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## G = C3×C17⋊C8order 408 = 23·3·17

### Direct product of C3 and C17⋊C8

Aliases: C3×C17⋊C8, C17⋊C24, C512C8, D17.C12, C17⋊C4.C6, (C3×D17).2C4, (C3×C17⋊C4).2C2, SmallGroup(408,33)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C3×C17⋊C8
 Chief series C1 — C17 — D17 — C17⋊C4 — C3×C17⋊C4 — C3×C17⋊C8
 Lower central C17 — C3×C17⋊C8
 Upper central C1 — C3

Generators and relations for C3×C17⋊C8
G = < a,b,c | a3=b17=c8=1, ab=ba, ac=ca, cbc-1=b2 >

Character table of C3×C17⋊C8

 class 1 2 3A 3B 4A 4B 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 17A 17B 24A 24B 24C 24D 24E 24F 24G 24H 51A 51B 51C 51D size 1 17 1 1 17 17 17 17 17 17 17 17 17 17 17 17 8 8 17 17 17 17 17 17 17 17 8 8 8 8 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 ζ32 ζ3 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ3 ζ32 ζ32 ζ3 1 1 ζ6 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ32 ζ3 ζ32 ζ3 linear of order 6 ρ4 1 1 ζ3 ζ32 1 1 ζ3 ζ32 1 1 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 ζ3 ζ32 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ32 ζ3 ζ3 ζ32 1 1 ζ65 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ3 ζ32 ζ3 ζ32 linear of order 6 ρ6 1 1 ζ32 ζ3 1 1 ζ32 ζ3 1 1 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ7 1 1 1 1 -1 -1 1 1 -i i i -i -1 -1 -1 -1 1 1 -i i i -i -i i i -i 1 1 1 1 linear of order 4 ρ8 1 1 1 1 -1 -1 1 1 i -i -i i -1 -1 -1 -1 1 1 i -i -i i i -i -i i 1 1 1 1 linear of order 4 ρ9 1 -1 1 1 -i i -1 -1 ζ8 ζ87 ζ83 ζ85 i -i i -i 1 1 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 1 1 1 1 linear of order 8 ρ10 1 -1 1 1 i -i -1 -1 ζ83 ζ85 ζ8 ζ87 -i i -i i 1 1 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 1 1 1 1 linear of order 8 ρ11 1 -1 1 1 -i i -1 -1 ζ85 ζ83 ζ87 ζ8 i -i i -i 1 1 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 1 1 1 1 linear of order 8 ρ12 1 -1 1 1 i -i -1 -1 ζ87 ζ8 ζ85 ζ83 -i i -i i 1 1 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 1 1 1 1 linear of order 8 ρ13 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 -i i i -i ζ65 ζ6 ζ6 ζ65 1 1 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 12 ρ14 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 i -i -i i ζ6 ζ65 ζ65 ζ6 1 1 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 12 ρ15 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 -i i i -i ζ6 ζ65 ζ65 ζ6 1 1 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 12 ρ16 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 i -i -i i ζ65 ζ6 ζ6 ζ65 1 1 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 12 ρ17 1 -1 ζ32 ζ3 i -i ζ6 ζ65 ζ83 ζ85 ζ8 ζ87 ζ86ζ3 ζ82ζ32 ζ86ζ32 ζ82ζ3 1 1 ζ83ζ32 ζ85ζ3 ζ8ζ3 ζ87ζ3 ζ83ζ3 ζ85ζ32 ζ8ζ32 ζ87ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 24 ρ18 1 -1 ζ32 ζ3 -i i ζ6 ζ65 ζ85 ζ83 ζ87 ζ8 ζ82ζ3 ζ86ζ32 ζ82ζ32 ζ86ζ3 1 1 ζ85ζ32 ζ83ζ3 ζ87ζ3 ζ8ζ3 ζ85ζ3 ζ83ζ32 