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G = C7×F5order 140 = 22·5·7

Direct product of C7 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — C7×F5
C1C5D5C7×D5 — C7×F5
C5 — C7×F5
C1C7

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

5C2
5C4
5C14
5C28

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

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

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

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

35 conjugacy classes

class 1  2 4A4B 5 7A···7F14A···14F28A···28L35A···35F
order124457···714···1428···2835···35
size155541···15···55···54···4

35 irreducible representations

dim11111144
type+++
imageC1C2C4C7C14C28F5C7×F5
kernelC7×F5C7×D5C35F5D5C5C7C1
# reps112661216

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

59000
05900
00590
00059
,
280280280280
1000
0100
0010
,
1000
0001
0100
280280280280
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

Export

Subgroup lattice of C7×F5 in TeX

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