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## G = S3×C16order 96 = 25·3

### Direct product of C16 and S3

Aliases: S3×C16, C485C2, Dic3C16, D6.2C8, C8.19D6, Dic3.2C8, C24.23C22, C16(C3⋊C8), C3⋊C166C2, C31(C2×C16), C3⋊C8.3C4, C16(C3⋊C16), C2.1(S3×C8), C6.1(C2×C8), (C4×S3).4C4, (S3×C8).3C2, C4.16(C4×S3), C12.21(C2×C4), SmallGroup(96,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C16
 Chief series C1 — C3 — C6 — C12 — C24 — S3×C8 — S3×C16
 Lower central C3 — S3×C16
 Upper central C1 — C16

Generators and relations for S3×C16
G = < a,b,c | a16=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of S3×C16
On 48 points
Generators in S48
(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 30 43)(2 31 44)(3 32 45)(4 17 46)(5 18 47)(6 19 48)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(14 27 40)(15 28 41)(16 29 42)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 33)(29 34)(30 35)(31 36)(32 37)

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

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

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

S3×C16 is a maximal subgroup of
C96⋊C2  D12.4C8  C16.12D6  D163S3  D6.2D8  D485C2  C24.60D6  D152C16  D15⋊C16
S3×C16 is a maximal quotient of
C96⋊C2  Dic3⋊C16  D6⋊C16  C24.60D6  D152C16  D15⋊C16

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 C16 S3 D6 C4×S3 S3×C8 S3×C16 kernel S3×C16 C3⋊C16 C48 S3×C8 C3⋊C8 C4×S3 Dic3 D6 S3 C16 C8 C4 C2 C1 # reps 1 1 1 1 2 2 4 4 16 1 1 2 4 8

Matrix representation of S3×C16 in GL2(𝔽17) generated by

 7 0 0 7
,
 0 9 15 16
,
 1 9 0 16
G:=sub<GL(2,GF(17))| [7,0,0,7],[0,15,9,16],[1,0,9,16] >;

S3×C16 in GAP, Magma, Sage, TeX

S_3\times C_{16}
% in TeX

G:=Group("S3xC16");
// GroupNames label

G:=SmallGroup(96,4);
// by ID

G=gap.SmallGroup(96,4);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,31,50,69,2309]);
// Polycyclic

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

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