direct product, metabelian, soluble, monomial, A-group
Aliases: S3×C52⋊C3, (C5×C15)⋊3C6, (S3×C52)⋊C3, C52⋊3(C3×S3), C3⋊(C2×C52⋊C3), (C3×C52⋊C3)⋊5C2, SmallGroup(450,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — S3×C52⋊C3 |
Generators and relations for S3×C52⋊C3
G = < a,b,c,d,e | a3=b2=c5=d5=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c3d3, ede-1=c-1d >
Smallest permutation representation of S3×C52⋊C3
►On 45 points
Generators in S45
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)
(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)
(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 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)
(1 34 18)(2 35 20)(3 31 17)(4 32 19)(5 33 16)(6 36 22)(7 37 24)(8 38 21)(9 39 23)(10 40 25)(11 41 27)(12 42 29)(13 43 26)(14 44 28)(15 45 30)
G:=sub<Sym(45)| (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45), (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), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28), (1,34,18)(2,35,20)(3,31,17)(4,32,19)(5,33,16)(6,36,22)(7,37,24)(8,38,21)(9,39,23)(10,40,25)(11,41,27)(12,42,29)(13,43,26)(14,44,28)(15,45,30)>;
G:=Group( (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45), (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), (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), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28), (1,34,18)(2,35,20)(3,31,17)(4,32,19)(5,33,16)(6,36,22)(7,37,24)(8,38,21)(9,39,23)(10,40,25)(11,41,27)(12,42,29)(13,43,26)(14,44,28)(15,45,30) );
G=PermutationGroup([[(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45)], [(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)], [(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)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,19,17,20,18),(21,24,22,25,23),(26,29,27,30,28)], [(1,34,18),(2,35,20),(3,31,17),(4,32,19),(5,33,16),(6,36,22),(7,37,24),(8,38,21),(9,39,23),(10,40,25),(11,41,27),(12,42,29),(13,43,26),(14,44,28),(15,45,30)]])
33 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | ··· | 5H | 6A | 6B | 10A | ··· | 10H | 15A | ··· | 15H |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 10 | ··· | 10 | 15 | ··· | 15 |
| size | 1 | 3 | 2 | 25 | 25 | 50 | 50 | 3 | ··· | 3 | 75 | 75 | 9 | ··· | 9 | 6 | ··· | 6 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | C52⋊C3 | C2×C52⋊C3 | S3×C52⋊C3 |
| kernel | S3×C52⋊C3 | C3×C52⋊C3 | S3×C52 | C5×C15 | C52⋊C3 | C52 | S3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 8 | 8 | 8 |
Matrix representation of S3×C52⋊C3 ►in GL5(𝔽31)
| 0 | 30 | 0 | 0 | 0 |
| 1 | 30 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 19 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 2 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 25 | 0 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 15 | 30 | 30 |
G:=sub<GL(5,GF(31))| [0,1,0,0,0,30,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,19,8,0,0,0,0,0,2],[1,0,0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,3,16,0,0,0,2,0,1],[25,0,0,0,0,0,25,0,0,0,0,0,1,0,15,0,0,0,0,30,0,0,0,1,30] >;
S3×C52⋊C3 in GAP, Magma, Sage, TeX
S_3\times C_5^2\rtimes C_3
% in TeX
G:=Group("S3xC5^2:C3"); // GroupNames label
G:=SmallGroup(450,23);
// by ID
G=gap.SmallGroup(450,23);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,5,182,2348,1359]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^5=d^5=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^3*d^3,e*d*e^-1=c^-1*d>;
// generators/relations
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