direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×D15, C5⋊1D30, C52⋊3D6, C15⋊3D10, C3⋊1D52, (C5×D5)⋊S3, (C3×D5)⋊D5, C5⋊2(S3×D5), C5⋊D15⋊2C2, (C5×D15)⋊2C2, (D5×C15)⋊1C2, (C5×C15)⋊3C22, SmallGroup(300,39)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — D5×D15 |
Generators and relations for D5×D15
G = < a,b,c,d | a5=b2=c15=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
5C2
15C2
75C2
2C5
2C5
75C22
5C6
5S3
25S3
3D5
5C10
15C10
15D5
15D5
30D5
30D5
2C15
2C15
25D6
15D10
15D10
5C30
5D15
5D15
10D15
10D15
5D30
3D52
Permutation representations of D5×D15
►On 30 points - transitive group 30T67
Generators in S30
(1 13 10 7 4)(2 14 11 8 5)(3 15 12 9 6)(16 19 22 25 28)(17 20 23 26 29)(18 21 24 27 30)
(1 28)(2 29)(3 30)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15)(16 17 18 19 20 21 22 23 24 25 26 27 28 29 30)
(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)
G:=sub<Sym(30)| (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)>;
G:=Group( (1,13,10,7,4)(2,14,11,8,5)(3,15,12,9,6)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30), (1,28)(2,29)(3,30)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16) );
G=PermutationGroup([[(1,13,10,7,4),(2,14,11,8,5),(3,15,12,9,6),(16,19,22,25,28),(17,20,23,26,29),(18,21,24,27,30)], [(1,28),(2,29),(3,30),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16)]])
G:=TransitiveGroup(30,67);
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 6 | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | ··· | 15N | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 30 | 30 | 30 | 30 |
| size | 1 | 5 | 15 | 75 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D5 | D5 | D6 | D10 | D15 | D30 | S3×D5 | D52 | D5×D15 |
| kernel | D5×D15 | D5×C15 | C5×D15 | C5⋊D15 | C5×D5 | C3×D5 | D15 | C52 | C15 | D5 | C5 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 2 | 4 | 8 |
Matrix representation of D5×D15 ►in GL4(𝔽31) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 30 |
| 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 18 | 30 |
| 0 | 0 | 13 | 13 |
| 26 | 7 | 0 | 0 |
| 3 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 15 | 0 | 0 |
| 29 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(31))| [1,0,0,0,0,1,0,0,0,0,18,1,0,0,30,0],[30,0,0,0,0,30,0,0,0,0,18,13,0,0,30,13],[26,3,0,0,7,8,0,0,0,0,1,0,0,0,0,1],[0,29,0,0,15,0,0,0,0,0,1,0,0,0,0,1] >;
D5×D15 in GAP, Magma, Sage, TeX
D_5\times D_{15} % in TeX
G:=Group("D5xD15"); // GroupNames label
G:=SmallGroup(300,39);
// by ID
G=gap.SmallGroup(300,39);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,122,488,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^15=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export