direct product, non-abelian, soluble, monomial, A-group
Aliases: C3×C52⋊S3, (C5×C15)⋊2S3, C52⋊C3⋊1C6, C52⋊1(C3×S3), (C3×C52⋊C3)⋊3C2, SmallGroup(450,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C3 — C3×C52⋊S3 |
Generators and relations for C3×C52⋊S3
G = < a,b,c,d,e | a3=b5=c5=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc3, be=eb, dcd-1=b-1c3, ece=b-1c-1, ede=d-1 >
Smallest permutation representation of C3×C52⋊S3
►On 45 points
Generators in S45
(1 34 19)(2 35 20)(3 31 16)(4 32 17)(5 33 18)(6 36 21)(7 37 22)(8 38 23)(9 39 24)(10 40 25)(11 41 26)(12 42 27)(13 43 28)(14 44 29)(15 45 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)(31 32 33 34 35)(36 37 38 39 40)(41 42 43 44 45)
(1 3 5 2 4)(6 10 9 8 7)(16 18 20 17 19)(21 25 24 23 22)(31 33 35 32 34)(36 40 39 38 37)
(1 12 7)(2 15 10)(3 13 8)(4 11 6)(5 14 9)(16 28 23)(17 26 21)(18 29 24)(19 27 22)(20 30 25)(31 43 38)(32 41 36)(33 44 39)(34 42 37)(35 45 40)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)
G:=sub<Sym(45)| (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22)(31,33,35,32,34)(36,40,39,38,37), (1,12,7)(2,15,10)(3,13,8)(4,11,6)(5,14,9)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25)(31,43,38)(32,41,36)(33,44,39)(34,42,37)(35,45,40), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(36,41)(37,42)(38,43)(39,44)(40,45)>;
G:=Group( (1,34,19)(2,35,20)(3,31,16)(4,32,17)(5,33,18)(6,36,21)(7,37,22)(8,38,23)(9,39,24)(10,40,25)(11,41,26)(12,42,27)(13,43,28)(14,44,29)(15,45,30), (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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45), (1,3,5,2,4)(6,10,9,8,7)(16,18,20,17,19)(21,25,24,23,22)(31,33,35,32,34)(36,40,39,38,37), (1,12,7)(2,15,10)(3,13,8)(4,11,6)(5,14,9)(16,28,23)(17,26,21)(18,29,24)(19,27,22)(20,30,25)(31,43,38)(32,41,36)(33,44,39)(34,42,37)(35,45,40), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)(36,41)(37,42)(38,43)(39,44)(40,45) );
G=PermutationGroup([[(1,34,19),(2,35,20),(3,31,16),(4,32,17),(5,33,18),(6,36,21),(7,37,22),(8,38,23),(9,39,24),(10,40,25),(11,41,26),(12,42,27),(13,43,28),(14,44,29),(15,45,30)], [(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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45)], [(1,3,5,2,4),(6,10,9,8,7),(16,18,20,17,19),(21,25,24,23,22),(31,33,35,32,34),(36,40,39,38,37)], [(1,12,7),(2,15,10),(3,13,8),(4,11,6),(5,14,9),(16,28,23),(17,26,21),(18,29,24),(19,27,22),(20,30,25),(31,43,38),(32,41,36),(33,44,39),(34,42,37),(35,45,40)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30),(36,41),(37,42),(38,43),(39,44),(40,45)]])
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 5C | 5D | 5E | 5F | 6A | 6B | 10A | 10B | 10C | 10D | 15A | ··· | 15H | 15I | 15J | 15K | 15L | 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 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 15 | 1 | 1 | 50 | 50 | 50 | 3 | 3 | 3 | 3 | 6 | 6 | 15 | 15 | 15 | 15 | 15 | 15 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 15 | ··· | 15 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | C52⋊S3 | C3×C52⋊S3 | C52⋊S3 | C3×C52⋊S3 |
| kernel | C3×C52⋊S3 | C3×C52⋊C3 | C52⋊S3 | C52⋊C3 | C5×C15 | C52 | C3 | C1 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 8 | 16 | 2 | 4 |
Matrix representation of C3×C52⋊S3 ►in GL3(𝔽31) generated by
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 8 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 2 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(31))| [5,0,0,0,5,0,0,0,5],[8,0,0,0,2,0,0,0,2],[2,0,0,0,16,0,0,0,1],[0,0,1,1,0,0,0,1,0],[1,0,0,0,0,1,0,1,0] >;
C3×C52⋊S3 in GAP, Magma, Sage, TeX
C_3\times C_5^2\rtimes S_3
% in TeX
G:=Group("C3xC5^2:S3"); // GroupNames label
G:=SmallGroup(450,20);
// by ID
G=gap.SmallGroup(450,20);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,182,973,10804,1439]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^5=c^5=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^3,b*e=e*b,d*c*d^-1=b^-1*c^3,e*c*e=b^-1*c^-1,e*d*e=d^-1>;
// generators/relations
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