D4:3Q8 - GroupNames

G = D₄⋊₃Q₈ order 64 = 2⁶

The semidirect product of D₄ and Q₈ acting through Inn(D₄)

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

Aliases: D₄⋊₃Q₈, C₄².₄₉C₂², C₂³.₄₈C₂³, C₂².₄₈C₂⁴, C₂.₁₇2₊¹⁺⁴, D₄ ○₃(C₄⋊C₄), C₄⋊Q₈⋊₁₅C₂, (C₄×Q₈)⋊₁₅C₂, C₄.₁₇(C₂×Q₈), (C₄×D₄).₁₂C₂, C₂²⋊Q₈⋊₁₇C₂, C₄.₃₇(C₄○D₄), C₂².₆(C₂×Q₈), C₄⋊C₄.₃₇C₂², (C₂×C₄).₃₁C₂³, C₄².C₂⋊₁₀C₂, C₂.₁₀(C₂²×Q₈), (C₂×D₄).₈₁C₂², (C₂×Q₈).₃₂C₂², C₂²⋊C₄.₂₃C₂², (C₂²×C₄).₇₃C₂², (C₂×C₄⋊C₄)⋊₂₃C₂, C₂.₂₇(C₂×C₄○D₄), SmallGroup(64,235)

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

Derived series

C₁ — C₂² — D₄⋊₃Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — D₄⋊₃Q₈

Lower central

C₁ — C₂² — D₄⋊₃Q₈

Upper central

C₁ — C₂² — D₄⋊₃Q₈

Jennings

C₁ — C₂² — D₄⋊₃Q₈

Generators and relations for D₄⋊₃Q₈
G = < a,b,c,d | a⁴=b²=c⁴=1, d²=c², bab=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a²b, dcd^-1=c^-1 >

Subgroups: 157 in 114 conjugacy classes, 83 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×C₄⋊C₄, C₄×D₄, C₄×D₄, C₄×Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, D₄⋊₃Q₈
Quotients: C₁, C₂, C₂², Q₈, C₂³, C₂×Q₈, C₄○D₄, C₂⁴, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, D₄⋊₃Q₈

Character table of D₄⋊₃Q₈

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	4Q
size	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	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	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	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₈	2	-2	2	-2	2	2	-2	-2	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₉	2	-2	2	-2	-2	2	-2	2	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₂₀	2	-2	2	-2	-2	-2	2	2	-2	0	0	2	0	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	0	0	0	2	-2	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	-2	-2	0	0	0	0	0	-2	2	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	2	-2	-2	0	0	0	0	0	-2	2	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	2	-2	-2	0	0	0	0	0	2	-2	0	2i	2i	-2i	-2i	0	0	0	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

Smallest permutation representation of D₄⋊₃Q₈
►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 13)(2 16)(3 15)(4 14)(5 17)(6 20)(7 19)(8 18)(9 27)(10 26)(11 25)(12 28)(21 32)(22 31)(23 30)(24 29)

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

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

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

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

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

Matrix representation of D₄⋊₃Q₈ ►in GL₄(𝔽₅) generated by

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

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

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

0	3	0	0
2	0	0	0
0	0	2	2
0	0	0	3

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

D₄⋊₃Q₈ in GAP, Magma, Sage, TeX

D_4\rtimes_3Q_8

% in TeX

G:=Group("D4:3Q8");

// GroupNames label

G:=SmallGroup(64,235);

// by ID

G=gap.SmallGroup(64,235);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,96,217,295,650,297,69]);

// Polycyclic

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

// generators/relations

Export

Character table of D₄⋊₃Q₈ in TeX

G = D4⋊3Q8 order 64 = 26

The semidirect product of D4 and Q8 acting through Inn(D4)

G = D₄⋊₃Q₈ order 64 = 2⁶

The semidirect product of D₄ and Q₈ acting through Inn(D₄)