Q8.14D6 - GroupNames

G = Q₈.₁₄D₆ order 96 = 2⁵·3

4^th non-split extension by Q₈ of D₆ acting via D₆/C₆=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄.₉D₆, C₁₂.₅₀D₄, Q₈.₁₄D₆, C₁₂.₁₉C₂³, Dic₆.₁₂C₂², D₄.S₃:₆C₂, C₄oD₄.₄S₃, C₃:Q₁₆:₆C₂, (C₂xC₆).₁₀D₄, (C₂xC₄).₂₃D₆, C₆.₆₁(C₂xD₄), C₃:C₈.₄C₂², C₃:₅(C₈.C₂²), (C₂xDic₆):₁₁C₂, C₄.₂₅(C₃:D₄), C₄.Dic₃:₁₀C₂, C₄.₁₉(C₂²xS₃), (C₃xD₄).₉C₂², (C₃xQ₈).₉C₂², (C₂xC₁₂).₄₄C₂², C₂².₆(C₃:D₄), (C₃xC₄oD₄).₃C₂, C₂.₂₅(C₂xC₃:D₄), SmallGroup(96,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — Q₈.₁₄D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — Dic₆ — C₂xDic₆ — Q₈.₁₄D₆

Lower central

C₃ — C₆ — C₁₂ — Q₈.₁₄D₆

Upper central

C₁ — C₂ — C₂xC₄ — C₄oD₄

Generators and relations for Q₈.₁₄D₆
G = < a,b,c,d | a⁴=c⁶=1, b²=d²=a², bab^-1=dad^-1=a^-1, ac=ca, cbc^-1=a²b, dbd^-1=ab, dcd^-1=c^-1 >

Subgroups: 122 in 60 conjugacy classes, 29 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, Dic₃, C₁₂, C₁₂, C₂xC₆, C₂xC₆, M₄(2), SD₁₆, Q₁₆, C₂xQ₈, C₄oD₄, C₃:C₈, Dic₆, Dic₆, C₂xDic₃, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xD₄, C₃xQ₈, C₈.C₂², C₄.Dic₃, D₄.S₃, C₃:Q₁₆, C₂xDic₆, C₃xC₄oD₄, Q₈.₁₄D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₈.C₂², C₂xC₃:D₄, Q₈.₁₄D₆

Character table of Q₈.₁₄D₆

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	6A	6B	6C	6D	8A	8B	12A	12B	12C	12D	12E
size	1	1	2	4	2	2	2	4	12	12	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	trivial
ρ₂	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	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	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	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	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	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	linear of order 2
ρ₉	2	2	2	-2	-1	2	2	-2	0	0	-1	-1	1	1	0	0	-1	-1	1	1	-1	orthogonal lifted from D₆
ρ₁₀	2	2	-2	2	-1	-2	2	-2	0	0	-1	1	-1	-1	0	0	1	1	1	1	-1	orthogonal lifted from D₆
ρ₁₁	2	2	2	2	-1	2	2	2	0	0	-1	-1	-1	-1	0	0	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₁₂	2	2	2	0	2	-2	-2	0	0	0	2	2	0	0	0	0	-2	-2	0	0	-2	orthogonal lifted from D₄
ρ₁₃	2	2	-2	0	2	2	-2	0	0	0	2	-2	0	0	0	0	2	2	0	0	-2	orthogonal lifted from D₄
ρ₁₄	2	2	-2	-2	-1	-2	2	2	0	0	-1	1	1	1	0	0	1	1	-1	-1	-1	orthogonal lifted from D₆
ρ₁₅	2	2	-2	0	-1	2	-2	0	0	0	-1	1	-√-3	√-3	0	0	-1	-1	√-3	-√-3	1	complex lifted from C₃:D₄
ρ₁₆	2	2	2	0	-1	-2	-2	0	0	0	-1	-1	-√-3	√-3	0	0	1	1	-√-3	√-3	1	complex lifted from C₃:D₄
ρ₁₇	2	2	-2	0	-1	2	-2	0	0	0	-1	1	√-3	-√-3	0	0	-1	-1	-√-3	√-3	1	complex lifted from C₃:D₄
ρ₁₈	2	2	2	0	-1	-2	-2	0	0	0	-1	-1	√-3	-√-3	0	0	1	1	√-3	-√-3	1	complex lifted from C₃:D₄
ρ₁₉	4	-4	0	0	4	0	0	0	0	0	-4	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₀	4	-4	0	0	-2	0	0	0	0	0	2	0	0	0	0	0	-2√3	2√3	0	0	0	symplectic faithful, Schur index 2
ρ₂₁	4	-4	0	0	-2	0	0	0	0	0	2	0	0	0	0	0	2√3	-2√3	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of Q₈.₁₄D₆
►On 48 points

