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G = S3×C16order 96 = 25·3

Direct product of C16 and S3

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C3 — S3×C16
C1C3C6C12C24S3×C8 — S3×C16
C3 — S3×C16
C1C16

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

3C2
3C2
3C22
3C4
3C2×C4
3C8
3C2×C8
3C16
3C2×C16

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

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

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

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

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 2A2B2C 3 4A4B4C4D 6 8A8B8C8D8E8F8G8H12A12B16A···16H16I···16P24A24B24C24D48A···48H
order122234444688888888121216···1616···162424242448···48
size113321133211113333221···13···322222···2

48 irreducible representations

dim11111111122222
type++++++
imageC1C2C2C2C4C4C8C8C16S3D6C4×S3S3×C8S3×C16
kernelS3×C16C3⋊C16C48S3×C8C3⋊C8C4×S3Dic3D6S3C16C8C4C2C1
# reps111122441611248

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

70
07
,
09
1516
,
19
016
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

Export

Subgroup lattice of S3×C16 in TeX

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