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G = C5×F7order 210 = 2·3·5·7

Direct product of C5 and F7

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

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

C1C7 — C5×F7
C1C7C35C5×C7⋊C3 — C5×F7
C7 — C5×F7
C1C5

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

7C2
7C3
7C6
7C10
7C15
7C30

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 3A3B5A5B5C5D6A6B 7 10A10B10C10D15A···15H30A···30H35A35B35C35D
order123355556671010101015···1530···3035353535
size1777111177677777···77···76666

35 irreducible representations

dim1111111166
type+++
imageC1C2C3C5C6C10C15C30F7C5×F7
kernelC5×F7C5×C7⋊C3C5×D7F7C35C7⋊C3D7C7C5C1
# reps1124248814

Matrix representation of C5×F7 in GL6(𝔽211)

5500000
0550000
0055000
0005500
0000550
0000055
,
210210210210210210
100000
010000
001000
000100
000010
,
100000
000001
000100
010000
210210210210210210
000010

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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Subgroup lattice of C5×F7 in TeX

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