direct product, metabelian, soluble, monomial, A-group
Aliases: C5×C52⋊C3, C5≀C3, AΣL1(𝔽125), C53⋊C3, C52⋊2C15, SmallGroup(375,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C5×C52⋊C3 |
Generators and relations for C5×C52⋊C3
G = < a,b,c,d | a5=b5=c5=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3c3, dcd-1=b-1c >
25C3
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
3C5
25C15
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52
3C52
Permutation representations of C5×C52⋊C3
►On 15 points - transitive group 15T25
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(1 3 5 2 4)(6 7 8 9 10)(11 13 15 12 14)
(1 3 5 2 4)(6 9 7 10 8)
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,3,5,2,4)(6,7,8,9,10)(11,13,15,12,14), (1,3,5,2,4)(6,9,7,10,8), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (1,3,5,2,4)(6,7,8,9,10)(11,13,15,12,14), (1,3,5,2,4)(6,9,7,10,8), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(1,3,5,2,4),(6,7,8,9,10),(11,13,15,12,14)], [(1,3,5,2,4),(6,9,7,10,8)], [(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10)]])
G:=TransitiveGroup(15,25);
Polynomial with Galois group C5×C52⋊C3 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T25 | x15+5x14-110x13-470x12+3330x11+13297x10-24280x9-114730x8+31690x7+333260x6+13487x5-436445x4-33660x3+263540x2+8415x-57107 | 524·710·414·1672·2232·2812·11354832 |
55 conjugacy classes
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | ··· | 5AR | 15A | ··· | 15H |
| order | 1 | 3 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 15 | ··· | 15 |
| size | 1 | 25 | 25 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 25 | ··· | 25 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C5 | C15 | C52⋊C3 | C5×C52⋊C3 |
| kernel | C5×C52⋊C3 | C53 | C52⋊C3 | C52 | C5 | C1 |
| # reps | 1 | 2 | 4 | 8 | 8 | 32 |
Matrix representation of C5×C52⋊C3 ►in GL3(𝔽11) generated by
| 3 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 4 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 9 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(11))| [3,0,0,0,3,0,0,0,3],[4,0,0,0,5,0,0,0,5],[9,0,0,0,5,0,0,0,1],[0,1,0,0,0,1,1,0,0] >;
C5×C52⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_5^2\rtimes C_3
% in TeX
G:=Group("C5xC5^2:C3"); // GroupNames label
G:=SmallGroup(375,6);
// by ID
G=gap.SmallGroup(375,6);
# by ID
G:=PCGroup([4,-3,-5,-5,5,2882,4563]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3*c^3,d*c*d^-1=b^-1*c>;
// generators/relations
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