D4:5D4 - GroupNames

G = D₄⋊₅D₄ order 64 = 2⁶

1^st semidirect product of D₄ and D₄ acting through Inn(D₄)

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

Aliases: D₄⋊₅D₄, C₄²⋊₉C₂², C₂².₄₀C₂⁴, C₂⁴.₁₉C₂², C₂³.₄₃C₂³, C₂.₁₃2₊¹⁺⁴, (C₄×D₄)⋊₁₅C₂, C₄⋊C₄⋊₅C₂², C₄.₃₆(C₂×D₄), C₂²≀C₂⋊₆C₂, C₄⋊D₄⋊₁₁C₂, D₄ ○₂(C₂²⋊C₄), (C₂²×D₄)⋊₉C₂, (C₂×D₄)⋊₆C₂², C₂²⋊Q₈⋊₁₁C₂, C₂².₃(C₂×D₄), C₄.₄D₄⋊₁₀C₂, C₂²⋊₂(C₄○D₄), C₂²⋊C₄⋊₇C₂², (C₂×C₄).₂₇C₂³, (C₂×Q₈)⋊₁₂C₂², C₂.₁₈(C₂²×D₄), (C₂²×C₄)⋊₁₁C₂², C₂².D₄⋊₈C₂, C₂²⋊C₄ ○(C₂×D₄), (C₂×C₄○D₄)⋊₆C₂, C₂.₂₀(C₂×C₄○D₄), (C₂×C₂²⋊C₄)⋊₁₄C₂, SmallGroup(64,227)

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

Derived series

C₁ — C₂² — D₄⋊₅D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — D₄⋊₅D₄

Lower central

C₁ — C₂² — D₄⋊₅D₄

Upper central

C₁ — C₂² — D₄⋊₅D₄

Jennings

C₁ — C₂² — D₄⋊₅D₄

Generators and relations for D₄⋊₅D₄
G = < a,b,c,d | a⁴=b²=c⁴=d²=1, bab=a^-1, ac=ca, ad=da, cbc^-1=dbd=a²b, dcd=c^-1 >

Subgroups: 285 in 167 conjugacy classes, 83 normal (31 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₂²×D₄, C₂×C₄○D₄, D₄⋊₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, D₄⋊₅D₄

Character table of D₄⋊₅D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	2L	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L
size	1	1	1	1	2	2	2	2	2	2	4	4	4	2	2	2	2	2	2	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	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	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	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	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	-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	-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	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	1	1	-1	linear of order 2
ρ₉	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
ρ₁₀	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
ρ₁₁	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
ρ₁₂	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
ρ₁₃	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
ρ₁₄	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
ρ₁₅	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
ρ₁₆	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
ρ₁₇	2	-2	-2	2	2	0	-2	-2	2	0	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	-2	2	2	0	-2	2	-2	0	0	0	0	0	0	0	-2	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	-2	0	2	-2	2	0	0	0	0	0	0	0	-2	2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	-2	2	-2	0	2	2	-2	0	0	0	0	0	0	0	2	-2	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	-2	2	-2	0	2	0	0	0	-2	0	0	0	2i	-2i	-2i	0	0	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	2	-2	0	2	0	0	0	-2	0	0	0	-2i	2i	2i	0	0	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	-2	2	-2	0	-2	0	0	0	2	0	0	0	-2i	-2i	2i	0	0	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	-2	2	-2	0	-2	0	0	0	2	0	0	0	2i	2i	-2i	0	0	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₅	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from 2₊¹⁺⁴

Permutation representations of D₄⋊₅D₄
►On 16 points - transitive group 16T115

Generators in S₁₆

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

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

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

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

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

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

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

G:=TransitiveGroup(16,115);

Matrix representation of D₄⋊₅D₄ ►in GL₄(𝔽₅) generated by

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

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

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

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

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

D₄⋊₅D₄ in GAP, Magma, Sage, TeX

D_4\rtimes_5D_4

% in TeX

G:=Group("D4:5D4");

// GroupNames label

G:=SmallGroup(64,227);

// by ID

G=gap.SmallGroup(64,227);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,650,297]);

// Polycyclic

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

// generators/relations

Export

Character table of D₄⋊₅D₄ in TeX

G = D4⋊5D4 order 64 = 26

1st semidirect product of D4 and D4 acting through Inn(D4)

G = D₄⋊₅D₄ order 64 = 2⁶

1^st semidirect product of D₄ and D₄ acting through Inn(D₄)