metabelian, supersoluble, monomial, A-group
Aliases: C53:6C4, C52:7F5, C52:6Dic5, C5:D5.2D5, C5:1(D5.D5), C5:2(C52:C4), (C5xC5:D5).3C2, SmallGroup(500,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C53:6C4 |
Generators and relations for C53:6C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b2, dcd-1=c3 >
Quotients: C1, C2, C4, D5, Dic5, F5, D5.D5, C52:C4, C53:6C4
25C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
125C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
25F5
25Dic5
25F5
10C5xD5
10C5xD5
Permutation representations of C53:6C4
►On 20 points - transitive group 20T125
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 4 2 5 3)(6 8 10 7 9)(11 12 13 14 15)(16 20 19 18 17)
(1 3 5 2 4)(6 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 17 8 12)(2 16 9 11)(3 20 10 15)(4 19 6 14)(5 18 7 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), (1,4,2,5,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,4,2,5,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,4,2,5,3),(6,8,10,7,9),(11,12,13,14,15),(16,20,19,18,17)], [(1,3,5,2,4),(6,9,7,10,8),(11,12,13,14,15),(16,20,19,18,17)], [(1,17,8,12),(2,16,9,11),(3,20,10,15),(4,19,6,14),(5,18,7,13)]])
G:=TransitiveGroup(20,125);
38 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | ··· | 5AF | 10A | 10B |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 10 | 10 |
| size | 1 | 25 | 125 | 125 | 2 | 2 | 4 | ··· | 4 | 50 | 50 |
38 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | + | |||
| image | C1 | C2 | C4 | D5 | Dic5 | F5 | D5.D5 | C52:C4 | C53:6C4 |
| kernel | C53:6C4 | C5xC5:D5 | C53 | C5:D5 | C52 | C52 | C5 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 16 |
Matrix representation of C53:6C4 ►in GL4(F41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 33 | 31 | 18 | 0 |
| 31 | 33 | 0 | 18 |
| 37 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 35 | 1 | 18 | 0 |
| 23 | 17 | 0 | 16 |
| 10 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 9 | 13 | 18 | 0 |
| 11 | 2 | 0 | 16 |
| 5 | 37 | 9 | 0 |
| 37 | 5 | 0 | 9 |
| 0 | 0 | 36 | 4 |
| 0 | 0 | 4 | 36 |
G:=sub<GL(4,GF(41))| [16,0,33,31,0,16,31,33,0,0,18,0,0,0,0,18],[37,0,35,23,0,10,1,17,0,0,18,0,0,0,0,16],[10,0,9,11,0,37,13,2,0,0,18,0,0,0,0,16],[5,37,0,0,37,5,0,0,9,0,36,4,0,9,4,36] >;
C53:6C4 in GAP, Magma, Sage, TeX
C_5^3\rtimes_6C_4
% in TeX
G:=Group("C5^3:6C4"); // GroupNames label
G:=SmallGroup(500,46);
// by ID
G=gap.SmallGroup(500,46);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1203,808,5004,5009]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations
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