⋊C₂
C₄²⋊D_2p: C₄²⋊₇D₄ C₄²⋊₈D₄ C₄²⋊₃D₆ C₄²⋊₆D₆ C₄²⋊D₁₀ C₄²⋊₄D₁₀ C₄²⋊D₁₄ C₄²⋊₄D₁₄ ...
C₂³.D_4p: C₂⁴.₆₀D₄ M₄(2)⋊₂₄D₆ C₂³.₂₀D₂₀ C₂³.₂₀D₂₈ ...
C₄○D₄.D_2p: C₂⁴.₆₆D₄ C₄².₁₀₂D₄ (C₆×D₄)⋊₉C₄ (D₄×C₁₀)⋊₂₁C₄ (D₄×C₁₄)⋊₉C₄ ...

Matrix representation of C₄²⋊C₂² ►in GL₄(𝔽₅) generated by

0	0	1	0
3	0	0	0
0	0	0	4
0	3	0	0

2	0	0	0
0	2	0	0
0	0	2	0
0	0	0	2

4	0	0	0
0	0	1	0
0	1	0	0
0	0	0	1

4	0	0	0
0	1	0	0
0	0	1	0
0	0	0	4

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

C₄²⋊C₂² in GAP, Magma, Sage, TeX

C_4^2\rtimes C_2^2

% in TeX

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

// GroupNames label

G:=SmallGroup(64,102);

// by ID

G=gap.SmallGroup(64,102);

# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,332,963,489,117,88]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄²⋊C₂² in TeX

G = C42⋊C22 order 64 = 26

1st semidirect product of C42 and C22 acting faithfully

G = C₄²⋊C₂² order 64 = 2⁶

1^st semidirect product of C₄² and C₂² acting faithfully