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## G = F5×C7⋊C3order 420 = 22·3·5·7

### Direct product of F5 and C7⋊C3

Aliases: F5×C7⋊C3, C352C12, (C7×F5)⋊C3, C72(C3×F5), (C7×D5).2C6, C5⋊(C4×C7⋊C3), (C5×C7⋊C3)⋊2C4, D5.(C2×C7⋊C3), (D5×C7⋊C3).2C2, SmallGroup(420,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — F5×C7⋊C3
 Chief series C1 — C7 — C35 — C7×D5 — D5×C7⋊C3 — F5×C7⋊C3
 Lower central C35 — F5×C7⋊C3
 Upper central C1

Generators and relations for F5×C7⋊C3
G = < a,b,c,d | a5=b4=c7=d3=1, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of F5×C7⋊C3

 class 1 2 3A 3B 4A 4B 5 6A 6B 7A 7B 12A 12B 12C 12D 14A 14B 15A 15B 28A 28B 28C 28D 35A 35B size 1 5 7 7 5 5 4 35 35 3 3 35 35 35 35 15 15 28 28 15 15 15 15 12 12 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 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 linear of order 2 ρ3 1 1 ζ32 ζ3 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ4 1 1 ζ32 ζ3 -1 -1 1 ζ3 ζ32 1 1 ζ6 ζ65 ζ65 ζ6 1 1 ζ3 ζ32 -1 -1 -1 -1 1 1 linear of order 6 ρ5 1 1 ζ3 ζ32 -1 -1 1 ζ32 ζ3 1 1 ζ65 ζ6 ζ6 ζ65 1 1 ζ32 ζ3 -1 -1 -1 -1 1 1 linear of order 6 ρ6 1 1 ζ3 ζ32 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ7 1 -1 1 1 i -i 1 -1 -1 1 1 i -i i -i -1 -1 1 1 i -i -i i 1 1 linear of order 4 ρ8 1 -1 1 1 -i i 1 -1 -1 1 1 -i i -i i -1 -1 1 1 -i i i -i 1 1 linear of order 4 ρ9 1 -1 ζ3 ζ32 -i i 1 ζ6 ζ65 1 1 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 -1 -1 ζ32 ζ3 -i i i -i 1 1 linear of order 12 ρ10 1 -1 ζ32 ζ3 -i i 1 ζ65 ζ6 1 1 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 -1 -1 ζ3 ζ32 -i i i -i 1 1 linear of order 12 ρ11 1 -1 ζ32 ζ3 i -i 1 ζ65 ζ6 1 1 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 -1 -1 ζ3 ζ32 i -i -i i 1 1 linear of order 12 ρ12 1 -1 ζ3 ζ32 i -i 1 ζ6 ζ65 1 1 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 -1 -1 ζ32 ζ3 i -i -i i 1 1 linear of order 12 ρ13 3 3 0 0 -3 -3 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ14 3 3 0 0 3 3 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ15 3 3 0 0 -3 -3 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 -1+√-7/2 -1-√-7/2 0 0 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ16 3 3 0 0 3 3 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 -1-√-7/2 -1+√-7/2 0 0 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ17 3 -3 0 0 3i -3i 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 1-√-7/2 1+√-7/2 0 0 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ4ζ74+ζ4ζ72+ζ4ζ7 -1-√-7/2 -1+√-7/2 complex lifted from C4×C7⋊C3 ρ18 3 -3 0 0 -3i 3i 3 0 0 -1-√-7/2 -1+√-7/2 0 0 0 0 1-√-7/2 1+√-7/2 0 0 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ43ζ74+ζ43ζ72+ζ43ζ7 -1-√-7/2 -1+√-7/2 complex lifted from C4×C7⋊C3 ρ19 3 -3 0 0 3i -3i 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 1+√-7/2 1-√-7/2 0 0 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ43ζ76+ζ43ζ75+ζ43ζ73 ζ4ζ76+ζ4ζ75+ζ4ζ73 -1+√-7/2 -1-√-7/2 complex lifted from C4×C7⋊C3 ρ20 3 -3 0 0 -3i 3i 3 0 0 -1+√-7/2 -1-√-7/2 0 0 0 0 1+√-7/2 1-√-7/2 0 0 ζ43ζ74+ζ43ζ72+ζ43ζ7 ζ4ζ74+ζ4ζ72+ζ4ζ7 ζ4ζ76+ζ4ζ75+ζ4ζ73 ζ43ζ76+ζ43ζ75+ζ43ζ73 -1+√-7/2 -1-√-7/2 complex lifted from C4×C7⋊C3 ρ21 4 0 4 4 0 0 -1 0 0 4 4 0 0 0 0 0 0 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from F5 ρ22 4 0 -2+2√-3 -2-2√-3 0 0 -1 0 0 4 4 0 0 0 0 0 0 ζ6 ζ65 0 0 0 0 -1 -1 complex lifted from C3×F5 ρ23 4 0 -2-2√-3 -2+2√-3 0 0 -1 0 0 4 4 0 0 0 0 0 0 ζ65 ζ6 0 0 0 0 -1 -1 complex lifted from C3×F5 ρ24 12 0 0 0 0 0 -3 0 0 -2-2√-7 -2+2√-7 0 0 0 0 0 0 0 0 0 0 0 0 1+√-7/2 1-√-7/2 complex faithful ρ25 12 0 0 0 0 0 -3 0 0 -2+2√-7 -2-2√-7 0 0 0 0 0 0 0 0 0 0 0 0 1-√-7/2 1+√-7/2 complex faithful

