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## G = A4×Q16order 192 = 26·3

### Direct product of A4 and Q16

Aliases: A4×Q16, C8.1(C2×A4), C2.8(D4×A4), (C22×C8).C6, (C22×Q16)⋊C3, C22⋊(C3×Q16), (C8×A4).2C2, (Q8×A4).2C2, Q8.3(C2×A4), (C2×A4).16D4, C4.3(C22×A4), C23.25(C3×D4), (C22×Q8).4C6, (C4×A4).19C22, (C22×C4).3(C2×C6), SmallGroup(192,1016)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C4 — A4×Q16
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — Q8×A4 — A4×Q16
 Lower central C22 — C23 — C22×C4 — A4×Q16
 Upper central C1 — C2 — C4 — Q16

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

Subgroups: 224 in 73 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, Q8, Q8, C23, C12, A4, C2×C8, Q16, Q16, C22×C4, C22×C4, C2×Q8, C24, C3×Q8, C2×A4, C22×C8, C2×Q16, C22×Q8, C3×Q16, C4×A4, C4×A4, C22×Q16, C8×A4, Q8×A4, A4×Q16
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, Q16, C3×D4, C2×A4, C3×Q16, C22×A4, D4×A4, A4×Q16

Character table of A4×Q16

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D size 1 1 3 3 4 4 2 4 4 6 12 12 4 4 2 2 6 6 8 8 16 16 16 16 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 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 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 -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 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 1 1 1 ζ32 ζ3 1 -1 1 1 -1 1 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ6 ζ65 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ7 1 1 1 1 ζ3 ζ32 1 -1 1 1 -1 1 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ65 ζ6 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ8 1 1 1 1 ζ3 ζ32 1 -1 -1 1 -1 -1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ9 1 1 1 1 ζ32 ζ3 1 1 -1 1 1 -1 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ10 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ11 1 1 1 1 ζ32 ζ3 1 -1 -1 1 -1 -1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ12 1 1 1 1 ζ3 ζ32 1 1 -1 1 1 -1 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ13 2 2 2 2 2 2 -2 0 0 -2 0 0 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 √2 -√2 -√2 √2 0 0 0 0 0 0 -√2 √2 -√2 √2 symplectic lifted from Q16, Schur index 2 ρ15 2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -√2 √2 √2 -√2 0 0 0 0 0 0 √2 -√2 √2 -√2 symplectic lifted from Q16, Schur index 2 ρ16 2 2 2 2 -1-√-3 -1+√-3 -2 0 0 -2 0 0 -1-√-3 -1+√-3 0 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ17 2 2 2 2 -1+√-3 -1-√-3 -2 0 0 -2 0 0 -1+√-3 -1-√-3 0 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 0 0 complex lifted from C3×D4 ρ18 2 -2 -2 2 -1+√-3 -1-√-3 0 0 0 0 0 0 1-√-3 1+√-3 √2 -√2 -√2 √2 0 0 0 0 0 0 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 complex lifted from C3×Q16 ρ19 2 -2 -2 2 -1-√-3 -1+√-3 0 0 0 0 0 0 1+√-3 1-√-3 -√2 √2 √2 -√2 0 0 0 0 0 0 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 complex lifted from C3×Q16 ρ20 2 -2 -2 2 -1+√-3 -1-√-3 0 0 0 0 0 0 1-√-3 1+√-3 -√2 √2 √2 -√2 0 0 0 0 0 0 ζ87ζ3-ζ85ζ3 ζ83ζ3-ζ8ζ3 ζ87ζ32-ζ85ζ32 ζ83ζ32-ζ8ζ32 complex lifted from C3×Q16 ρ21 2 -2 -2 2 -1-√-3 -1+√-3 0 0 0 0 0 0 1+√-3 1-√-3 √2 -√2 -√2 √2 0 0 0 0 0 0 ζ83ζ32-ζ8ζ32 ζ87ζ32-ζ85ζ32 ζ83ζ3-ζ8ζ3 ζ87ζ3-ζ85ζ3 complex lifted from C3×Q16 ρ22 3 3 -1 -1 0 0 3 3 -3 -1 -1 1 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ23 3 3 -1 -1 0 0 3 3 3 -1 -1 -1 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ24 3 3 -1 -1 0 0 3 -3 3 -1 1 -1 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ25 3 3 -1 -1 0 0 3 -3 -3 -1 1 1 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ26 6 6 -2 -2 0 0 -6 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4×A4 ρ27 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 3√2 -3√2 √2 -√2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2 ρ28 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 -3√2 3√2 -√2 √2 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of A4×Q16
On 48 points
Generators in S48
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(41 45)(42 46)(43 47)(44 48)
(1 34 43)(2 35 44)(3 36 45)(4 37 46)(5 38 47)(6 39 48)(7 40 41)(8 33 42)(9 19 32)(10 20 25)(11 21 26)(12 22 27)(13 23 28)(14 24 29)(15 17 30)(16 18 31)
(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 41 13 45)(10 48 14 44)(11 47 15 43)(12 46 16 42)(25 39 29 35)(26 38 30 34)(27 37 31 33)(28 36 32 40)

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

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

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

Matrix representation of A4×Q16 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 72 72
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 72 72 72 0 0 1 0 0
,
 8 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 65 65 65 0 0 0 8 0
,
 45 54 0 0 0 27 60 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 72
,
 19 58 0 0 0 29 54 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,1,0,72,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,72,0,0,0,1,72,0],[8,0,0,0,0,0,8,0,0,0,0,0,8,65,0,0,0,0,65,8,0,0,0,65,0],[45,27,0,0,0,54,60,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72],[19,29,0,0,0,58,54,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×Q16 in GAP, Magma, Sage, TeX

A_4\times Q_{16}
% in TeX

G:=Group("A4xQ16");
// GroupNames label

G:=SmallGroup(192,1016);
// by ID

G=gap.SmallGroup(192,1016);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,168,197,176,1011,514,80,1027,1784]);
// 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=a*b=b*a,a*d=d*a,a*e=e*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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