ζ87ζ32 ζ8ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 24 ρ19 1 -1 ζ32 ζ3 -i i ζ6 ζ65 ζ8 ζ87 ζ83 ζ85 ζ82ζ3 ζ86ζ32 ζ82ζ32 ζ86ζ3 1 1 ζ8ζ32 ζ87ζ3 ζ83ζ3 ζ85ζ3 ζ8ζ3 ζ87ζ32 ζ83ζ32 ζ85ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 24 ρ20 1 -1 ζ3 ζ32 -i i ζ65 ζ6 ζ85 ζ83 ζ87 ζ8 ζ82ζ32 ζ86ζ3 ζ82ζ3 ζ86ζ32 1 1 ζ85ζ3 ζ83ζ32 ζ87ζ32 ζ8ζ32 ζ85ζ32 ζ83ζ3 ζ87ζ3 ζ8ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 24 ρ21 1 -1 ζ3 ζ32 i -i ζ65 ζ6 ζ87 ζ8 ζ85 ζ83 ζ86ζ32 ζ82ζ3 ζ86ζ3 ζ82ζ32 1 1 ζ87ζ3 ζ8ζ32 ζ85ζ32 ζ83ζ32 ζ87ζ32 ζ8ζ3 ζ85ζ3 ζ83ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 24 ρ22 1 -1 ζ32 ζ3 i -i ζ6 ζ65 ζ87 ζ8 ζ85 ζ83 ζ86ζ3 ζ82ζ32 ζ86ζ32 ζ82ζ3 1 1 ζ87ζ32 ζ8ζ3 ζ85ζ3 ζ83ζ3 ζ87ζ3 ζ8ζ32 ζ85ζ32 ζ83ζ32 ζ32 ζ3 ζ32 ζ3 linear of order 24 ρ23 1 -1 ζ3 ζ32 i -i ζ65 ζ6 ζ83 ζ85 ζ8 ζ87 ζ86ζ32 ζ82ζ3 ζ86ζ3 ζ82ζ32 1 1 ζ83ζ3 ζ85ζ32 ζ8ζ32 ζ87ζ32 ζ83ζ32 ζ85ζ3 ζ8ζ3 ζ87ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 24 ρ24 1 -1 ζ3 ζ32 -i i ζ65 ζ6 ζ8 ζ87 ζ83 ζ85 ζ82ζ32 ζ86ζ3 ζ82ζ3 ζ86ζ32 1 1 ζ8ζ3 ζ87ζ32 ζ83ζ32 ζ85ζ32 ζ8ζ32 ζ87ζ3 ζ83ζ3 ζ85ζ3 ζ3 ζ32 ζ3 ζ32 linear of order 24 ρ25 8 0 8 8 0 0 0 0 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 -1+√17/2 -1-√17/2 orthogonal lifted from C17⋊C8 ρ26 8 0 8 8 0 0 0 0 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 -1-√17/2 -1+√17/2 orthogonal lifted from C17⋊C8 ρ27 8 0 -4-4√-3 -4+4√-3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 0 0 0 0 0 0 0 0 ζ32ζ1716+ζ32ζ1715+ζ32ζ1713+ζ32ζ179+ζ32ζ178+ζ32ζ174+ζ32ζ172+ζ32ζ17 ζ3ζ1714+ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176+ζ3ζ175+ζ3ζ173 ζ32ζ1714+ζ32ζ1712+ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176+ζ32ζ175+ζ32ζ173 ζ3ζ1716+ζ3ζ1715+ζ3ζ1713+ζ3ζ179+ζ3ζ178+ζ3ζ174+ζ3ζ172+ζ3ζ17 complex faithful ρ28 8 0 -4-4√-3 -4+4√-3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 0 0 0 0 0 0 0 0 ζ32ζ1714+ζ32ζ1712+ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176+ζ32ζ175+ζ32ζ173 ζ3ζ1716+ζ3ζ1715+ζ3ζ1713+ζ3ζ179+ζ3ζ178+ζ3ζ174+ζ3ζ172+ζ3ζ17 ζ32ζ1716+ζ32ζ1715+ζ32ζ1713+ζ32ζ179+ζ32ζ178+ζ32ζ174+ζ32ζ172+ζ32ζ17 ζ3ζ1714+ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176+ζ3ζ175+ζ3ζ173 complex faithful ρ29 8 0 -4+4√-3 -4-4√-3 0 0 0 0 0 0 0 0 0 0 0 0 -1-√17/2 -1+√17/2 0 0 0 0 0 0 0 0 ζ3ζ1716+ζ3ζ1715+ζ3ζ1713+ζ3ζ179+ζ3ζ178+ζ3ζ174+ζ3ζ172+ζ3ζ17 ζ32ζ1714+ζ32ζ1712+ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176+ζ32ζ175+ζ32ζ173 ζ3ζ1714+ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176+ζ3ζ175+ζ3ζ173 ζ32ζ1716+ζ32ζ1715+ζ32ζ1713+ζ32ζ179+ζ32ζ178+ζ32ζ174+ζ32ζ172+ζ32ζ17 complex faithful ρ30 8 0 -4+4√-3 -4-4√-3 0 0 0 0 0 0 0 0 0 0 0 0 -1+√17/2 -1-√17/2 0 0 0 0 0 0 0 0 ζ3ζ1714+ζ3ζ1712+ζ3ζ1711+ζ3ζ1710+ζ3ζ177+ζ3ζ176+ζ3ζ175+ζ3ζ173 ζ32ζ1716+ζ32ζ1715+ζ32ζ1713+ζ32ζ179+ζ32ζ178+ζ32ζ174+ζ32ζ172+ζ32ζ17 ζ3ζ1716+ζ3ζ1715+ζ3ζ1713+ζ3ζ179+ζ3ζ178+ζ3ζ174+ζ3ζ172+ζ3ζ17 ζ32ζ1714+ζ32ζ1712+ζ32ζ1711+ζ32ζ1710+ζ32ζ177+ζ32ζ176+ζ32ζ175+ζ32ζ173 complex faithful

