direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×D5.D5, C53⋊2C4, C52⋊5C20, C52⋊11F5, C52⋊4Dic5, D5.(C5×D5), C5⋊(C5×Dic5), C5⋊3(C5×F5), (C5×D5).1D5, (C5×D5).3C10, (D5×C52).2C2, SmallGroup(500,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C5×D5.D5 |
Generators and relations for C5×D5.D5
G = < a,b,c,d,e | a5=b5=c2=d5=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b2, cd=dc, ece-1=bc, ede-1=d-1 >
5C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
25C4
5C10
5C10
10C10
10C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
5F5
5Dic5
25C20
Permutation representations of C5×D5.D5
►On 20 points - transitive group 20T127
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 19 7 14)(2 20 8 15)(3 16 9 11)(4 17 10 12)(5 18 6 13)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,9)(2,10)(3,6)(4,7)(5,8)(11,17)(12,18)(13,19)(14,20)(15,16), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,19,7,14)(2,20,8,15)(3,16,9,11)(4,17,10,12)(5,18,6,13)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,9)(2,10)(3,6)(4,7)(5,8)(11,17)(12,18)(13,19)(14,20)(15,16), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,19,7,14)(2,20,8,15)(3,16,9,11)(4,17,10,12)(5,18,6,13) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,9,7,10,8),(11,12,13,14,15),(16,20,19,18,17)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,19,7,14),(2,20,8,15),(3,16,9,11),(4,17,10,12),(5,18,6,13)]])
G:=TransitiveGroup(20,127);
65 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 5O | ··· | 5AM | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 5 | 25 | 25 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 25 | ··· | 25 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D5 | Dic5 | C5×D5 | C5×Dic5 | F5 | C5×F5 | D5.D5 | C5×D5.D5 |
| kernel | C5×D5.D5 | D5×C52 | C53 | D5.D5 | C5×D5 | C52 | C5×D5 | C52 | D5 | C5 | C52 | C5 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 2 | 8 | 8 | 1 | 4 | 4 | 16 |
Matrix representation of C5×D5.D5 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 10 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 2 | 15 | 16 | 18 |
| 0 | 37 | 0 | 0 |
| 10 | 0 | 0 | 0 |
| 39 | 2 | 18 | 33 |
| 39 | 26 | 25 | 23 |
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 17 | 24 | 0 | 37 |
| 0 | 0 | 37 | 0 |
| 5 | 36 | 37 | 20 |
| 0 | 10 | 0 | 0 |
| 30 | 11 | 0 | 5 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[10,0,0,2,0,37,0,15,0,0,16,16,0,0,0,18],[0,10,39,39,37,0,2,26,0,0,18,25,0,0,33,23],[10,0,0,17,0,10,0,24,0,0,37,0,0,0,0,37],[0,5,0,30,0,36,10,11,37,37,0,0,0,20,0,5] >;
C5×D5.D5 in GAP, Magma, Sage, TeX
C_5\times D_5.D_5
% in TeX
G:=Group("C5xD5.D5"); // GroupNames label
G:=SmallGroup(500,42);
// by ID
G=gap.SmallGroup(500,42);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,7504,1014]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^5=c^2=d^5=1,e^2=b^-1*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^2,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
Export