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G = S3×D17  order 204 = 22·3·17

Direct product of S3 and D17

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

Aliases: S3×D17, D51⋊C2, C3⋊1D34, C17⋊1D6, C51⋊C22, (S3×C17)⋊C2, (C3×D17)⋊C2, SmallGroup(204,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C51 — S3×D17
Chief series
C1 — C17 — C51 — C3×D17 — S3×D17
Lower central
C51 — S3×D17
Upper central
C1

Generators and relations for S3×D17
 G = < a,b,c,d | a3=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
3C2
17C2
51C2
C3
C17
51C22
S3
17C6
17S3
D17
3C34
3D17
C51
17D6
3D34
C3×D17
D51
S3×C17
S3×D17

Character table of S3×D17

 class 12A2B2C3617A17B17C17D17E17F17G17H34A34B34C34D34E34F34G34H51A51B51C51D51E51F51G51H
 size 131751234222222226666666644444444
ρ1111111111111111111111111111111    trivial
ρ21-11-11111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ311-1-11-1111111111111111111111111    linear of order 2
ρ41-1-111-111111111-1-1-1-1-1-1-1-111111111    linear of order 2
ρ52020-1-12222222200000000-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ620-20-112222222200000000-1-1-1-1-1-1-1-1    orthogonal lifted from D6
ρ72-20020ζ1714+ζ173ζ1716+ζ17ζ1711+ζ176ζ179+ζ178ζ1712+ζ175ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172-ζ1711-ζ176-ζ1712-ζ175-ζ1710-ζ177-ζ1715-ζ172-ζ1714-ζ173-ζ179-ζ178-ζ1713-ζ174-ζ1716-ζ17ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177ζ1715+ζ172ζ1714+ζ173    orthogonal lifted from D34
ρ82-20020ζ1716+ζ17ζ1711+ζ176ζ1715+ζ172ζ1714+ζ173ζ1713+ζ174ζ179+ζ178ζ1710+ζ177ζ1712+ζ175-ζ1715-ζ172-ζ1713-ζ174-ζ179-ζ178-ζ1712-ζ175-ζ1716-ζ17-ζ1714-ζ173-ζ1710-ζ177-ζ1711-ζ176ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1715+ζ172ζ1713+ζ174ζ179+ζ178ζ1712+ζ175ζ1716+ζ17    orthogonal lifted from D34
ρ9220020ζ1712+ζ175ζ1713+ζ174ζ1710+ζ177ζ1715+ζ172ζ1714+ζ173ζ1711+ζ176ζ1716+ζ17ζ179+ζ178ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176ζ179+ζ178ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176ζ179+ζ178ζ1712+ζ175    orthogonal lifted from D17
ρ10220020ζ1714+ζ173ζ1716+ζ17ζ1711+ζ176ζ179+ζ178ζ1712+ζ175ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177ζ1715+ζ172ζ1714+ζ173ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ179+ζ178ζ1713+ζ174ζ1716+ζ17ζ1711+ζ176ζ1712+ζ175ζ1710+ζ177ζ1715+ζ172ζ1714+ζ173    orthogonal lifted from D17
ρ112-20020ζ1711+ζ176ζ1715+ζ172ζ1712+ζ175ζ1716+ζ17ζ1710+ζ177ζ1714+ζ173ζ179+ζ178ζ1713+ζ174-ζ1712-ζ175-ζ1710-ζ177-ζ1714-ζ173-ζ1713-ζ174-ζ1711-ζ176-ζ1716-ζ17-ζ179-ζ178-ζ1715-ζ172ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173ζ1713+ζ174ζ1711+ζ176    orthogonal lifted from D34
