direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C14×F5, C10⋊C28, D5⋊C28, C70⋊2C4, D10.C14, C5⋊(C2×C28), C35⋊3(C2×C4), (C7×D5)⋊3C4, D5.(C2×C14), (D5×C14).3C2, (C7×D5).3C22, SmallGroup(280,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C14×F5 |
Generators and relations for C14×F5
G = < a,b,c | a14=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C14×F5
►On 70 points
Generators in S70
(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 36 37 38 39 40 41 42)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)
(1 53 64 16 37)(2 54 65 17 38)(3 55 66 18 39)(4 56 67 19 40)(5 43 68 20 41)(6 44 69 21 42)(7 45 70 22 29)(8 46 57 23 30)(9 47 58 24 31)(10 48 59 25 32)(11 49 60 26 33)(12 50 61 27 34)(13 51 62 28 35)(14 52 63 15 36)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 45 63 29)(16 46 64 30)(17 47 65 31)(18 48 66 32)(19 49 67 33)(20 50 68 34)(21 51 69 35)(22 52 70 36)(23 53 57 37)(24 54 58 38)(25 55 59 39)(26 56 60 40)(27 43 61 41)(28 44 62 42)
G:=sub<Sym(70)| (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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,53,64,16,37)(2,54,65,17,38)(3,55,66,18,39)(4,56,67,19,40)(5,43,68,20,41)(6,44,69,21,42)(7,45,70,22,29)(8,46,57,23,30)(9,47,58,24,31)(10,48,59,25,32)(11,49,60,26,33)(12,50,61,27,34)(13,51,62,28,35)(14,52,63,15,36), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,45,63,29)(16,46,64,30)(17,47,65,31)(18,48,66,32)(19,49,67,33)(20,50,68,34)(21,51,69,35)(22,52,70,36)(23,53,57,37)(24,54,58,38)(25,55,59,39)(26,56,60,40)(27,43,61,41)(28,44,62,42)>;
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)(29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,53,64,16,37)(2,54,65,17,38)(3,55,66,18,39)(4,56,67,19,40)(5,43,68,20,41)(6,44,69,21,42)(7,45,70,22,29)(8,46,57,23,30)(9,47,58,24,31)(10,48,59,25,32)(11,49,60,26,33)(12,50,61,27,34)(13,51,62,28,35)(14,52,63,15,36), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,45,63,29)(16,46,64,30)(17,47,65,31)(18,48,66,32)(19,49,67,33)(20,50,68,34)(21,51,69,35)(22,52,70,36)(23,53,57,37)(24,54,58,38)(25,55,59,39)(26,56,60,40)(27,43,61,41)(28,44,62,42) );
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),(29,30,31,32,33,34,35,36,37,38,39,40,41,42),(43,44,45,46,47,48,49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64,65,66,67,68,69,70)], [(1,53,64,16,37),(2,54,65,17,38),(3,55,66,18,39),(4,56,67,19,40),(5,43,68,20,41),(6,44,69,21,42),(7,45,70,22,29),(8,46,57,23,30),(9,47,58,24,31),(10,48,59,25,32),(11,49,60,26,33),(12,50,61,27,34),(13,51,62,28,35),(14,52,63,15,36)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,45,63,29),(16,46,64,30),(17,47,65,31),(18,48,66,32),(19,49,67,33),(20,50,68,34),(21,51,69,35),(22,52,70,36),(23,53,57,37),(24,54,58,38),(25,55,59,39),(26,56,60,40),(27,43,61,41),(28,44,62,42)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 7A | ··· | 7F | 10 | 14A | ··· | 14F | 14G | ··· | 14R | 28A | ··· | 28X | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 7 | ··· | 7 | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 1 | ··· | 1 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C28 | C28 | F5 | C2×F5 | C7×F5 | C14×F5 |
| kernel | C14×F5 | C7×F5 | D5×C14 | C7×D5 | C70 | C2×F5 | F5 | D10 | D5 | C10 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 12 | 6 | 12 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C14×F5 ►in GL5(𝔽281)
| 59 | 0 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 | 0 |
| 0 | 0 | 280 | 0 | 0 |
| 0 | 0 | 0 | 280 | 0 |
| 0 | 0 | 0 | 0 | 280 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 280 | 280 | 280 | 280 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 53 | 0 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 280 |
| 0 | 0 | 280 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(5,GF(281))| [59,0,0,0,0,0,280,0,0,0,0,0,280,0,0,0,0,0,280,0,0,0,0,0,280],[1,0,0,0,0,0,280,1,0,0,0,280,0,1,0,0,280,0,0,1,0,280,0,0,0],[53,0,0,0,0,0,280,0,0,1,0,0,0,280,1,0,0,0,0,1,0,0,280,0,1] >;
C14×F5 in GAP, Magma, Sage, TeX
C_{14}\times F_5 % in TeX
G:=Group("C14xF5"); // GroupNames label
G:=SmallGroup(280,34);
// by ID
G=gap.SmallGroup(280,34);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-5,140,2804,219]);
// Polycyclic
G:=Group<a,b,c|a^14=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export