direct product, metabelian, supersoluble, monomial, A-group
Aliases: C5×S3×D5, D15⋊C10, C52⋊5D6, C15⋊5D10, (C5×S3)⋊C10, C15⋊(C2×C10), (C3×D5)⋊C10, C3⋊1(D5×C10), C5⋊1(S3×C10), (C5×D15)⋊1C2, (D5×C15)⋊3C2, (C5×C15)⋊1C22, (S3×C52)⋊1C2, SmallGroup(300,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C5×S3×D5 |
Generators and relations for C5×S3×D5
G = < a,b,c,d,e | a5=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
3C2
5C2
15C2
2C5
2C5
15C22
5S3
5C6
3D5
3C10
3C10
5C10
6C10
6C10
15C10
2C15
2C15
5D6
3D10
15C2×C10
5C30
Permutation representations of C5×S3×D5
►On 30 points - transitive group 30T75
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 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 20)(2 16)(3 17)(4 18)(5 19)(6 25)(7 21)(8 22)(9 23)(10 24)(11 30)(12 26)(13 27)(14 28)(15 29)
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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,16)(3,17)(4,18)(5,19)(6,25)(7,21)(8,22)(9,23)(10,24)(11,30)(12,26)(13,27)(14,28)(15,29)>;
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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,20)(2,16)(3,17)(4,18)(5,19)(6,25)(7,21)(8,22)(9,23)(10,24)(11,30)(12,26)(13,27)(14,28)(15,29) );
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)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30)], [(1,5,4,3,2),(6,10,9,8,7),(11,15,14,13,12),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,20),(2,16),(3,17),(4,18),(5,19),(6,25),(7,21),(8,22),(9,23),(10,24),(11,30),(12,26),(13,27),(14,28),(15,29)]])
G:=TransitiveGroup(30,75);
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | ··· | 10R | 10S | 10T | 10U | 10V | 15A | 15B | 15C | 15D | 15E | ··· | 15N | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 30 | 30 | 30 | 30 |
| size | 1 | 3 | 5 | 15 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | S3 | D5 | D6 | D10 | C5×S3 | C5×D5 | S3×C10 | D5×C10 | S3×D5 | C5×S3×D5 |
| kernel | C5×S3×D5 | D5×C15 | S3×C52 | C5×D15 | S3×D5 | C5×S3 | C3×D5 | D15 | C5×D5 | C5×S3 | C52 | C15 | D5 | S3 | C5 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 1 | 2 | 4 | 8 | 4 | 8 | 2 | 8 |
Matrix representation of C5×S3×D5 ►in GL4(𝔽31) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 0 | 1 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 30 |
| 0 | 0 | 0 | 30 |
| 4 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 8 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
G:=sub<GL(4,GF(31))| [2,0,0,0,0,2,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,30,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,30,30],[4,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,4,0,0,8,0,0,0,0,0,30,0,0,0,0,30] >;
C5×S3×D5 in GAP, Magma, Sage, TeX
C_5\times S_3\times D_5
% in TeX
G:=Group("C5xS3xD5"); // GroupNames label
G:=SmallGroup(300,37);
// by ID
G=gap.SmallGroup(300,37);
# by ID
G:=PCGroup([5,-2,-2,-5,-3,-5,408,6004]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations
Export