Q8:3Dic3 - GroupNames

G = Q₈:₃Dic₃ order 96 = 2⁵·3

2^nd semidirect product of Q₈ and Dic₃ acting via Dic₃/C₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q₈:₃Dic₃, D₄:₂Dic₃, C₁₂.₅₆D₄, C₃:₃C₄wrC₂, (C₃xD₄):₂C₄, (C₃xQ₈):₂C₄, (C₂xC₆).₃D₄, C₁₂.₉(C₂xC₄), C₄oD₄.₃S₃, (C₂xC₄).₄₁D₆, (C₄xDic₃):₂C₂, C₄.Dic₃:₄C₂, C₄.₃(C₂xDic₃), C₄.₃₁(C₃:D₄), C₆.₁₈(C₂²:C₄), (C₂xC₁₂).₂₀C₂², C₂².₃(C₃:D₄), C₂.₈(C₆.D₄), (C₃xC₄oD₄).₁C₂, SmallGroup(96,44)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — Q₈:₃Dic₃

Chief series

C₁ — C₃ — C₆ — C₁₂ — C₂xC₁₂ — C₄.Dic₃ — Q₈:₃Dic₃

Lower central

C₃ — C₆ — C₁₂ — Q₈:₃Dic₃

Upper central

C₁ — C₄ — C₂xC₄ — C₄oD₄

Generators and relations for Q₈:₃Dic₃
G = < a,b,c,d | a⁴=c⁶=1, b²=a², d²=c³, bab^-1=a^-1, ac=ca, ad=da, cbc^-1=a²b, dbd^-1=a^-1b, dcd^-1=c^-1 >

Subgroups: 90 in 44 conjugacy classes, 21 normal (all characteristic)
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, D₄, Dic₃, D₆, C₂²:C₄, C₂xDic₃, C₃:D₄, C₄wrC₂, C₆.D₄, Q₈:₃Dic₃

C₁

C₂

₂C₂

₄C₂

C₃

C₄

C₂²

C₄

₂C₄

₂C₂²

₆C₄

C₆

₂C₆

₄C₆

D₄

Q₈

C₂xC₄

₂D₄

₂C₂xC₄

₆C₂xC₄

₆C₈

C₁₂

C₂xC₆

C₁₂

₂C₂xC₆

₂Dic₃

₂C₁₂

₂Dic₃

C₄oD₄

₃C₄²

₃M₄(2)

C₃xQ₈

C₂xC₁₂

C₃xD₄

₂C₃:C₈

₂C₂xC₁₂

₂C₃xD₄

₂C₂xDic₃

₃C₄wrC₂

C₄.Dic₃

C₄xDic₃

C₃xC₄oD₄

Q₈:₃Dic₃

Character table of Q₈:₃Dic₃

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	4G	4H	6A	6B	6C	6D	8A	8B	12A	12B	12C	12D	12E
size	1	1	2	4	2	1	1	2	4	6	6	6	6	2	4	4	4	12	12	2	2	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	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	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	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	linear of order 2
ρ₅	1	1	-1	-1	1	-1	-1	1	1	-i	i	-i	i	1	-1	-1	-1	-i	i	-1	-1	1	1	1	linear of order 4
ρ₆	1	1	-1	-1	1	-1	-1	1	1	i	-i	i	-i	1	-1	-1	-1	i	-i	-1	-1	1	1	1	linear of order 4
ρ₇	1	1	-1	1	1	-1	-1	1	-1	i	-i	i	-i	1	1	-1	1	-i	i	-1	-1	-1	1	-1	linear of order 4
ρ₈	1	1	-1	1	1	-1	-1	1	-1	-i	i	-i	i	1	1	-1	1	i	-i	-1	-1	-1	1	-1	linear of order 4
ρ₉	2	2	-2	0	2	2	2	-2	0	0	0	0	0	2	0	-2	0	0	0	2	2	0	-2	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	-2	-1	2	2	2	-2	0	0	0	0	-1	1	-1	1	0	0	-1	-1	1	-1	1	orthogonal lifted from D₆
ρ₁₁	2	2	2	0	2	-2	-2	-2	0	0	0	0	0	2	0	2	0	0	0	-2	-2	0	-2	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-1	2	2	2	2	0	0	0	0	-1	-1	-1	-1	0	0	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₃	2	2	-2	-2	-1	-2	-2	2	2	0	0	0	0	-1	1	1	1	0	0	1	1	-1	-1	-1	symplectic lifted from Dic₃, Schur index 2
ρ₁₄	2	2	-2	2	-1	-2	-2	2	-2	0	0	0	0	-1	-1	1	-1	0	0	1	1	1	-1	1	symplectic lifted from Dic₃, Schur index 2
ρ₁₅	2	2	-2	0	-1	2	2	-2	0	0	0	0	0	-1	√-3	1	-√-3	0	0	-1	-1	-√-3	1	√-3	complex lifted from C₃:D₄
ρ₁₆	2	2	2	0	-1	-2	-2	-2	0	0	0	0	0	-1	√-3	-1	-√-3	0	0	1	1	√-3	1	-√-3	complex lifted from C₃:D₄
ρ₁₇	2	2	2	0	-1	-2	-2	-2	0	0	0	0	0	-1	-√-3	-1	√-3	0	0	1	1	-√-3	1	√-3	complex lifted from C₃:D₄
ρ₁₈	2	2	-2	0	-1	2	2	-2	0	0	0	0	0	-1	-√-3	1	√-3	0	0	-1	-1	√-3	1	-√-3	complex lifted from C₃:D₄
ρ₁₉	2	-2	0	0	2	-2i	2i	0	0	1-i	-1-i	-1+i	1+i	-2	0	0	0	0	0	2i	-2i	0	0	0	complex lifted from C₄wrC₂
ρ₂₀	2	-2	0	0	2	-2i	2i	0	0	-1+i	1+i	1-i	-1-i	-2	0	0	0	0	0	2i	-2i	0	0	0	complex lifted from C₄wrC₂
ρ₂₁	2	-2	0	0	2	2i	-2i	0	0	-1-i	1-i	1+i	-1+i	-2	0	0	0	0	0	-2i	2i	0	0	0	complex lifted from C₄wrC₂
ρ₂₂	2	-2	0	0	2	2i	-2i	0	0	1+i	-1+i	-1-i	1-i	-2	0	0	0	0	0	-2i	2i	0	0	0	complex lifted from C₄wrC₂
ρ₂₃	4	-4	0	0	-2	4i	-4i	0	0	0	0	0	0	2	0	0	0	0	0	2i	-2i	0	0	0	complex faithful
ρ₂₄	4	-4	0	0	-2	-4i	4i	0	0	0	0	0	0	2	0	0	0	0	0	-2i	2i	0	0	0	complex faithful