ρ12220020ζ1710+ζ177ζ179+ζ178ζ1714+ζ173ζ1713+ζ174ζ1711+ζ176ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175ζ1716+ζ17ζ1710+ζ177ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175ζ1716+ζ17ζ1710+ζ177    orthogonal lifted from D17
ρ132-20020ζ1710+ζ177ζ179+ζ178ζ1714+ζ173ζ1713+ζ174ζ1711+ζ176ζ1712+ζ175ζ1715+ζ172ζ1716+ζ17-ζ1714-ζ173-ζ1711-ζ176-ζ1712-ζ175-ζ1716-ζ17-ζ1710-ζ177-ζ1713-ζ174-ζ1715-ζ172-ζ179-ζ178ζ1713+ζ174ζ1715+ζ172ζ179+ζ178ζ1714+ζ173ζ1711+ζ176ζ1712+ζ175ζ1716+ζ17ζ1710+ζ177    orthogonal lifted from D34
ρ14220020ζ1716+ζ17ζ1711+ζ176ζ1715+ζ172ζ1714+ζ173ζ1713+ζ174ζ179+ζ178ζ1710+ζ177ζ1712+ζ175ζ1715+ζ172ζ1713+ζ174ζ179+ζ178ζ1712+ζ175ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1714+ζ173ζ1710+ζ177ζ1711+ζ176ζ1715+ζ172ζ1713+ζ174ζ179+ζ178ζ1712+ζ175ζ1716+ζ17    orthogonal lifted from D17
ρ152-20020ζ1712+ζ175ζ1713+ζ174ζ1710+ζ177ζ1715+ζ172ζ1714+ζ173ζ1711+ζ176ζ1716+ζ17ζ179+ζ178-ζ1710-ζ177-ζ1714-ζ173-ζ1711-ζ176-ζ179-ζ178-ζ1712-ζ175-ζ1715-ζ172-ζ1716-ζ17-ζ1713-ζ174ζ1715+ζ172ζ1716+ζ17ζ1713+ζ174ζ1710+ζ177ζ1714+ζ173ζ1711+ζ176ζ179+ζ178ζ1712+ζ175    orthogonal lifted from D34
ρ16220020ζ1713+ζ174ζ1710+ζ177ζ179+ζ178ζ1712+ζ175ζ1716+ζ17ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173ζ179+ζ178ζ1716+ζ17ζ1715+ζ172ζ1714+ζ173ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ179+ζ178ζ1716+ζ17ζ1715+ζ172ζ1714+ζ173ζ1713+ζ174    orthogonal lifted from D17
ρ172-20020ζ1715+ζ172ζ1712+ζ175ζ1713+ζ174ζ1711+ζ176ζ179+ζ178ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177-ζ1713-ζ174-ζ179-ζ178-ζ1716-ζ17-ζ1710-ζ177-ζ1715-ζ172-ζ1711-ζ176-ζ1714-ζ173-ζ1712-ζ175ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1713+ζ174ζ179+ζ178ζ1716+ζ17ζ1710+ζ177ζ1715+ζ172    orthogonal lifted from D34
ρ18220020ζ179+ζ178ζ1714+ζ173ζ1716+ζ17ζ1710+ζ177ζ1715+ζ172ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174ζ1711+ζ176ζ179+ζ178ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174ζ1711+ζ176ζ179+ζ178    orthogonal lifted from D17
ρ19220020ζ1711+ζ176ζ1715+ζ172ζ1712+ζ175ζ1716+ζ17ζ1710+ζ177ζ1714+ζ173ζ179+ζ178ζ1713+ζ174ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173ζ1713+ζ174ζ1711+ζ176ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1716+ζ17ζ179+ζ178ζ1715+ζ172ζ1712+ζ175ζ1710+ζ177ζ1714+ζ173ζ1713+ζ174ζ1711+ζ176    orthogonal lifted from D17