Generators in S₄₈

(1 30 19 7)(2 25 20 8)(3 26 21 9)(4 27 22 10)(5 28 23 11)(6 29 24 12)(13 32 42 47)(14 33 37 48)(15 34 38 43)(16 35 39 44)(17 36 40 45)(18 31 41 46)

(1 22 19 4)(2 5 20 23)(3 24 21 6)(7 10 30 27)(8 28 25 11)(9 12 26 29)(13 44 42 35)(14 36 37 45)(15 46 38 31)(16 32 39 47)(17 48 40 33)(18 34 41 43)

(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 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 46 19 31)(2 45 20 36)(3 44 21 35)(4 43 22 34)(5 48 23 33)(6 47 24 32)(7 18 30 41)(8 17 25 40)(9 16 26 39)(10 15 27 38)(11 14 28 37)(12 13 29 42)

G:=sub<Sym(48)| (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,22,19,4)(2,5,20,23)(3,24,21,6)(7,10,30,27)(8,28,25,11)(9,12,26,29)(13,44,42,35)(14,36,37,45)(15,46,38,31)(16,32,39,47)(17,48,40,33)(18,34,41,43), (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)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,46,19,31)(2,45,20,36)(3,44,21,35)(4,43,22,34)(5,48,23,33)(6,47,24,32)(7,18,30,41)(8,17,25,40)(9,16,26,39)(10,15,27,38)(11,14,28,37)(12,13,29,42)>;

G:=Group( (1,30,19,7)(2,25,20,8)(3,26,21,9)(4,27,22,10)(5,28,23,11)(6,29,24,12)(13,32,42,47)(14,33,37,48)(15,34,38,43)(16,35,39,44)(17,36,40,45)(18,31,41,46), (1,22,19,4)(2,5,20,23)(3,24,21,6)(7,10,30,27)(8,28,25,11)(9,12,26,29)(13,44,42,35)(14,36,37,45)(15,46,38,31)(16,32,39,47)(17,48,40,33)(18,34,41,43), (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)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,46,19,31)(2,45,20,36)(3,44,21,35)(4,43,22,34)(5,48,23,33)(6,47,24,32)(7,18,30,41)(8,17,25,40)(9,16,26,39)(10,15,27,38)(11,14,28,37)(12,13,29,42) );

G=PermutationGroup([[(1,30,19,7),(2,25,20,8),(3,26,21,9),(4,27,22,10),(5,28,23,11),(6,29,24,12),(13,32,42,47),(14,33,37,48),(15,34,38,43),(16,35,39,44),(17,36,40,45),(18,31,41,46)], [(1,22,19,4),(2,5,20,23),(3,24,21,6),(7,10,30,27),(8,28,25,11),(9,12,26,29),(13,44,42,35),(14,36,37,45),(15,46,38,31),(16,32,39,47),(17,48,40,33),(18,34,41,43)], [(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),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,46,19,31),(2,45,20,36),(3,44,21,35),(4,43,22,34),(5,48,23,33),(6,47,24,32),(7,18,30,41),(8,17,25,40),(9,16,26,39),(10,15,27,38),(11,14,28,37),(12,13,29,42)]])