Permutation representations of Q₈:₃Dic₃
►On 24 points - transitive group 24T109

Generators in S₂₄

(1 10 6 7)(2 11 4 8)(3 12 5 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)

(1 22 6 19)(2 20 4 23)(3 24 5 21)(7 16 10 13)(8 14 11 17)(9 18 12 15)

(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(2 3)(4 5)(8 9)(11 12)(13 19 16 22)(14 24 17 21)(15 23 18 20)

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

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

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

G:=TransitiveGroup(24,109);

Q₈:₃Dic₃ is a maximal subgroup of
S₃xC₄wrC₂ C₄²:₃D₆ C₂₄.₁₀₀D₄ C₂₄.₅₄D₄ D₈:₅Dic₃ D₈:₄Dic₃ D₁₂:₁₈D₄ D₁₂.₃₈D₄ D₁₂.₃₉D₄ D₁₂.₄₀D₄ (C₆xD₄):₉C₄ 2₊¹⁺⁴:₆S₃ 2₊¹⁺⁴.₄S₃ 2_-¹⁺⁴:₄S₃ 2_-¹⁺⁴.₂S₃ Q₈:₃Dic₉ C₁₂.₉S₄ D₁₂:₄Dic₃ D₁₂:₂Dic₃ C₆².₃₉D₄ C₃:U₂(F₃) C₆₀.₉₆D₄ C₆₀.₉₇D₄ Q₈:₃Dic₁₅ Dic₁₀:Dic₃ D₂₀:₂Dic₃
Q₈:₃Dic₃ is a maximal quotient of
C₁₂.₂C₄² C₁₂.₅₇D₈ C₁₂.₂₆Q₁₆ (C₆xD₄):C₄ (C₆xQ₈):C₄ C₄².₇D₆ C₄².₈D₆ Q₈:₃Dic₉ D₁₂:₄Dic₃ D₁₂:₂Dic₃ C₆².₃₉D₄ C₆₀.₉₆D₄ C₆₀.₉₇D₄ Q₈:₃Dic₁₅ Dic₁₀:Dic₃ D₂₀:₂Dic₃

Matrix representation of Q₈:₃Dic₃ ►in GL₄(F₅) generated by

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

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

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

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

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

Q₈:₃Dic₃ in GAP, Magma, Sage, TeX

Q_8\rtimes_3{\rm Dic}_3

% in TeX

G:=Group("Q8:3Dic3");

// GroupNames label

G:=SmallGroup(96,44);

// by ID

G=gap.SmallGroup(96,44);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,86,579,297,69,2309]);

// Polycyclic

G:=Group<a,b,c,d|a^4=c^6=1,b^2=a^2,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;

// generators/relations

Export

Subgroup lattice of Q₈:₃Dic₃ in TeX
Character table of Q₈:₃Dic₃ in TeX

G = Q8:3Dic3 order 96 = 25·3

2nd semidirect product of Q8 and Dic3 acting via Dic3/C6=C2

G = Q₈:₃Dic₃ order 96 = 2⁵·3

2^nd semidirect product of Q₈ and Dic₃ acting via Dic₃/C₆=C₂