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## G = C5×C7⋊C3order 105 = 3·5·7

### Direct product of C5 and C7⋊C3

Aliases: C5×C7⋊C3, C7⋊C15, C35⋊C3, SmallGroup(105,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C5×C7⋊C3
 Chief series C1 — C7 — C35 — C5×C7⋊C3
 Lower central C7 — C5×C7⋊C3
 Upper central C1 — C5

Generators and relations for C5×C7⋊C3
G = < a,b,c | a5=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Character table of C5×C7⋊C3

 class 1 3A 3B 5A 5B 5C 5D 7A 7B 15A 15B 15C 15D 15E 15F 15G 15H 35A 35B 35C 35D 35E 35F 35G 35H size 1 7 7 1 1 1 1 3 3 7 7 7 7 7 7 7 7 3 3 3 3 3 3 3 3 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 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 1 1 ζ54 ζ5 ζ52 ζ53 1 1 ζ54 ζ52 ζ53 ζ54 ζ5 ζ5 ζ52 ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ5 ζ52 ζ52 linear of order 5 ρ5 1 1 1 ζ5 ζ54 ζ53 ζ52 1 1 ζ5 ζ53 ζ52 ζ5 ζ54 ζ54 ζ53 ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ54 ζ53 ζ53 linear of order 5 ρ6 1 1 1 ζ52 ζ53 ζ5 ζ54 1 1 ζ52 ζ5 ζ54 ζ52 ζ53 ζ53 ζ5 ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ53 ζ5 ζ5 linear of order 5 ρ7 1 1 1 ζ53 ζ52 ζ54 ζ5 1 1 ζ53 ζ54 ζ5 ζ53 ζ52 ζ52 ζ54 ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ52 ζ54 ζ54 linear of order 5 ρ8 1 ζ32 ζ3 ζ54 ζ5 ζ52 ζ53 1 1 ζ32ζ54 ζ3ζ52 ζ3ζ53 ζ3ζ54 ζ3ζ5 ζ32ζ5 ζ32ζ52 ζ32ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ5 ζ52 ζ52 linear of order 15 ρ9 1 ζ3 ζ32 ζ5 ζ54 ζ53 ζ52 1 1 ζ3ζ5 ζ32ζ53 ζ32ζ52 ζ32ζ5 ζ32ζ54 ζ3ζ54 ζ3ζ53 ζ3ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ54 ζ53 ζ53 linear of order 15 ρ10 1 ζ32 ζ3 ζ52 ζ53 ζ5 ζ54 1 1 ζ32ζ52 ζ3ζ5 ζ3ζ54 ζ3ζ52 ζ3ζ53 ζ32ζ53 ζ32ζ5 ζ32ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ53 ζ5 ζ5 linear of order 15 ρ11 1 ζ32 ζ3 ζ53 ζ52 ζ54 ζ5 1 1 ζ32ζ53 ζ3ζ54 ζ3ζ5 ζ3ζ53 ζ3ζ52 ζ32ζ52 ζ32ζ54 ζ32ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ52 ζ54 ζ54 linear of order 15 ρ12 1 ζ3 ζ32 ζ53 ζ52 ζ54 ζ5 1 1 ζ3ζ53 ζ32ζ54 ζ32ζ5 ζ32ζ53 ζ32ζ52 ζ3ζ52 ζ3ζ54 ζ3ζ5 ζ5 ζ5 ζ52 ζ53 ζ53 ζ52 ζ54 ζ54 linear of order 15 ρ13 1 ζ32 ζ3 ζ5 ζ54 ζ53 ζ52 1 1 ζ32ζ5 ζ3ζ53 ζ3ζ52 ζ3ζ5 ζ3ζ54 ζ32ζ54 ζ32ζ53 ζ32ζ52 ζ52 ζ52 ζ54 ζ5 ζ5 ζ54 ζ53 ζ53 linear of order 15 ρ14 1 ζ3 ζ32 ζ52 ζ53 ζ5 ζ54 1 1 ζ3ζ52 ζ32ζ5 ζ32ζ54 ζ32ζ52 ζ32ζ53 ζ3ζ53 ζ3ζ5 ζ3ζ54 ζ54 ζ54 ζ53 ζ52 ζ52 ζ53 ζ5 ζ5 linear of order 15 ρ15 1 ζ3 ζ32 ζ54 ζ5 ζ52 ζ53 1 1 ζ3ζ54 ζ32ζ52 ζ32ζ53 ζ32ζ54 ζ32ζ5 ζ3ζ5 ζ3ζ52 ζ3ζ53 ζ53 ζ53 ζ5 ζ54 ζ54 ζ5 ζ52 ζ52 linear of order 15 ρ16 3 0 0 3 3 3 3 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -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 