Smallest permutation representation of C3×C17⋊C8
On 51 points
Generators in S51
(1 35 18)(2 36 19)(3 37 20)(4 38 21)(5 39 22)(6 40 23)(7 41 24)(8 42 25)(9 43 26)(10 44 27)(11 45 28)(12 46 29)(13 47 30)(14 48 31)(15 49 32)(16 50 33)(17 51 34)
(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)
(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(19 27 31 33 34 26 22 20)(21 28 23 29 32 25 30 24)(36 44 48 50 51 43 39 37)(38 45 40 46 49 42 47 41)

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

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

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

Matrix representation of C3×C17⋊C8 in GL9(𝔽409)

 53 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 40 38 37 79 37 38 40 408 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
,
 343 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 408 40 38 37 79 37 38 40 0 407 40 39 407 40 408 408 1 0 370 332 293 294 294 293 332 370 0 1 408 408 40 407 39 40 407

G:=sub<GL(9,GF(409))| [53,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,0,38,0,1,0,0,0,0,0,0,37,0,0,1,0,0,0,0,0,79,0,0,0,1,0,0,0,0,37,0,0,0,0,1,0,0,0,38,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,1,0,408,0,0,0,0,0,0,0],[343,0,0,0,0,0,0,0,0,0,1,0,0,0,408,407,370,1,0,0,0,0,0,40,40,332,408,0,0,1,0,0,38,39,293,408,0,0,0,0,0,37,407,294,40,0,0,0,1,0,79,40,294,407,0,0,0,0,0,37,408,293,39,0,0,0,0,1,38,408,332,40,0,0,0,0,0,40,1,370,407] >;

C3×C17⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_{17}\rtimes C_8
% in TeX

G:=Group("C3xC17:C8");
// GroupNames label

G:=SmallGroup(408,33);
// by ID

G=gap.SmallGroup(408,33);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-17,30,42,5404,1314,819]);
// Polycyclic

G:=Group<a,b,c|a^3=b^17=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations

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