C4^2:4D4 - GroupNames

G = C₄²⋊₄D₄ order 128 = 2⁷

4^th semidirect product of C₄² and D₄ acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial

Aliases: C₄²⋊₄D₄, C₂⁴.₄₀D₄, 2₊¹⁺⁴.₁C₂², C₂≀C₄⋊₂C₂, (C₂×Q₈)⋊₃D₄, C₂≀C₂².C₂, C₂²⋊C₄⋊₂D₄, (C₂²×C₄)⋊₂D₄, C₄²⋊₃C₄⋊₅C₂, C₂.₂₁C₂≀C₂², D₄.₉D₄⋊₁C₂, (C₂×D₄).₂C₂³, C₂³.₁₄(C₂×D₄), C₂³.₇D₄⋊₁C₂, C₂³.D₄⋊₁C₂, C₂³⋊C₄.₁C₂², C₂²≀C₂.₄C₂², C₂².₄₅C₂²≀C₂, C₄.D₄.₁C₂², C₂².₄₅C₂⁴⋊₁C₂, C₄.₄D₄.₁₈C₂², C₂².D₄.₃C₂², (C₂×C₄).₁₄(C₂×D₄), SmallGroup(128,929)

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

Derived series

C₁ — C₂ — C₂×D₄ — C₄²⋊₄D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂×D₄ — C₂²≀C₂ — C₂².₄₅C₂⁴ — C₄²⋊₄D₄

Lower central

C₁ — C₂ — C₂² — C₂×D₄ — C₄²⋊₄D₄

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₄²⋊₄D₄

Jennings

C₁ — C₂ — C₂² — C₂×D₄ — C₄²⋊₄D₄

Generators and relations for C₄²⋊₄D₄
G = < a,b,c,d | a⁴=b⁴=c⁴=d²=1, ab=ba, cac^-1=dad=a^-1b^-1, cbc^-1=a²b^-1, bd=db, dcd=c^-1 >

Subgroups: 368 in 129 conjugacy classes, 28 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂³⋊C₄, C₂³⋊C₄, C₄.D₄, C₄≀C₂, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₈.C₂², 2₊¹⁺⁴, C₂≀C₄, C₂³.D₄, C₄²⋊₃C₄, D₄.₉D₄, C₂≀C₂², C₂³.₇D₄, C₂².₄₅C₂⁴, C₄²⋊₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₂≀C₂², C₄²⋊₄D₄

Character table of C₄²⋊₄D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8
size	1	1	2	4	4	4	4	8	4	4	4	4	4	8	8	8	8	16	16	16
ρ₁	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	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	0	0	-2	-2	0	2	2	2	0	0	0	-2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	-2	0	0	-2	0	0	0	0	0	0	-2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	0	0	-2	2	0	0	-2	0	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	0	0	-2	2	0	0	-2	0	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	-2	-2	2	-2	0	0	-2	0	0	0	0	0	0	2	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	0	0	-2	-2	0	-2	2	-2	0	0	0	2	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	4	4	-4	0	0	0	0	2	0	0	0	0	0	0	0	-2	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₁₆	4	4	-4	0	0	0	0	-2	0	0	0	0	0	0	0	2	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₁₇	4	-4	0	2	-2	0	0	0	2i	0	-2i	-2i	2i	0	0	0	0	0	0	0	complex faithful
ρ₁₈	4	-4	0	2	-2	0	0	0	-2i	0	2i	2i	-2i	0	0	0	0	0	0	0	complex faithful
ρ₁₉	4	-4	0	-2	2	0	0	0	2i	0	-2i	2i	-2i	0	0	0	0	0	0	0	complex faithful
ρ₂₀	4	-4	0	-2	2	0	0	0	-2i	0	2i	-2i	2i	0	0	0	0	0	0	0	complex faithful

Permutation representations of C₄²⋊₄D₄
►On 16 points - transitive group 16T345

Generators in S₁₆

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

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

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

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

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

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

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

G:=TransitiveGroup(16,345);

►On 16 points - transitive group 16T399

Generators in S₁₆

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

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

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

(2 11)(3 13)(4 8)(5 10)(6 16)(9 14)

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

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

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

G:=TransitiveGroup(16,399);

►On 16 points - transitive group 16T400

Generators in S₁₆

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

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

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

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

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

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

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

G:=TransitiveGroup(16,400);

►On 16 points - transitive group 16T410

Generators in S₁₆

(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

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

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

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

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

G:=TransitiveGroup(16,410);

Matrix representation of C₄²⋊₄D₄ ►in GL₄(𝔽₅) generated by

2	0	0	4
0	2	0	0
1	2	0	0
4	3	2	0

0	0	3	0
3	0	1	1
3	0	0	0
2	4	1	0

2	4	0	4
4	3	2	0
4	0	0	0
0	3	0	0

0	0	4	0
0	0	0	2
4	0	0	0
0	3	0	0

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

C₄²⋊₄D₄ in GAP, Magma, Sage, TeX

C_4^2\rtimes_4D_4

% in TeX

G:=Group("C4^2:4D4");

// GroupNames label

G:=SmallGroup(128,929);

// by ID

G=gap.SmallGroup(128,929);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,297,1971,375,4037]);

// Polycyclic

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

// generators/relations

Export

Character table of C₄²⋊₄D₄ in TeX

G = C42⋊4D4 order 128 = 27

4th semidirect product of C42 and D4 acting faithfully

G = C₄²⋊₄D₄ order 128 = 2⁷

4^th semidirect product of C₄² and D₄ acting faithfully