Q₈.₁₄D₆ is a maximal subgroup of
C₄²:₅D₆ Q₈.₁₄D₁₂ Q₈.₈D₁₂ Q₈.₁₀D₁₂ M₄(2).₁₃D₆ M₄(2).₁₆D₆ 2₊¹⁺⁴.₄S₃ 2_-¹⁺⁴.₂S₃ D₈:₁₁D₆ D₈.₁₀D₆ D₈:₄D₆ S₃xC₈.C₂² C₁₂.C₂⁴ D₁₂.₃₃C₂³ D₁₂.₃₅C₂³ D₄.D₁₈ C₁₂.₃S₄ D₁₂.₃₂D₆ D₁₂.₂₉D₆ Dic₆.₁₉D₆ D₁₂.₁₁D₆ C₆².₇₅D₄ C₁₂.₆S₄ D₂₀.₃₇D₆ C₆₀.₆₃D₄ C₆₀.₈C₂³ D₂₀.₁₃D₆ D₄.₉D₃₀
Q₈.₁₄D₆ is a maximal quotient of
C₄:C₄.₂₃₂D₆ C₄:C₄.₂₃₃D₆ C₄:C₄.₂₃₇D₆ C₁₂.₅₀D₈ C₄².₅₁D₆ D₄.₂D₁₂ Q₈:₄Dic₆ C₄².₅₉D₆ C₁₂:₇Q₁₆ (C₂xC₆).D₈ Dic₆:₁₇D₄ C₃:C₈:₅D₄ (C₂xQ₈).₄₉D₆ Dic₆.₃₇D₄ C₃:C₈.₆D₄ C₄².₆₁D₆ C₄².₆₂D₆ C₄².₆₅D₆ Dic₆.₄Q₈ C₄².₆₈D₆ C₄².₇₁D₆ C₄oD₄:₃Dic₃ (C₃xD₄).₃₂D₄ D₄.D₁₈ D₁₂.₃₂D₆ D₁₂.₂₉D₆ Dic₆.₁₉D₆ D₁₂.₁₁D₆ C₆².₇₅D₄ D₂₀.₃₇D₆ C₆₀.₆₃D₄ C₆₀.₈C₂³ D₂₀.₁₃D₆ D₄.₉D₃₀

Matrix representation of Q₈.₁₄D₆ ►in GL₆(F₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	10	0	0
0	0	29	72	0	0
0	0	32	14	1	71
0	0	16	14	1	72

72	0	0	0	0	0
0	72	0	0	0	0
0	0	52	51	3	0
0	0	0	46	14	59
0	0	23	13	2	19
0	0	23	13	48	46

8	0	0	0	0	0
71	64	0	0	0	0
0	0	52	51	3	0
0	0	48	46	0	14
0	0	35	13	21	54
0	0	29	13	21	27

72	28	0	0	0	0
0	1	0	0	0	0
0	0	40	14	0	36
0	0	69	41	11	0
0	0	56	38	32	33
0	0	24	5	16	33

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,29,32,16,0,0,10,72,14,14,0,0,0,0,1,1,0,0,0,0,71,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,52,0,23,23,0,0,51,46,13,13,0,0,3,14,2,48,0,0,0,59,19,46],[8,71,0,0,0,0,0,64,0,0,0,0,0,0,52,48,35,29,0,0,51,46,13,13,0,0,3,0,21,21,0,0,0,14,54,27],[72,0,0,0,0,0,28,1,0,0,0,0,0,0,40,69,56,24,0,0,14,41,38,5,0,0,0,11,32,16,0,0,36,0,33,33] >;

Q₈.₁₄D₆ in GAP, Magma, Sage, TeX

Q_8._{14}D_6

% in TeX

G:=Group("Q8.14D6");

// GroupNames label

G:=SmallGroup(96,158);

// by ID

G=gap.SmallGroup(96,158);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,188,579,159,69,2309]);

// Polycyclic

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

// generators/relations

Export

Character table of Q₈.₁₄D₆ in TeX

G = Q8.14D6 order 96 = 25·3

4th non-split extension by Q8 of D6 acting via D6/C6=C2

G = Q₈.₁₄D₆ order 96 = 2⁵·3

4^th non-split extension by Q₈ of D₆ acting via D₆/C₆=C₂