D4:6D4 - GroupNames

G = D₄:₆D₄ order 64 = 2⁶

2^nd semidirect product of D₄ and D₄ acting through Inn(D₄)

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

Aliases: D₄:₆D₄, C₂².₄₁C₂⁴, C₂³.₄₄C₂³, C₄².₄₃C₂², C₂.₉2_-¹⁺⁴, D₄ o₂(C₄:C₄), C₄:Q₈:₁₃C₂, (C₄xD₄):₁₆C₂, C₄:₂(C₄oD₄), C₄.₃₇(C₂xD₄), C₄:D₄:₁₂C₂, C₂²:Q₈:₁₂C₂, C₂².₄(C₂xD₄), C₄:C₄.₃₃C₂², (C₂xC₄).₂₈C₂³, C₂.₁₉(C₂²xD₄), (C₂xD₄).₇₉C₂², C₂².D₄:₉C₂, C₂²:C₄.₅C₂², (C₂xQ₈).₆₁C₂², (C₂²xC₄).₆₈C₂², (C₂xC₄:C₄):₂₀C₂, (C₂xC₄oD₄):₇C₂, C₂.₂₁(C₂xC₄oD₄), SmallGroup(64,228)

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

Derived series

C₁ — C₂² — D₄:₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²xC₄ — C₂xC₄oD₄ — D₄:₆D₄

Lower central

C₁ — C₂² — D₄:₆D₄

Upper central

C₁ — C₂² — D₄:₆D₄

Jennings

C₁ — C₂² — D₄:₆D₄

Generators and relations for D₄:₆D₄
G = < a,b,c,d | a⁴=b²=c⁴=d²=1, bab=a^-1, ac=ca, ad=da, bc=cb, dbd=a²b, dcd=c^-1 >

Subgroups: 213 in 146 conjugacy classes, 83 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂xC₄, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, C₂³, C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂xC₄:C₄, C₄xD₄, C₄:D₄, C₂²:Q₈, C₂².D₄, C₄:Q₈, C₂xC₄oD₄, D₄:₆D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₄oD₄, C₂⁴, C₂²xD₄, C₂xC₄oD₄, 2_-¹⁺⁴, D₄:₆D₄

Character table of D₄:₆D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O
size	1	1	1	1	2	2	2	2	4	4	2	2	2	2	2	2	2	2	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	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	2	2	-2	0	0	0	0	0	2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₈	2	-2	-2	2	2	2	-2	-2	0	0	0	0	0	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₉	2	-2	-2	2	2	-2	-2	2	0	0	0	0	0	2	0	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₀	2	-2	-2	2	-2	-2	2	2	0	0	0	0	0	-2	0	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₂₁	2	-2	2	-2	0	0	0	0	0	0	-2	2	2i	0	-2i	-2i	0	2i	0	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₂	2	-2	2	-2	0	0	0	0	0	0	-2	2	-2i	0	2i	2i	0	-2i	0	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₃	2	-2	2	-2	0	0	0	0	0	0	2	-2	-2i	0	2i	-2i	0	2i	0	0	0	0	0	0	0	complex lifted from C₄oD₄
ρ₂₄	2	-2	2	-2	0	0	0	0	0	0	2	-2	2i	0	-2i	2i	0	-2i	0	0	0	0	0	0	0	complex lifted from C₄oD₄
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	symplectic lifted from 2_-¹⁺⁴, Schur index 2

Smallest permutation representation of D₄:₆D₄
►On 32 points

Generators in S₃₂

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)

(1 18)(2 17)(3 20)(4 19)(5 16)(6 15)(7 14)(8 13)(9 23)(10 22)(11 21)(12 24)(25 31)(26 30)(27 29)(28 32)

(1 27 13 11)(2 28 14 12)(3 25 15 9)(4 26 16 10)(5 22 19 30)(6 23 20 31)(7 24 17 32)(8 21 18 29)

(1 11)(2 12)(3 9)(4 10)(5 32)(6 29)(7 30)(8 31)(13 27)(14 28)(15 25)(16 26)(17 22)(18 23)(19 24)(20 21)

G:=sub<Sym(32)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18)(2,17)(3,20)(4,19)(5,16)(6,15)(7,14)(8,13)(9,23)(10,22)(11,21)(12,24)(25,31)(26,30)(27,29)(28,32), (1,27,13,11)(2,28,14,12)(3,25,15,9)(4,26,16,10)(5,22,19,30)(6,23,20,31)(7,24,17,32)(8,21,18,29), (1,11)(2,12)(3,9)(4,10)(5,32)(6,29)(7,30)(8,31)(13,27)(14,28)(15,25)(16,26)(17,22)(18,23)(19,24)(20,21)>;

G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18)(2,17)(3,20)(4,19)(5,16)(6,15)(7,14)(8,13)(9,23)(10,22)(11,21)(12,24)(25,31)(26,30)(27,29)(28,32), (1,27,13,11)(2,28,14,12)(3,25,15,9)(4,26,16,10)(5,22,19,30)(6,23,20,31)(7,24,17,32)(8,21,18,29), (1,11)(2,12)(3,9)(4,10)(5,32)(6,29)(7,30)(8,31)(13,27)(14,28)(15,25)(16,26)(17,22)(18,23)(19,24)(20,21) );

G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,18),(2,17),(3,20),(4,19),(5,16),(6,15),(7,14),(8,13),(9,23),(10,22),(11,21),(12,24),(25,31),(26,30),(27,29),(28,32)], [(1,27,13,11),(2,28,14,12),(3,25,15,9),(4,26,16,10),(5,22,19,30),(6,23,20,31),(7,24,17,32),(8,21,18,29)], [(1,11),(2,12),(3,9),(4,10),(5,32),(6,29),(7,30),(8,31),(13,27),(14,28),(15,25),(16,26),(17,22),(18,23),(19,24),(20,21)]])

Matrix representation of D₄:₆D₄ ►in GL₄(F₅) generated by

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

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

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

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

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

D₄:₆D₄ in GAP, Magma, Sage, TeX

D_4\rtimes_6D_4

% in TeX

G:=Group("D4:6D4");

// GroupNames label

G:=SmallGroup(64,228);

// by ID

G=gap.SmallGroup(64,228);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,650,86,297]);

// 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,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;

// generators/relations

Export

Character table of D₄:₆D₄ in TeX

G = D4:6D4 order 64 = 26

2nd semidirect product of D4 and D4 acting through Inn(D4)

G = D₄:₆D₄ order 64 = 2⁶

2^nd semidirect product of D₄ and D₄ acting through Inn(D₄)