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## G = S3×D16order 192 = 26·3

### Direct product of S3 and D16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — S3×D16
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×D8 — S3×D16
 Lower central C3 — C6 — C12 — C24 — S3×D16
 Upper central C1 — C2 — C4 — C8 — D16

Generators and relations for S3×D16
G = < a,b,c,d | a3=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 476 in 98 conjugacy classes, 33 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, C23, Dic3, C12, D6, D6, C2×C6, C16, C16, C2×C8, D8, D8, C2×D4, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C2×C16, D16, D16, C2×D8, C3⋊C16, C48, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C2×D16, S3×C16, D48, C3⋊D16, C3×D16, S3×D8, S3×D16
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C22×S3, D16, C2×D8, S3×D4, C2×D16, S3×D8, S3×D16

Smallest permutation representation of S3×D16
On 48 points
Generators in S48
(1 37 21)(2 38 22)(3 39 23)(4 40 24)(5 41 25)(6 42 26)(7 43 27)(8 44 28)(9 45 29)(10 46 30)(11 47 31)(12 48 32)(13 33 17)(14 34 18)(15 35 19)(16 36 20)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 25)(18 24)(19 23)(20 22)(26 32)(27 31)(28 30)(33 41)(34 40)(35 39)(36 38)(42 48)(43 47)(44 46)

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 6A 6B 6C 8A 8B 8C 8D 12 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 48A 48B 48C 48D order 1 2 2 2 2 2 2 2 3 4 4 6 6 6 8 8 8 8 12 16 16 16 16 16 16 16 16 24 24 48 48 48 48 size 1 1 3 3 8 8 24 24 2 2 6 2 16 16 2 2 6 6 4 2 2 2 2 6 6 6 6 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D8 D8 D16 S3×D4 S3×D8 S3×D16 kernel S3×D16 S3×C16 D48 C3⋊D16 C3×D16 S3×D8 D16 C3⋊C8 C4×S3 C16 D8 Dic3 D6 S3 C4 C2 C1 # reps 1 1 1 2 1 2 1 1 1 1 2 2 2 8 1 2 4

Matrix representation of S3×D16 in GL4(𝔽97) generated by

 0 96 0 0 1 96 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 96 0 0 0 0 96
,
 96 0 0 0 0 96 0 0 0 0 95 71 0 0 26 95
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 96
G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,96,0,0,0,0,96],[96,0,0,0,0,96,0,0,0,0,95,26,0,0,71,95],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,96] >;

S3×D16 in GAP, Magma, Sage, TeX

S_3\times D_{16}
% in TeX

G:=Group("S3xD16");
// GroupNames label

G:=SmallGroup(192,469);
// by ID

G=gap.SmallGroup(192,469);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,135,346,185,192,851,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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