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## G = S3×D17order 204 = 22·3·17

### Direct product of S3 and D17

Aliases: S3×D17, D51⋊C2, C31D34, C171D6, 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 >

3C2
17C2
51C2
51C22
17C6
17S3
3C34
3D17
17D6
3D34

Character table of S3×D17

 class 1 2A 2B 2C 3 6 17A 17B 17C 17D 17E 17F 17G 17H 34A 34B 34C 34D 34E 34F 34G 34H 51A 51B 51C 51D 51E 51F 51G 51H size 1 3 17 51 2 34 2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 0 2 0 -1 -1 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 0 -2 0 -1 1 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 -2 0 0 2 0 ζ1714+ζ173 ζ1716+ζ17 ζ1711+ζ176 ζ179+ζ178 ζ1712+ζ175 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 -ζ1711-ζ176 -ζ1712-ζ175 -ζ1710-ζ177 -ζ1715-ζ172 -ζ1714-ζ173 -ζ179-ζ178 -ζ1713-ζ174 -ζ1716-ζ17 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 ζ1715+ζ172 ζ1714+ζ173 orthogonal lifted from D34 ρ8 2 -2 0 0 2 0 ζ1716+ζ17 ζ1711+ζ176 ζ1715+ζ172 ζ1714+ζ173 ζ1713+ζ174 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 -ζ1715-ζ172 -ζ1713-ζ174 -ζ179-ζ178 -ζ1712-ζ175 -ζ1716-ζ17 -ζ1714-ζ173 -ζ1710-ζ177 -ζ1711-ζ176 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 ζ1712+ζ175 ζ1716+ζ17 orthogonal lifted from D34 ρ9 2 2 0 0 2 0 ζ1712+ζ175 ζ1713+ζ174 ζ1710+ζ177 ζ1715+ζ172 ζ1714+ζ173 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 ζ179+ζ178 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 ζ179+ζ178 ζ1712+ζ175 orthogonal lifted from D17 ρ10 2 2 0 0 2 0 ζ1714+ζ173 ζ1716+ζ17 ζ1711+ζ176 ζ179+ζ178 ζ1712+ζ175 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 ζ1715+ζ172 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ179+ζ178 ζ1713+ζ174 ζ1716+ζ17 ζ1711+ζ176 ζ1712+ζ175 ζ1710+ζ177 ζ1715+ζ172 ζ1714+ζ173 orthogonal lifted from D17 ρ11 2 -2 0 0 2 0 ζ1711+ζ176 ζ1715+ζ172 ζ1712+ζ175 ζ1716+ζ17 ζ1710+ζ177 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 -ζ1712-ζ175 -ζ1710-ζ177 -ζ1714-ζ173 -ζ1713-ζ174 -ζ1711-ζ176 -ζ1716-ζ17 -ζ179-ζ178 -ζ1715-ζ172 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 ζ1713+ζ174 ζ1711+ζ176 orthogonal lifted from D34 ρ12 2 2 0 0 2 0 ζ1710+ζ177 ζ179+ζ178 ζ1714+ζ173 ζ1713+ζ174 ζ1711+ζ176 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 ζ1716+ζ17 ζ1710+ζ177 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 ζ1716+ζ17 ζ1710+ζ177 orthogonal lifted from D17 ρ13 2 -2 0 0 2 0 ζ1710+ζ177 ζ179+ζ178 ζ1714+ζ173 ζ1713+ζ174 ζ1711+ζ176 ζ1712+ζ175 ζ1715+ζ172 ζ1716+ζ17 -ζ1714-ζ173 -ζ1711-ζ176 -ζ1712-ζ175 -ζ1716-ζ17 -ζ1710-ζ177 -ζ1713-ζ174 -ζ1715-ζ172 -ζ179-ζ178 ζ1713+ζ174 