direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C15×D15, C15≀C2, C15⋊1C30, C152⋊3C2, C5⋊(S3×C15), C3⋊(D5×C15), C15⋊2(C5×S3), (C5×C15)⋊5S3, (C5×C15)⋊6C6, (C3×C15)⋊4D5, C15⋊3(C3×D5), C52⋊4(C3×S3), C32⋊1(C5×D5), (C3×C15)⋊2C10, SmallGroup(450,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C15×D15 |
Generators and relations for C15×D15
G = < a,b,c | a15=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C15×D15
►On 30 points - transitive group 30T104
Generators in S30
(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 9 2 10 3 11 4 12 5 13 6 14 7 15 8)(16 23 30 22 29 21 28 20 27 19 26 18 25 17 24)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 16)(14 17)(15 18)
G:=sub<Sym(30)| (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), (1,9,2,10,3,11,4,12,5,13,6,14,7,15,8)(16,23,30,22,29,21,28,20,27,19,26,18,25,17,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,16)(14,17)(15,18)>;
G:=Group( (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), (1,9,2,10,3,11,4,12,5,13,6,14,7,15,8)(16,23,30,22,29,21,28,20,27,19,26,18,25,17,24), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,16)(14,17)(15,18) );
G=PermutationGroup([[(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)], [(1,9,2,10,3,11,4,12,5,13,6,14,7,15,8),(16,23,30,22,29,21,28,20,27,19,26,18,25,17,24)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(30,104);
135 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | ··· | 15CV | 30A | ··· | 30H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 15 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 15 | 15 | 1 | ··· | 1 | 2 | ··· | 2 | 15 | ··· | 15 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | S3 | D5 | C3×S3 | C5×S3 | C3×D5 | D15 | C5×D5 | S3×C15 | C3×D15 | D5×C15 | C5×D15 | C15×D15 |
| kernel | C15×D15 | C152 | C5×D15 | C3×D15 | C5×C15 | C3×C15 | D15 | C15 | C5×C15 | C3×C15 | C52 | C15 | C15 | C15 | C32 | C5 | C5 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×D15 ►in GL2(𝔽31) generated by
| 10 | 0 |
| 0 | 10 |
| 14 | 0 |
| 12 | 20 |
| 11 | 21 |
| 12 | 20 |
G:=sub<GL(2,GF(31))| [10,0,0,10],[14,12,0,20],[11,12,21,20] >;
C15×D15 in GAP, Magma, Sage, TeX
C_{15}\times D_{15} % in TeX
G:=Group("C15xD15"); // GroupNames label
G:=SmallGroup(450,29);
// by ID
G=gap.SmallGroup(450,29);
# by ID
G:=PCGroup([5,-2,-3,-5,-3,-5,1203,9004]);
// Polycyclic
G:=Group<a,b,c|a^15=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export