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## G = C7×F5order 140 = 22·5·7

### Direct product of C7 and F5

Aliases: C7×F5, C5⋊C28, C352C4, D5.C14, (C7×D5).2C2, SmallGroup(140,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C7×F5
 Chief series C1 — C5 — D5 — C7×D5 — C7×F5
 Lower central C5 — C7×F5
 Upper central C1 — C7

Generators and relations for C7×F5
G = < a,b,c | a7=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

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

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

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

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

35 conjugacy classes

 class 1 2 4A 4B 5 7A ··· 7F 14A ··· 14F 28A ··· 28L 35A ··· 35F order 1 2 4 4 5 7 ··· 7 14 ··· 14 28 ··· 28 35 ··· 35 size 1 5 5 5 4 1 ··· 1 5 ··· 5 5 ··· 5 4 ··· 4

35 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C7 C14 C28 F5 C7×F5 kernel C7×F5 C7×D5 C35 F5 D5 C5 C7 C1 # reps 1 1 2 6 6 12 1 6

Matrix representation of C7×F5 in GL4(𝔽281) generated by

 59 0 0 0 0 59 0 0 0 0 59 0 0 0 0 59
,
 280 280 280 280 1 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 1 0 0 280 280 280 280
G:=sub<GL(4,GF(281))| [59,0,0,0,0,59,0,0,0,0,59,0,0,0,0,59],[280,1,0,0,280,0,1,0,280,0,0,1,280,0,0,0],[1,0,0,280,0,0,1,280,0,0,0,280,0,1,0,280] >;

C7×F5 in GAP, Magma, Sage, TeX

C_7\times F_5
% in TeX

G:=Group("C7xF5");
// GroupNames label

G:=SmallGroup(140,5);
// by ID

G=gap.SmallGroup(140,5);
# by ID

G:=PCGroup([4,-2,-7,-2,-5,56,899,139]);
// Polycyclic

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

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