ζ1715+ζ172 ζ179+ζ178 ζ1714+ζ173 ζ1711+ζ176 ζ1712+ζ175 ζ1716+ζ17 ζ1710+ζ177 orthogonal lifted from D34 ρ14 2 2 0 0 2 0 ζ1716+ζ17 ζ1711+ζ176 ζ1715+ζ172 ζ1714+ζ173 ζ1713+ζ174 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 ζ1712+ζ175 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1714+ζ173 ζ1710+ζ177 ζ1711+ζ176 ζ1715+ζ172 ζ1713+ζ174 ζ179+ζ178 ζ1712+ζ175 ζ1716+ζ17 orthogonal lifted from D17 ρ15 2 -2 0 0 2 0 ζ1712+ζ175 ζ1713+ζ174 ζ1710+ζ177 ζ1715+ζ172 ζ1714+ζ173 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 -ζ1710-ζ177 -ζ1714-ζ173 -ζ1711-ζ176 -ζ179-ζ178 -ζ1712-ζ175 -ζ1715-ζ172 -ζ1716-ζ17 -ζ1713-ζ174 ζ1715+ζ172 ζ1716+ζ17 ζ1713+ζ174 ζ1710+ζ177 ζ1714+ζ173 ζ1711+ζ176 ζ179+ζ178 ζ1712+ζ175 orthogonal lifted from D34 ρ16 2 2 0 0 2 0 ζ1713+ζ174 ζ1710+ζ177 ζ179+ζ178 ζ1712+ζ175 ζ1716+ζ17 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 ζ1714+ζ173 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 ζ1714+ζ173 ζ1713+ζ174 orthogonal lifted from D17 ρ17 2 -2 0 0 2 0 ζ1715+ζ172 ζ1712+ζ175 ζ1713+ζ174 ζ1711+ζ176 ζ179+ζ178 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 -ζ1713-ζ174 -ζ179-ζ178 -ζ1716-ζ17 -ζ1710-ζ177 -ζ1715-ζ172 -ζ1711-ζ176 -ζ1714-ζ173 -ζ1712-ζ175 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 ζ1710+ζ177 ζ1715+ζ172 orthogonal lifted from D34 ρ18 2 2 0 0 2 0 ζ179+ζ178 ζ1714+ζ173 ζ1716+ζ17 ζ1710+ζ177 ζ1715+ζ172 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 ζ1711+ζ176 ζ179+ζ178 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 ζ1711+ζ176 ζ179+ζ178 orthogonal lifted from D17 ρ19 2 2 0 0 2 0 ζ1711+ζ176 ζ1715+ζ172 ζ1712+ζ175 ζ1716+ζ17 ζ1710+ζ177 ζ1714+ζ173 ζ179+ζ178 ζ1713+ζ174 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 ζ1713+ζ174 ζ1711+ζ176 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1716+ζ17 ζ179+ζ178 ζ1715+ζ172 ζ1712+ζ175 ζ1710+ζ177 ζ1714+ζ173 ζ1713+ζ174 ζ1711+ζ176 orthogonal lifted from D17 ρ20 2 2 0 0 2 0 ζ1715+ζ172 ζ1712+ζ175 ζ1713+ζ174 ζ1711+ζ176 ζ179+ζ178 ζ1716+ζ17 ζ1714+ζ173 ζ1710+ζ177 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 ζ1710+ζ177 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1711+ζ176 ζ1714+ζ173 ζ1712+ζ175 ζ1713+ζ174 ζ179+ζ178 ζ1716+ζ17 ζ1710+ζ177 ζ1715+ζ172 orthogonal lifted from D17 ρ21 2 -2 0 0 2 0 ζ179+ζ178 ζ1714+ζ173 ζ1716+ζ17 ζ1710+ζ177 ζ1715+ζ172 ζ1713+ζ174 ζ1712+ζ175 ζ1711+ζ176 -ζ1716-ζ17 -ζ1715-ζ172 -ζ1713-ζ174 -ζ1711-ζ176 -ζ179-ζ178 -ζ1710-ζ177 -ζ1712-ζ175 -ζ1714-ζ173 ζ1710+ζ177 ζ1712+ζ175 ζ1714+ζ173 ζ1716+ζ17 ζ1715+ζ172 ζ1713+ζ174 ζ1711+ζ176 ζ179+ζ178 orthogonal lifted from D34 ρ22 2 -2 0 0 2 0 ζ1713+ζ174 ζ1710+ζ177 