ρ20220020ζ1715+ζ172ζ1712+ζ175ζ1713+ζ174ζ1711+ζ176ζ179+ζ178ζ1716+ζ17ζ1714+ζ173ζ1710+ζ177ζ1713+ζ174ζ179+ζ178ζ1716+ζ17ζ1710+ζ177ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1711+ζ176ζ1714+ζ173ζ1712+ζ175ζ1713+ζ174ζ179+ζ178ζ1716+ζ17ζ1710+ζ177ζ1715+ζ172    orthogonal lifted from D17
ρ212-20020ζ179+ζ178ζ1714+ζ173ζ1716+ζ17ζ1710+ζ177ζ1715+ζ172ζ1713+ζ174ζ1712+ζ175ζ1711+ζ176-ζ1716-ζ17-ζ1715-ζ172-ζ1713-ζ174-ζ1711-ζ176-ζ179-ζ178-ζ1710-ζ177-ζ1712-ζ175-ζ1714-ζ173ζ1710+ζ177ζ1712+ζ175ζ1714+ζ173ζ1716+ζ17ζ1715+ζ172ζ1713+ζ174ζ1711+ζ176ζ179+ζ178    orthogonal lifted from D34
ρ222-20020ζ1713+ζ174ζ1710+ζ177ζ179+ζ178ζ1712+ζ175ζ1716+ζ17ζ1715+ζ172ζ1711+ζ176ζ1714+ζ173-ζ179-ζ178-ζ1716-ζ17-ζ1715-ζ172-ζ1714-ζ173-ζ1713-ζ174-ζ1712-ζ175-ζ1711-ζ176-ζ1710-ζ177ζ1712+ζ175ζ1711+ζ176ζ1710+ζ177ζ179+ζ178ζ1716+ζ17ζ1715+ζ172ζ1714+ζ173ζ1713+ζ174    orthogonal lifted from D34
ρ234000-202ζ1715+2ζ1722ζ1712+2ζ1752ζ1713+2ζ1742ζ1711+2ζ1762ζ179+2ζ1782ζ1716+2ζ172ζ1714+2ζ1732ζ1710+2ζ17700000000-ζ1711-ζ176-ζ1714-ζ173-ζ1712-ζ175-ζ1713-ζ174-ζ179-ζ178-ζ1716-ζ17-ζ1710-ζ177-ζ1715-ζ172    orthogonal faithful
ρ244000-202ζ1714+2ζ1732ζ1716+2ζ172ζ1711+2ζ1762ζ179+2ζ1782ζ1712+2ζ1752ζ1710+2ζ1772ζ1713+2ζ1742ζ1715+2ζ17200000000-ζ179-ζ178-ζ1713-ζ174-ζ1716-ζ17-ζ1711-ζ176-ζ1712-ζ175-ζ1710-ζ177-ζ1715-ζ172-ζ1714-ζ173    orthogonal faithful
ρ254000-202ζ1713+2ζ1742ζ1710+2ζ1772ζ179+2ζ1782ζ1712+2ζ1752ζ1716+2ζ172ζ1715+2ζ1722ζ1711+2ζ1762ζ1714+2ζ17300000000-ζ1712-ζ175-ζ1711-ζ176-ζ1710-ζ177-ζ179-ζ178-ζ1716-ζ17-ζ1715-ζ172-ζ1714-ζ173-ζ1713-ζ174    orthogonal faithful
ρ264000-202ζ1711+2ζ1762ζ1715+2ζ1722ζ1712+2ζ1752ζ1716+2ζ172ζ1710+2ζ1772ζ1714+2ζ1732ζ179+2ζ1782ζ1713+2ζ17400000000-ζ1716-ζ17-ζ179-ζ178-ζ1715-ζ172-ζ1712-ζ175-ζ1710-ζ177-ζ1714-ζ173-ζ1713-ζ174-ζ1711-ζ176    orthogonal faithful
ρ274000-202ζ1712+2ζ1752ζ1713+2ζ1742ζ1710+2ζ1772ζ1715+2ζ1722ζ1714+2ζ1732ζ1711+2ζ1762ζ1716+2ζ172ζ179+2ζ17800000000-ζ1715-ζ172-ζ1716-ζ17-ζ1713-ζ174-ζ1710-ζ177-ζ1714-ζ173-ζ1711-ζ176-ζ179-ζ178-ζ1712-ζ175    orthogonal faithful
ρ284000-202ζ1716+2ζ172ζ1711+2ζ1762ζ1715+2ζ1722ζ1714+2ζ1732ζ1713+2ζ1742ζ179+2ζ1782ζ1710+2ζ1772ζ1712+2ζ17500000000-ζ1714-ζ173-ζ1710-ζ177-ζ1711-ζ176-ζ1715-ζ172-ζ1713-ζ174-ζ179-ζ178-ζ1712-ζ175-ζ1716-ζ17    orthogonal faithful
ρ294000-202ζ179+2ζ1782ζ1714+2ζ1732ζ1716+2ζ172ζ1710+2ζ1772ζ1715+2ζ1722ζ1713+2ζ1742ζ1712+2ζ1752ζ1711+2ζ17600000000-ζ1710-ζ177-ζ1712-ζ175-ζ1714-ζ173-ζ1716-ζ17-ζ1715-ζ172-ζ1713-ζ174-ζ1711-ζ176-ζ179-ζ178    orthogonal faithful
ρ304000-202ζ1710+2ζ1772ζ179+2ζ1782ζ1714+2ζ1732ζ1713+2ζ1742ζ1711+2ζ1762ζ1712+2ζ1752ζ1715+2ζ1722ζ1716+2ζ1700000000-ζ1713-ζ174-ζ1715-ζ172-ζ179-ζ178-ζ1714-ζ173-ζ1711-ζ176-ζ1712-ζ175-ζ1716-ζ17-ζ1710-ζ177    orthogonal faithful

