direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×C52⋊C4, C53⋊4C4, C52⋊6F5, C52⋊7C20, C5⋊2(C5×F5), C5⋊D5.3C10, (C5×C5⋊D5).2C2, SmallGroup(500,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C5×C52⋊C4 |
Generators and relations for C5×C52⋊C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >
25C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
25C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
5F5
5F5
25C20
10C5×D5
10C5×D5
Permutation representations of C5×C52⋊C4
►On 20 points - transitive group 20T126
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 2 3 4 5)(6 10 9 8 7)(11 13 15 12 14)(16 19 17 20 18)
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)
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), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12)>;
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), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,18,8,13)(2,19,9,14)(3,20,10,15)(4,16,6,11)(5,17,7,12) );
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)], [(1,2,3,4,5),(6,10,9,8,7),(11,13,15,12,14),(16,19,17,20,18)], [(1,18,8,13),(2,19,9,14),(3,20,10,15),(4,16,6,11),(5,17,7,12)]])
G:=TransitiveGroup(20,126);
50 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5AH | 10A | 10B | 10C | 10D | 20A | ··· | 20H |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 25 | 25 | 25 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 25 | 25 | 25 | 25 | 25 | ··· | 25 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | F5 | C5×F5 | C52⋊C4 | C5×C52⋊C4 |
| kernel | C5×C52⋊C4 | C5×C5⋊D5 | C53 | C52⋊C4 | C5⋊D5 | C52 | C52 | C5 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 2 | 8 | 4 | 16 |
Matrix representation of C5×C52⋊C4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 37 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 16 |
| 18 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[37,0,0,0,0,10,0,0,0,0,18,0,0,0,0,16],[18,0,0,0,0,16,0,0,0,0,37,0,0,0,0,10],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C5×C52⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_5^2\rtimes C_4
% in TeX
G:=Group("C5xC5^2:C4"); // GroupNames label
G:=SmallGroup(500,44);
// by ID
G=gap.SmallGroup(500,44);
# by ID
G:=PCGroup([5,-2,-5,-2,-5,-5,50,1203,173,5004,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations
Export