direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×F5, C5⋊C28, C35⋊2C4, D5.C14, (C7×D5).2C2, SmallGroup(140,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C7×F5 |
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 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 | 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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