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G = S3×D16  order 192 = 26·3

Direct product of S3 and D16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: S3×D16, D8⋊1D6, C16⋊4D6, D48⋊4C2, C48⋊2C22, D6.12D8, D24⋊5C22, Dic3.3D8, C24.13C23, C3⋊2(C2×D16), (S3×D8)⋊3C2, C3⋊C8.11D4, C4.1(S3×D4), (S3×C16)⋊1C2, (C3×D16)⋊2C2, C3⋊D16⋊1C2, C3⋊C16⋊5C22, C12.7(C2×D4), C6.32(C2×D8), C2.16(S3×D8), (C4×S3).18D4, (C3×D8)⋊5C22, (S3×C8).9C22, C8.19(C22×S3), SmallGroup(192,469)

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 2A2B2C2D2E2F2G 3 4A4B6A6B6C8A8B8C8D 12 16A16B16C16D16E16F16G16H24A24B48A48B48C48D
order122222223446668888121616161616161616242448484848
size1133882424226216162266422226666444444

33 irreducible representations

dim11111122222222444
type+++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8D8D16S3×D4S3×D8S3×D16
kernelS3×D16S3×C16D48C3⋊D16C3×D16S3×D8D16C3⋊C8C4×S3C16D8Dic3D6S3C4C2C1
# reps11121211112228124

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

09600
19600
0010
0001
,
0100
1000
00960
00096
,
96000
09600
009571
002695
,
1000
0100
0010
00096
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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