metabelian, supersoluble, monomial, A-group
Aliases: C25⋊2F5, C52.5F5, (C5×C25)⋊6C4, C5⋊2(C25⋊C4), C25⋊D5.2C2, C5.(C52⋊C4), SmallGroup(500,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C25 — C25⋊2F5 |
Generators and relations for C25⋊2F5
G = < a,b,c | a25=b5=c4=1, ab=ba, cac-1=a7, cbc-1=b3 >
Smallest permutation representation of C25⋊2F5
►On 50 points
Generators in S50
(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)
(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 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)
(1 26)(2 44 25 33)(3 37 24 40)(4 30 23 47)(5 48 22 29)(6 41 21 36)(7 34 20 43)(8 27 19 50)(9 45 18 32)(10 38 17 39)(11 31 16 46)(12 49 15 28)(13 42 14 35)
G:=sub<Sym(50)| (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), (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)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35), (1,26)(2,44,25,33)(3,37,24,40)(4,30,23,47)(5,48,22,29)(6,41,21,36)(7,34,20,43)(8,27,19,50)(9,45,18,32)(10,38,17,39)(11,31,16,46)(12,49,15,28)(13,42,14,35)>;
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), (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)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35), (1,26)(2,44,25,33)(3,37,24,40)(4,30,23,47)(5,48,22,29)(6,41,21,36)(7,34,20,43)(8,27,19,50)(9,45,18,32)(10,38,17,39)(11,31,16,46)(12,49,15,28)(13,42,14,35) );
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)], [(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),(26,46,41,36,31),(27,47,42,37,32),(28,48,43,38,33),(29,49,44,39,34),(30,50,45,40,35)], [(1,26),(2,44,25,33),(3,37,24,40),(4,30,23,47),(5,48,22,29),(6,41,21,36),(7,34,20,43),(8,27,19,50),(9,45,18,32),(10,38,17,39),(11,31,16,46),(12,49,15,28),(13,42,14,35)]])
35 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | ··· | 5F | 25A | ··· | 25Y |
| order | 1 | 2 | 4 | 4 | 5 | ··· | 5 | 25 | ··· | 25 |
| size | 1 | 125 | 125 | 125 | 4 | ··· | 4 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C4 | F5 | F5 | C25⋊C4 | C52⋊C4 | C25⋊2F5 |
| kernel | C25⋊2F5 | C25⋊D5 | C5×C25 | C25 | C52 | C5 | C5 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 5 | 4 | 20 |
Matrix representation of C25⋊2F5 ►in GL4(𝔽101) generated by
| 99 | 50 | 0 | 0 |
| 51 | 88 | 0 | 0 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 69 | 8 |
| 100 | 22 | 0 | 0 |
| 79 | 79 | 0 | 0 |
| 0 | 0 | 22 | 100 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 93 | 0 | 0 |
| 90 | 69 | 0 | 0 |
G:=sub<GL(4,GF(101))| [99,51,0,0,50,88,0,0,0,0,11,69,0,0,32,8],[100,79,0,0,22,79,0,0,0,0,22,1,0,0,100,0],[0,0,32,90,0,0,93,69,1,0,0,0,0,1,0,0] >;
C25⋊2F5 in GAP, Magma, Sage, TeX
C_{25}\rtimes_2F_5 % in TeX
G:=Group("C25:2F5"); // GroupNames label
G:=SmallGroup(500,24);
// by ID
G=gap.SmallGroup(500,24);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,1682,2377,762,803,808,7504,5009]);
// Polycyclic
G:=Group<a,b,c|a^25=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^7,c*b*c^-1=b^3>;
// generators/relations
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