0 0 3 3 3 3 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ18 3 0 0 3ζ54 3ζ5 3ζ52 3ζ53 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 0 0 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ52ζ76+ζ52ζ75+ζ52ζ73 complex faithful ρ19 3 0 0 3ζ54 3ζ5 3ζ52 3ζ53 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 0 0 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ52ζ74+ζ52ζ72+ζ52ζ7 complex faithful ρ20 3 0 0 3ζ53 3ζ52 3ζ54 3ζ5 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 0 0 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ54ζ76+ζ54ζ75+ζ54ζ73 complex faithful ρ21 3 0 0 3ζ5 3ζ54 3ζ53 3ζ52 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 0 0 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ53ζ74+ζ53ζ72+ζ53ζ7 complex faithful ρ22 3 0 0 3ζ52 3ζ53 3ζ5 3ζ54 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 0 0 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ5ζ76+ζ5ζ75+ζ5ζ73 complex faithful ρ23 3 0 0 3ζ52 3ζ53 3ζ5 3ζ54 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 0 0 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ5ζ74+ζ5ζ72+ζ5ζ7 complex faithful ρ24 3 0 0 3ζ53 3ζ52 3ζ54 3ζ5 -1-√-7/2 -1+√-7/2 0 0 0 0 0 0 0 0 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ53ζ76+ζ53ζ75+ζ53ζ73 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ54ζ74+ζ54ζ72+ζ54ζ7 complex faithful ρ25 3 0 0 3ζ5 3ζ54 3ζ53 3ζ52 -1+√-7/2 -1-√-7/2 0 0 0 0 0 0 0 0 ζ52ζ76+ζ52ζ75+ζ52ζ73 ζ52ζ74+ζ52ζ72+ζ52ζ7 ζ54ζ76+ζ54ζ75+ζ54ζ73 ζ5ζ76+ζ5ζ75+ζ5ζ73 ζ5ζ74+ζ5ζ72+ζ5ζ7 ζ54ζ74+ζ54ζ72+ζ54ζ7 ζ53ζ74+ζ53ζ72+ζ53ζ7 ζ53ζ76+ζ53ζ75+ζ53ζ73 complex faithful

Smallest permutation representation of C5×C7⋊C3
On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)
(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)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (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([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(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)]])

C5×C7⋊C3 is a maximal subgroup of   C5⋊F7

Matrix representation of C5×C7⋊C3 in GL3(𝔽11) generated by

 3 0 0 0 3 0 0 0 3
,
 8 5 0 4 7 9 10 7 0
,
 1 5 10 0 3 5 0 4 7
G:=sub<GL(3,GF(11))| [3,0,0,0,3,0,0,0,3],[8,4,10,5,7,7,0,9,0],[1,0,0,5,3,4,10,5,7] >;

C5×C7⋊C3 in GAP, Magma, Sage, TeX

C_5\times C_7\rtimes C_3
% in TeX

G:=Group("C5xC7:C3");
// GroupNames label

G:=SmallGroup(105,1);
// by ID

G=gap.SmallGroup(105,1);
# by ID

G:=PCGroup([3,-3,-5,-7,272]);
// Polycyclic

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

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