direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×D15, C15⋊3D5, C52⋊2S3, C15⋊1C10, C3⋊(C5×D5), C5⋊(C5×S3), (C5×C15)⋊2C2, SmallGroup(150,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C5×D15 |
Generators and relations for C5×D15
G = < a,b,c | a5=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of C5×D15
►On 30 points - transitive group 30T36
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 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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15),(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(30,36);
C5×D15 is a maximal subgroup of
C5×S3×D5 D15⋊D5 C52⋊D9 C52⋊(C3⋊S3)
45 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 15A | ··· | 15X |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 15 | ··· | 15 |
| size | 1 | 15 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 15 | 15 | 15 | 15 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C5 | C10 | S3 | D5 | C5×S3 | D15 | C5×D5 | C5×D15 |
| kernel | C5×D15 | C5×C15 | D15 | C15 | C52 | C15 | C5 | C5 | C3 | C1 |
| # reps | 1 | 1 | 4 | 4 | 1 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C5×D15 ►in GL2(𝔽31) generated by
| 8 | 0 |
| 0 | 8 |
| 7 | 0 |
| 29 | 9 |
| 22 | 9 |
| 29 | 9 |
G:=sub<GL(2,GF(31))| [8,0,0,8],[7,29,0,9],[22,29,9,9] >;
C5×D15 in GAP, Magma, Sage, TeX
C_5\times D_{15} % in TeX
G:=Group("C5xD15"); // GroupNames label
G:=SmallGroup(150,11);
// by ID
G=gap.SmallGroup(150,11);
# by ID
G:=PCGroup([4,-2,-5,-3,-5,242,1923]);
// Polycyclic
G:=Group<a,b,c|a^5=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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