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## G = A4×SD16order 192 = 26·3

### Direct product of A4 and SD16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C4 — A4×SD16
 Chief series C1 — C22 — C23 — C22×C4 — C4×A4 — D4×A4 — A4×SD16
 Lower central C22 — C23 — C22×C4 — A4×SD16
 Upper central C1 — C2 — C4 — SD16

Generators and relations for A4×SD16
G = < a,b,c,d,e | a2=b2=c3=d8=e2=1, cac-1=ab=ba, ad=da, ae=ea, cbc-1=a, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 324 in 83 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C8, C8, C2×C4, D4, D4, Q8, Q8, C23, C23, C12, A4, C2×C6, C2×C8, SD16, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C24, C3×D4, C3×Q8, C2×A4, C2×A4, C22×C8, C2×SD16, C22×D4, C22×Q8, C3×SD16, C4×A4, C4×A4, C22×A4, C22×SD16, C8×A4, D4×A4, Q8×A4, A4×SD16
Quotients: C1, C2, C3, C22, C6, D4, A4, C2×C6, SD16, C3×D4, C2×A4, C3×SD16, C22×A4, D4×A4, A4×SD16

Character table of A4×SD16

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 3 3 4 12 4 4 2 4 6 12 4 4 16 16 2 2 6 6 8 8 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 -1 -1 ζ3 ζ32 1 -1 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 1 1 ζ32 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 linear of order 6 ρ6 1 1 1 1 -1 -1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ6 ζ65 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ7 1 1 1 1 1 1 ζ3 ζ32 1 -1 1 -1 ζ3 ζ32 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ3 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ8 1 1 1 1 -1 -1 ζ32 ζ3 1 -1 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 1 1 ζ3 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 linear of order 6 ρ9 1 1 1 1 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 1 1 1 1 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ11 1 1 1 1 -1 -1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ65 ζ6 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 linear of order 6 ρ12 1 1 1 1 1 1 ζ32 ζ3 1 -1 1 -1 ζ32 ζ3 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ32 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 linear of order 6 ρ13 2 2 2 2 0 0 2 2 -2 0 -2 0 2 2 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 0 0 -1+√-3 -1-√-3 -2 0 -2 0 -1+√-3 -1-√-3 0 0 0 0 0 0 1+√-3 1-√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ15 2 2 2 2 0 0 -1-√-3 -1+√-3 -2 0 -2 0 -1-√-3 -1+√-3 0 0 0 0 0 0 1-√-3 1+√-3 0 0 0 0 0 0 complex lifted from C3×D4 ρ16 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 √-2 -√-2 √-2 -√-2 complex lifted from SD16 ρ17 2 -2 -2 2 0 0 2 2 0 0 0 0 -2 -2 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 -√-2 √-2 -√-2 √-2 complex lifted from SD16 ρ18 2 -2 -2 2 0 0 -1-√-3 -1+√-3 0 0 0 0 1+√-3 1-√-3 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 complex lifted from C3×SD16 ρ19 2 -2 -2 2 0 0 -1+√-3 -1-√-3 0 0 0 0 1-√-3 1+√-3 0 0 -√-2 √-2 -√-2 √-2 0 0 0 0 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 complex lifted from C3×SD16 ρ20 2 -2 -2 2 0 0 -1+√-3 -1-√-3 0 0 0 0 1-√-3 1+√-3 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 complex lifted from C3×SD16 ρ21 2 -2 -2 2 0 0 -1-√-3 -1+√-3 0 0 0 0 1+√-3 1-√-3 0 0 √-2 -√-2 √-2 -√-2 0 0 0 0 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 complex lifted from C3×SD16 ρ22 3 3 -1 -1 3 -1 0 0 3 -3 -1 1 0 0 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ23 3 3 -1 -1 -3 1 0 0 3 3 -1 -1 0 0 0 0 -3 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ24 3 3 -1 -1 3 -1 0 0 3 3 -1 -1 0 0 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from A4 ρ25 3 3 -1 -1 -3 1 0 0 3 -3 -1 1 0 0 0 0 3 3 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ26 6 6 -2 -2 0 0 0 0 -6 0 2 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 0 0 3√-2 -3√-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex faithful ρ28 6 -6 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -3√-2 3√-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of A4×SD16
On 24 points - transitive group 24T329
Generators in S24
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)
(1 5)(2 6)(3 7)(4 8)(17 21)(18 22)(19 23)(20 24)
(1 15 17)(2 16 18)(3 9 19)(4 10 20)(5 11 21)(6 12 22)(7 13 23)(8 14 24)
(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 4)(3 7)(6 8)(9 13)(10 16)(12 14)(18 20)(19 23)(22 24)

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

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

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

G:=TransitiveGroup(24,329);

Matrix representation of A4×SD16 in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 72 0 0 0 0 72 0 1 0 0 72 1 0
,
 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
,
 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 72 0 0 0 72 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 61 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,1,0],[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],[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],[0,72,0,0,0,72,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,61,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A4×SD16 in GAP, Magma, Sage, TeX

A_4\times {\rm SD}_{16}
% in TeX

G:=Group("A4xSD16");
// GroupNames label

G:=SmallGroup(192,1015);
// by ID

G=gap.SmallGroup(192,1015);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,2,197,176,1011,514,80,1027,1784]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^8=e^2=1,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=d^3>;
// generators/relations

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