direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: C5×F7, C7⋊C30, D7⋊C15, C35⋊2C6, C7⋊C3⋊C10, (C5×D7)⋊C3, (C5×C7⋊C3)⋊2C2, SmallGroup(210,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C5×F7 |
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
Export