ζ179+ζ178 ζ1712+ζ175 ζ1716+ζ17 ζ1715+ζ172 ζ1711+ζ176 ζ1714+ζ173 -ζ179-ζ178 -ζ1716-ζ17 -ζ1715-ζ172 -ζ1714-ζ173 -ζ1713-ζ174 -ζ1712-ζ175 -ζ1711-ζ176 -ζ1710-ζ177 ζ1712+ζ175 ζ1711+ζ176 ζ1710+ζ177 ζ179+ζ178 ζ1716+ζ17 ζ1715+ζ172 ζ1714+ζ173 ζ1713+ζ174 orthogonal lifted from D34 ρ23 4 0 0 0 -2 0 2ζ1715+2ζ172 2ζ1712+2ζ175 2ζ1713+2ζ174 2ζ1711+2ζ176 2ζ179+2ζ178 2ζ1716+2ζ17 2ζ1714+2ζ173 2ζ1710+2ζ177 0 0 0 0 0 0 0 0 -ζ1711-ζ176 -ζ1714-ζ173 -ζ1712-ζ175 -ζ1713-ζ174 -ζ179-ζ178 -ζ1716-ζ17 -ζ1710-ζ177 -ζ1715-ζ172 orthogonal faithful ρ24 4 0 0 0 -2 0 2ζ1714+2ζ173 2ζ1716+2ζ17 2ζ1711+2ζ176 2ζ179+2ζ178 2ζ1712+2ζ175 2ζ1710+2ζ177 2ζ1713+2ζ174 2ζ1715+2ζ172 0 0 0 0 0 0 0 0 -ζ179-ζ178 -ζ1713-ζ174 -ζ1716-ζ17 -ζ1711-ζ176 -ζ1712-ζ175 -ζ1710-ζ177 -ζ1715-ζ172 -ζ1714-ζ173 orthogonal faithful ρ25 4 0 0 0 -2 0 2ζ1713+2ζ174 2ζ1710+2ζ177 2ζ179+2ζ178 2ζ1712+2ζ175 2ζ1716+2ζ17 2ζ1715+2ζ172 2ζ1711+2ζ176 2ζ1714+2ζ173 0 0 0 0 0 0 0 0 -ζ1712-ζ175 -ζ1711-ζ176 -ζ1710-ζ177 -ζ179-ζ178 -ζ1716-ζ17 -ζ1715-ζ172 -ζ1714-ζ173 -ζ1713-ζ174 orthogonal faithful ρ26 4 0 0 0 -2 0 2ζ1711+2ζ176 2ζ1715+2ζ172 2ζ1712+2ζ175 2ζ1716+2ζ17 2ζ1710+2ζ177 2ζ1714+2ζ173 2ζ179+2ζ178 2ζ1713+2ζ174 0 0 0 0 0 0 0 0 -ζ1716-ζ17 -ζ179-ζ178 -ζ1715-ζ172 -ζ1712-ζ175 -ζ1710-ζ177 -ζ1714-ζ173 -ζ1713-ζ174 -ζ1711-ζ176 orthogonal faithful ρ27 4 0 0 0 -2 0 2ζ1712+2ζ175 2ζ1713+2ζ174 2ζ1710+2ζ177 2ζ1715+2ζ172 2ζ1714+2ζ173 2ζ1711+2ζ176 2ζ1716+2ζ17 2ζ179+2ζ178 0 0 0 0 0 0 0 0 -ζ1715-ζ172 -ζ1716-ζ17 -ζ1713-ζ174 -ζ1710-ζ177 -ζ1714-ζ173 -ζ1711-ζ176 -ζ179-ζ178 -ζ1712-ζ175 orthogonal faithful ρ28 4 0 0 0 -2 0 2ζ1716+2ζ17 2ζ1711+2ζ176 2ζ1715+2ζ172 2ζ1714+2ζ173 2ζ1713+2ζ174 2ζ179+2ζ178 2ζ1710+2ζ177 2ζ1712+2ζ175 0 0 0 0 0 0 0 0 -ζ1714-ζ173 -ζ1710-ζ177 -ζ1711-ζ176 -ζ1715-ζ172 -ζ1713-ζ174 -ζ179-ζ178 -ζ1712-ζ175 -ζ1716-ζ17 orthogonal faithful ρ29 4 0 0 0 -2 0 2ζ179+2ζ178 2ζ1714+2ζ173 2ζ1716+2ζ17 2ζ1710+2ζ177 2ζ1715+2ζ172 2ζ1713+2ζ174 2ζ1712+2ζ175 2ζ1711+2ζ176 0 0 0 0 0 0 0 0 -ζ1710-ζ177 -ζ1712-ζ175 -ζ1714-ζ173 -ζ1716-ζ17 -ζ1715-ζ172 -ζ1713-ζ174 -ζ1711-ζ176 -ζ179-ζ178 orthogonal faithful ρ30 4 0 0 0 -2 0 2ζ1710+2ζ177 2ζ179+2ζ178 2ζ1714+2ζ173 2ζ1713+2ζ174 2ζ1711+2ζ176 2ζ1712+2ζ175 2ζ1715+2ζ172 2ζ1716+2ζ17 0 0 0 0 0 0 0 0 -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   D512C4  C51⋊D4  C3⋊D68  C17⋊D12  C51⋊Q8

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

 1 0 0 0 0 1 0 0 0 0 0 102 0 0 1 102
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 102 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
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

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