A4:Q16 - GroupNames

G = A₄⋊Q₁₆ order 192 = 2⁶·3

The semidirect product of A₄ and Q₁₆ acting via Q₁₆/C₈=C₂

Aliases: C₈.₁S₄, A₄⋊₁Q₁₆, C₂²⋊Dic₁₂, C₂³.₈D₁₂, C₂.₆(C₄⋊S₄), C₄.₁₆(C₂×S₄), (C₈×A₄).₁C₂, (C₂×A₄).₁D₄, A₄⋊Q₈.₁C₂, (C₂²×C₈).₂S₃, (C₄×A₄).₈C₂², (C₂²×C₄).₁₃D₆, SmallGroup(192,957)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₄×A₄ — A₄⋊Q₁₆

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₄×A₄ — A₄⋊Q₈ — A₄⋊Q₁₆

Lower central

A₄ — C₂×A₄ — C₄×A₄ — A₄⋊Q₁₆

Upper central

C₁ — C₂ — C₄ — C₈

Generators and relations for A₄⋊Q₁₆
G = < a,b,c,d,e | a²=b²=c³=d⁸=1, e²=d⁴, cac^-1=eae^-1=ab=ba, ad=da, cbc^-1=a, bd=db, be=eb, cd=dc, ece^-1=c^-1, ede^-1=d^-1 >

Subgroups: 258 in 65 conjugacy classes, 15 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², C₂², C₆, C₈, C₈, C₂×C₄, Q₈, C₂³, Dic₃, C₁₂, A₄, C₂²⋊C₄, C₄⋊C₄, C₂×C₈, Q₁₆, C₂²×C₄, C₂×Q₈, C₂₄, Dic₆, C₂×A₄, Q₈⋊C₄, C₂.D₈, C₂²⋊Q₈, C₂²×C₈, C₂×Q₁₆, Dic₁₂, A₄⋊C₄, C₄×A₄, C₈.₁₈D₄, C₈×A₄, A₄⋊Q₈, A₄⋊Q₁₆
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, Q₁₆, D₁₂, S₄, Dic₁₂, C₂×S₄, C₄⋊S₄, A₄⋊Q₁₆

Character table of A₄⋊Q₁₆

class	1	2A	2B	2C	3	4A	4B	4C	4D	4E	4F	6	8A	8B	8C	8D	12A	12B	24A	24B	24C	24D
size	1	1	3	3	8	2	6	24	24	24	24	8	2	2	6	6	8	8	8	8	8	8
ρ₁	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	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	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	linear of order 2
ρ₅	2	2	2	2	-1	2	2	0	0	0	0	-1	2	2	2	2	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	2	2	2	2	-2	-2	0	0	0	0	2	0	0	0	0	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₇	2	2	2	2	-1	2	2	0	0	0	0	-1	-2	-2	-2	-2	-1	-1	1	1	1	1	orthogonal lifted from D₆
ρ₈	2	2	2	2	-1	-2	-2	0	0	0	0	-1	0	0	0	0	1	1	-√3	√3	-√3	√3	orthogonal lifted from D₁₂
ρ₉	2	2	2	2	-1	-2	-2	0	0	0	0	-1	0	0	0	0	1	1	√3	-√3	√3	-√3	orthogonal lifted from D₁₂
ρ₁₀	2	-2	2	-2	2	0	0	0	0	0	0	-2	√2	-√2	-√2	√2	0	0	√2	√2	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₁	2	-2	2	-2	2	0	0	0	0	0	0	-2	-√2	√2	√2	-√2	0	0	-√2	-√2	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₂	2	-2	2	-2	-1	0	0	0	0	0	0	1	-√2	√2	√2	-√2	√3	-√3	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	symplectic lifted from Dic₁₂, Schur index 2
ρ₁₃	2	-2	2	-2	-1	0	0	0	0	0	0	1	√2	-√2	-√2	√2	√3	-√3	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	symplectic lifted from Dic₁₂, Schur index 2
ρ₁₄	2	-2	2	-2	-1	0	0	0	0	0	0	1	-√2	√2	√2	-√2	-√3	√3	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	symplectic lifted from Dic₁₂, Schur index 2
ρ₁₅	2	-2	2	-2	-1	0	0	0	0	0	0	1	√2	-√2	-√2	√2	-√3	√3	ζ₈⁷ζ₃²+ζ₈⁵ζ₃²+ζ₈⁵	ζ₈³ζ₃²+ζ₈³+ζ₈ζ₃²	ζ₈⁷ζ₃+ζ₈⁷+ζ₈⁵ζ₃	ζ₈³ζ₃+ζ₈ζ₃+ζ₈	symplectic lifted from Dic₁₂, Schur index 2
ρ₁₆	3	3	-1	-1	0	3	-1	1	-1	-1	1	0	-3	-3	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×S₄
ρ₁₇	3	3	-1	-1	0	3	-1	1	1	-1	-1	0	3	3	-1	-1	0	0	0	0	0	0	orthogonal lifted from S₄
ρ₁₈	3	3	-1	-1	0	3	-1	-1	-1	1	1	0	3	3	-1	-1	0	0	0	0	0	0	orthogonal lifted from S₄
ρ₁₉	3	3	-1	-1	0	3	-1	-1	1	1	-1	0	-3	-3	1	1	0	0	0	0	0	0	orthogonal lifted from C₂×S₄
ρ₂₀	6	6	-2	-2	0	-6	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₄⋊S₄
ρ₂₁	6	-6	-2	2	0	0	0	0	0	0	0	0	3√2	-3√2	√2	-√2	0	0	0	0	0	0	symplectic faithful, Schur index 2
ρ₂₂	6	-6	-2	2	0	0	0	0	0	0	0	0	-3√2	3√2	-√2	√2	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of A₄⋊Q₁₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

Matrix representation of A₄⋊Q₁₆ ►in GL₅(𝔽₇₃)

1	0	0	0	0
0	1	0	0	0
0	0	72	0	0
0	0	0	1	0
0	0	0	0	72

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	72	0
0	0	0	0	72

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1
0	0	1	0	0

0	16	0	0	0
41	32	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

15	17	0	0	0
64	58	0	0	0
0	0	72	0	0
0	0	0	0	72
0	0	0	72	0

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[0,41,0,0,0,16,32,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[15,64,0,0,0,17,58,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,72,0] >;

A₄⋊Q₁₆ in GAP, Magma, Sage, TeX

A_4\rtimes Q_{16}

% in TeX

G:=Group("A4:Q16");

// GroupNames label

G:=SmallGroup(192,957);

// by ID

G=gap.SmallGroup(192,957);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,56,85,92,254,58,1124,4037,285,2358,475]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=1,e^2=d^4,c*a*c^-1=e*a*e^-1=a*b=b*a,a*d=d*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;

// generators/relations

Export

Character table of A₄⋊Q₁₆ in TeX

G = A4⋊Q16 order 192 = 26·3

The semidirect product of A4 and Q16 acting via Q16/C8=C2

G = A₄⋊Q₁₆ order 192 = 2⁶·3

The semidirect product of A₄ and Q₁₆ acting via Q₁₆/C₈=C₂