metabelian, supersoluble, monomial, A-group
Aliases: C53⋊6C4, C52⋊7F5, C52⋊6Dic5, C5⋊D5.2D5, C5⋊1(D5.D5), C5⋊2(C52⋊C4), (C5×C5⋊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 >
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
10C5×D5
10C5×D5
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 | C5×C5⋊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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