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## G = C5×F7order 210 = 2·3·5·7

### Direct product of C5 and F7

Aliases: C5×F7, C7⋊C30, D7⋊C15, C352C6, C7⋊C3⋊C10, (C5×D7)⋊C3, (C5×C7⋊C3)⋊2C2, SmallGroup(210,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C5×F7
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C5×F7
 Lower central C7 — C5×F7
 Upper central C1 — C5

Generators and relations for C5×F7
G = < a,b,c | a5=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Smallest permutation representation of C5×F7
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 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)(23 25 24 28 26 27)(30 32 31 35 33 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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34)>;

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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34) );

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

35 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B 7 10A 10B 10C 10D 15A ··· 15H 30A ··· 30H 35A 35B 35C 35D order 1 2 3 3 5 5 5 5 6 6 7 10 10 10 10 15 ··· 15 30 ··· 30 35 35 35 35 size 1 7 7 7 1 1 1 1 7 7 6 7 7 7 7 7 ··· 7 7 ··· 7 6 6 6 6

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 6 6 type + + + image C1 C2 C3 C5 C6 C10 C15 C30 F7 C5×F7 kernel C5×F7 C5×C7⋊C3 C5×D7 F7 C35 C7⋊C3 D7 C7 C5 C1 # reps 1 1 2 4 2 4 8 8 1 4

Matrix representation of C5×F7 in GL6(𝔽211)

 55 0 0 0 0 0 0 55 0 0 0 0 0 0 55 0 0 0 0 0 0 55 0 0 0 0 0 0 55 0 0 0 0 0 0 55
,
 210 210 210 210 210 210 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 210 210 210 210 210 210 0 0 0 0 1 0

G:=sub<GL(6,GF(211))| [55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55,0,0,0,0,0,0,55],[210,1,0,0,0,0,210,0,1,0,0,0,210,0,0,1,0,0,210,0,0,0,1,0,210,0,0,0,0,1,210,0,0,0,0,0],[1,0,0,0,210,0,0,0,0,1,210,0,0,0,0,0,210,0,0,0,1,0,210,0,0,0,0,0,210,1,0,1,0,0,210,0] >;

C5×F7 in GAP, Magma, Sage, TeX

C_5\times F_7
% in TeX

G:=Group("C5xF7");
// GroupNames label

G:=SmallGroup(210,1);
// by ID

G=gap.SmallGroup(210,1);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,2883,967]);
// Polycyclic

G:=Group<a,b,c|a^5=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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