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## G = SD16.A4order 192 = 26·3

### The non-split extension by SD16 of A4 acting through Inn(SD16)

Aliases: SD16.A4, 2- 1+4.C6, 2+ 1+42C6, SL2(𝔽3).12D4, D4○SD16⋊C3, C8○D41C6, C8.A47C2, C8.4(C2×A4), D4.A44C2, D4.2(C2×A4), C2.10(D4×A4), Q8.A45C2, Q8.5(C2×A4), Q8.4(C3×D4), C4.5(C22×A4), C4.A4.16C22, C4○D4.2(C2×C6), SmallGroup(192,1018)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4 — SD16.A4
 Chief series C1 — C2 — Q8 — C4○D4 — C4.A4 — D4.A4 — SD16.A4
 Lower central Q8 — C4○D4 — SD16.A4
 Upper central C1 — C2 — C4 — SD16

Generators and relations for SD16.A4
G = < a,b,c,d,e | a8=b2=e3=1, c2=d2=a4, bab=a3, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a4c, ece-1=a4cd, ede-1=c >

Subgroups: 275 in 73 conjugacy classes, 19 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×4], C6 [×2], C8, C8, C2×C4 [×4], D4, D4 [×5], Q8 [×2], Q8 [×2], C23, C12 [×2], C2×C6, C2×C8, M4(2), D8, SD16, SD16 [×3], Q16, C2×D4 [×2], C2×Q8 [×2], C4○D4, C4○D4 [×4], C24, SL2(𝔽3), C3×D4, C3×Q8, C8○D4, C2×SD16, C4○D8, C8⋊C22, C8.C22, 2+ 1+4, 2- 1+4, C3×SD16, C2×SL2(𝔽3), C4.A4, C4.A4, D4○SD16, C8.A4, Q8.A4, D4.A4, SD16.A4
Quotients: C1, C2 [×3], C3, C22, C6 [×3], D4, A4, C2×C6, C3×D4, C2×A4 [×3], C22×A4, D4×A4, SD16.A4

Character table of SD16.A4

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

Smallest permutation representation of SD16.A4
On 32 points
Generators in S32
(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)
(2 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)
(1 18 5 22)(2 19 6 23)(3 20 7 24)(4 21 8 17)(9 30 13 26)(10 31 14 27)(11 32 15 28)(12 25 16 29)
(1 12 5 16)(2 13 6 9)(3 14 7 10)(4 15 8 11)(17 28 21 32)(18 29 22 25)(19 30 23 26)(20 31 24 27)
(9 30 23)(10 31 24)(11 32 17)(12 25 18)(13 26 19)(14 27 20)(15 28 21)(16 29 22)

G:=sub<Sym(32)| (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32), (1,18,5,22)(2,19,6,23)(3,20,7,24)(4,21,8,17)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (9,30,23)(10,31,24)(11,32,17)(12,25,18)(13,26,19)(14,27,20)(15,28,21)(16,29,22)>;

G:=Group( (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32), (1,18,5,22)(2,19,6,23)(3,20,7,24)(4,21,8,17)(9,30,13,26)(10,31,14,27)(11,32,15,28)(12,25,16,29), (1,12,5,16)(2,13,6,9)(3,14,7,10)(4,15,8,11)(17,28,21,32)(18,29,22,25)(19,30,23,26)(20,31,24,27), (9,30,23)(10,31,24)(11,32,17)(12,25,18)(13,26,19)(14,27,20)(15,28,21)(16,29,22) );

G=PermutationGroup([(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32)], [(1,18,5,22),(2,19,6,23),(3,20,7,24),(4,21,8,17),(9,30,13,26),(10,31,14,27),(11,32,15,28),(12,25,16,29)], [(1,12,5,16),(2,13,6,9),(3,14,7,10),(4,15,8,11),(17,28,21,32),(18,29,22,25),(19,30,23,26),(20,31,24,27)], [(9,30,23),(10,31,24),(11,32,17),(12,25,18),(13,26,19),(14,27,20),(15,28,21),(16,29,22)])

Matrix representation of SD16.A4 in GL4(𝔽3) generated by

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

SD16.A4 in GAP, Magma, Sage, TeX

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

G:=Group("SD16.A4");
// GroupNames label

G:=SmallGroup(192,1018);
// by ID

G=gap.SmallGroup(192,1018);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-2,197,680,3027,1522,248,438,172,775,285,124]);
// Polycyclic

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

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