Smallest permutation representation of F5×C7⋊C3
On 35 points
Generators in S35
(1 8 15 22 29)(2 9 16 23 30)(3 10 17 24 31)(4 11 18 25 32)(5 12 19 26 33)(6 13 20 27 34)(7 14 21 28 35)
(8 15 29 22)(9 16 30 23)(10 17 31 24)(11 18 32 25)(12 19 33 26)(13 20 34 27)(14 21 35 28)
(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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)

G:=sub<Sym(35)| (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)>;

G:=Group( (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,33)(6,13,20,27,34)(7,14,21,28,35), (8,15,29,22)(9,16,30,23)(10,17,31,24)(11,18,32,25)(12,19,33,26)(13,20,34,27)(14,21,35,28), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34) );

G=PermutationGroup([[(1,8,15,22,29),(2,9,16,23,30),(3,10,17,24,31),(4,11,18,25,32),(5,12,19,26,33),(6,13,20,27,34),(7,14,21,28,35)], [(8,15,29,22),(9,16,30,23),(10,17,31,24),(11,18,32,25),(12,19,33,26),(13,20,34,27),(14,21,35,28)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34)]])

Matrix representation of F5×C7⋊C3 in GL7(𝔽421)

 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 420 0 0 0 1 0 0 420 0 0 0 0 1 0 420 0 0 0 0 0 1 420
,
 420 0 0 0 0 0 0 0 420 0 0 0 0 0 0 0 420 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0
,
 0 0 1 0 0 0 0 1 0 177 0 0 0 0 0 1 176 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 176 244 0 0 0 0 0 420 245 1 0 0 0 0 420 1 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 20 0 0 0 0 0 0 0 20

G:=sub<GL(7,GF(421))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,420,420,420,420],[420,0,0,0,0,0,0,0,420,0,0,0,0,0,0,0,420,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,177,176,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[176,420,420,0,0,0,0,244,245,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,20] >;

F5×C7⋊C3 in GAP, Magma, Sage, TeX

F_5\times C_7\rtimes C_3
% in TeX

G:=Group("F5xC7:C3");
// GroupNames label

G:=SmallGroup(420,14);
// by ID

G=gap.SmallGroup(420,14);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-7,30,483,173,1509]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=c^7=d^3=1,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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