direct product, metacyclic, supersoluble, monomial, A-group
Aliases: F5×C25, C5⋊C100, D5.C50, C52.2C20, (C5×C25)⋊1C4, (C5×F5).C5, C5.4(C5×F5), (D5×C25).1C2, (C5×D5).1C10, SmallGroup(500,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C25 |
Generators and relations for F5×C25
G = < a,b,c | a25=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of F5×C25
►On 100 points
Generators in S100
(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 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 6 11 16 21)(2 7 12 17 22)(3 8 13 18 23)(4 9 14 19 24)(5 10 15 20 25)(26 41 31 46 36)(27 42 32 47 37)(28 43 33 48 38)(29 44 34 49 39)(30 45 35 50 40)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)(76 86 96 81 91)(77 87 97 82 92)(78 88 98 83 93)(79 89 99 84 94)(80 90 100 85 95)
(1 42 60 77)(2 43 61 78)(3 44 62 79)(4 45 63 80)(5 46 64 81)(6 47 65 82)(7 48 66 83)(8 49 67 84)(9 50 68 85)(10 26 69 86)(11 27 70 87)(12 28 71 88)(13 29 72 89)(14 30 73 90)(15 31 74 91)(16 32 75 92)(17 33 51 93)(18 34 52 94)(19 35 53 95)(20 36 54 96)(21 37 55 97)(22 38 56 98)(23 39 57 99)(24 40 58 100)(25 41 59 76)
G:=sub<Sym(100)| (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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,86,96,81,91)(77,87,97,82,92)(78,88,98,83,93)(79,89,99,84,94)(80,90,100,85,95), (1,42,60,77)(2,43,61,78)(3,44,62,79)(4,45,63,80)(5,46,64,81)(6,47,65,82)(7,48,66,83)(8,49,67,84)(9,50,68,85)(10,26,69,86)(11,27,70,87)(12,28,71,88)(13,29,72,89)(14,30,73,90)(15,31,74,91)(16,32,75,92)(17,33,51,93)(18,34,52,94)(19,35,53,95)(20,36,54,96)(21,37,55,97)(22,38,56,98)(23,39,57,99)(24,40,58,100)(25,41,59,76)>;
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)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,86,96,81,91)(77,87,97,82,92)(78,88,98,83,93)(79,89,99,84,94)(80,90,100,85,95), (1,42,60,77)(2,43,61,78)(3,44,62,79)(4,45,63,80)(5,46,64,81)(6,47,65,82)(7,48,66,83)(8,49,67,84)(9,50,68,85)(10,26,69,86)(11,27,70,87)(12,28,71,88)(13,29,72,89)(14,30,73,90)(15,31,74,91)(16,32,75,92)(17,33,51,93)(18,34,52,94)(19,35,53,95)(20,36,54,96)(21,37,55,97)(22,38,56,98)(23,39,57,99)(24,40,58,100)(25,41,59,76) );
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),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,6,11,16,21),(2,7,12,17,22),(3,8,13,18,23),(4,9,14,19,24),(5,10,15,20,25),(26,41,31,46,36),(27,42,32,47,37),(28,43,33,48,38),(29,44,34,49,39),(30,45,35,50,40),(51,71,66,61,56),(52,72,67,62,57),(53,73,68,63,58),(54,74,69,64,59),(55,75,70,65,60),(76,86,96,81,91),(77,87,97,82,92),(78,88,98,83,93),(79,89,99,84,94),(80,90,100,85,95)], [(1,42,60,77),(2,43,61,78),(3,44,62,79),(4,45,63,80),(5,46,64,81),(6,47,65,82),(7,48,66,83),(8,49,67,84),(9,50,68,85),(10,26,69,86),(11,27,70,87),(12,28,71,88),(13,29,72,89),(14,30,73,90),(15,31,74,91),(16,32,75,92),(17,33,51,93),(18,34,52,94),(19,35,53,95),(20,36,54,96),(21,37,55,97),(22,38,56,98),(23,39,57,99),(24,40,58,100),(25,41,59,76)]])
125 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 10A | 10B | 10C | 10D | 20A | ··· | 20H | 25A | ··· | 25T | 25U | ··· | 25AN | 50A | ··· | 50T | 100A | ··· | 100AN |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 25 | ··· | 25 | 25 | ··· | 25 | 50 | ··· | 50 | 100 | ··· | 100 |
| size | 1 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 1 | ··· | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C25 | C50 | C100 | F5 | C5×F5 | F5×C25 |
| kernel | F5×C25 | D5×C25 | C5×C25 | C5×F5 | C5×D5 | C52 | F5 | D5 | C5 | C25 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 40 | 1 | 4 | 20 |
Matrix representation of F5×C25 ►in GL4(𝔽101) generated by
| 81 | 0 | 0 | 0 |
| 0 | 81 | 0 | 0 |
| 0 | 0 | 81 | 0 |
| 0 | 0 | 0 | 81 |
| 84 | 0 | 0 | 0 |
| 0 | 95 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(101))| [81,0,0,0,0,81,0,0,0,0,81,0,0,0,0,81],[84,0,0,0,0,95,0,0,0,0,87,0,0,0,0,36],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
F5×C25 in GAP, Magma, Sage, TeX
F_5\times C_{25} % in TeX
G:=Group("F5xC25"); // GroupNames label
G:=SmallGroup(500,15);
// by ID
G=gap.SmallGroup(500,15);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-5,50,106,5004,1014]);
// Polycyclic
G:=Group<a,b,c|a^25=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export