Smallest permutation representation of S3×D17
►On 51 points
Generators in S51
(1 20 43)(2 21 44)(3 22 45)(4 23 46)(5 24 47)(6 25 48)(7 26 49)(8 27 50)(9 28 51)(10 29 35)(11 30 36)(12 31 37)(13 32 38)(14 33 39)(15 34 40)(16 18 41)(17 19 42)

(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 35)(30 36)(31 37)(32 38)(33 39)(34 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 49 50 51)

(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 21)(19 20)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)
 
G:=sub<Sym(51)| (1,20,43)(2,21,44)(3,22,45)(4,23,46)(5,24,47)(6,25,48)(7,26,49)(8,27,50)(9,28,51)(10,29,35)(11,30,36)(12,31,37)(13,32,38)(14,33,39)(15,34,40)(16,18,41)(17,19,42), (18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,35)(30,36)(31,37)(32,38)(33,39)(34,40), (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)>;
 
G:=Group( (1,20,43)(2,21,44)(3,22,45)(4,23,46)(5,24,47)(6,25,48)(7,26,49)(8,27,50)(9,28,51)(10,29,35)(11,30,36)(12,31,37)(13,32,38)(14,33,39)(15,34,40)(16,18,41)(17,19,42), (18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,35)(30,36)(31,37)(32,38)(33,39)(34,40), (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)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,21)(19,20)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43) );
 
G=PermutationGroup([[(1,20,43),(2,21,44),(3,22,45),(4,23,46),(5,24,47),(6,25,48),(7,26,49),(8,27,50),(9,28,51),(10,29,35),(11,30,36),(12,31,37),(13,32,38),(14,33,39),(15,34,40),(16,18,41),(17,19,42)], [(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,35),(30,36),(31,37),(32,38),(33,39),(34,40)], [(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),(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,10),(18,21),(19,20),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)]])
 

S3×D17 is a maximal quotient of   D51⋊2C4  C51⋊D4  C3⋊D68  C17⋊D12  C51⋊Q8

Matrix representation of S3×D17 ►in GL4(𝔽103) generated by

1000
0100
000102
001102
,
1000
0100
0001
0010
,
0100
1029500
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(103))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,102,102],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,102,0,0,1,95,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
 

S3×D17 in GAP, Magma, Sage, TeX 

S_3\times D_{17}
 
% in TeX
 
G:=Group("S3xD17");
 
// GroupNames label
 
G:=SmallGroup(204,7);
 
// by ID
 
G=gap.SmallGroup(204,7);
 
# by ID
 
G:=PCGroup([4,-2,-2,-3,-17,54,3075]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^17=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
 

Export

Subgroup lattice of S3×D17 in TeX
Character table of S3×D17 in TeX

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