direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C52, C15⋊3C10, C3⋊(C5×C10), (C5×C15)⋊5C2, SmallGroup(150,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C52 |
Generators and relations for S3×C52
G = < a,b,c,d | a5=b5=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×C52
►On 75 points
Generators in S75
(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)(46 47 48 49 50)(51 52 53 54 55)(56 57 58 59 60)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)
(1 36 8 48 31)(2 37 9 49 32)(3 38 10 50 33)(4 39 6 46 34)(5 40 7 47 35)(11 25 55 43 28)(12 21 51 44 29)(13 22 52 45 30)(14 23 53 41 26)(15 24 54 42 27)(16 71 60 65 66)(17 72 56 61 67)(18 73 57 62 68)(19 74 58 63 69)(20 75 59 64 70)
(1 27 72)(2 28 73)(3 29 74)(4 30 75)(5 26 71)(6 22 64)(7 23 65)(8 24 61)(9 25 62)(10 21 63)(11 57 37)(12 58 38)(13 59 39)(14 60 40)(15 56 36)(16 35 41)(17 31 42)(18 32 43)(19 33 44)(20 34 45)(46 52 70)(47 53 66)(48 54 67)(49 55 68)(50 51 69)
(11 57)(12 58)(13 59)(14 60)(15 56)(16 41)(17 42)(18 43)(19 44)(20 45)(21 63)(22 64)(23 65)(24 61)(25 62)(26 71)(27 72)(28 73)(29 74)(30 75)(51 69)(52 70)(53 66)(54 67)(55 68)
G:=sub<Sym(75)| (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)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75), (1,36,8,48,31)(2,37,9,49,32)(3,38,10,50,33)(4,39,6,46,34)(5,40,7,47,35)(11,25,55,43,28)(12,21,51,44,29)(13,22,52,45,30)(14,23,53,41,26)(15,24,54,42,27)(16,71,60,65,66)(17,72,56,61,67)(18,73,57,62,68)(19,74,58,63,69)(20,75,59,64,70), (1,27,72)(2,28,73)(3,29,74)(4,30,75)(5,26,71)(6,22,64)(7,23,65)(8,24,61)(9,25,62)(10,21,63)(11,57,37)(12,58,38)(13,59,39)(14,60,40)(15,56,36)(16,35,41)(17,31,42)(18,32,43)(19,33,44)(20,34,45)(46,52,70)(47,53,66)(48,54,67)(49,55,68)(50,51,69), (11,57)(12,58)(13,59)(14,60)(15,56)(16,41)(17,42)(18,43)(19,44)(20,45)(21,63)(22,64)(23,65)(24,61)(25,62)(26,71)(27,72)(28,73)(29,74)(30,75)(51,69)(52,70)(53,66)(54,67)(55,68)>;
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)(31,32,33,34,35)(36,37,38,39,40)(41,42,43,44,45)(46,47,48,49,50)(51,52,53,54,55)(56,57,58,59,60)(61,62,63,64,65)(66,67,68,69,70)(71,72,73,74,75), (1,36,8,48,31)(2,37,9,49,32)(3,38,10,50,33)(4,39,6,46,34)(5,40,7,47,35)(11,25,55,43,28)(12,21,51,44,29)(13,22,52,45,30)(14,23,53,41,26)(15,24,54,42,27)(16,71,60,65,66)(17,72,56,61,67)(18,73,57,62,68)(19,74,58,63,69)(20,75,59,64,70), (1,27,72)(2,28,73)(3,29,74)(4,30,75)(5,26,71)(6,22,64)(7,23,65)(8,24,61)(9,25,62)(10,21,63)(11,57,37)(12,58,38)(13,59,39)(14,60,40)(15,56,36)(16,35,41)(17,31,42)(18,32,43)(19,33,44)(20,34,45)(46,52,70)(47,53,66)(48,54,67)(49,55,68)(50,51,69), (11,57)(12,58)(13,59)(14,60)(15,56)(16,41)(17,42)(18,43)(19,44)(20,45)(21,63)(22,64)(23,65)(24,61)(25,62)(26,71)(27,72)(28,73)(29,74)(30,75)(51,69)(52,70)(53,66)(54,67)(55,68) );
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),(31,32,33,34,35),(36,37,38,39,40),(41,42,43,44,45),(46,47,48,49,50),(51,52,53,54,55),(56,57,58,59,60),(61,62,63,64,65),(66,67,68,69,70),(71,72,73,74,75)], [(1,36,8,48,31),(2,37,9,49,32),(3,38,10,50,33),(4,39,6,46,34),(5,40,7,47,35),(11,25,55,43,28),(12,21,51,44,29),(13,22,52,45,30),(14,23,53,41,26),(15,24,54,42,27),(16,71,60,65,66),(17,72,56,61,67),(18,73,57,62,68),(19,74,58,63,69),(20,75,59,64,70)], [(1,27,72),(2,28,73),(3,29,74),(4,30,75),(5,26,71),(6,22,64),(7,23,65),(8,24,61),(9,25,62),(10,21,63),(11,57,37),(12,58,38),(13,59,39),(14,60,40),(15,56,36),(16,35,41),(17,31,42),(18,32,43),(19,33,44),(20,34,45),(46,52,70),(47,53,66),(48,54,67),(49,55,68),(50,51,69)], [(11,57),(12,58),(13,59),(14,60),(15,56),(16,41),(17,42),(18,43),(19,44),(20,45),(21,63),(22,64),(23,65),(24,61),(25,62),(26,71),(27,72),(28,73),(29,74),(30,75),(51,69),(52,70),(53,66),(54,67),(55,68)]])
75 conjugacy classes
| class | 1 | 2 | 3 | 5A | ··· | 5X | 10A | ··· | 10X | 15A | ··· | 15X |
| order | 1 | 2 | 3 | 5 | ··· | 5 | 10 | ··· | 10 | 15 | ··· | 15 |
| size | 1 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C5 | C10 | S3 | C5×S3 |
| kernel | S3×C52 | C5×C15 | C5×S3 | C15 | C52 | C5 |
| # reps | 1 | 1 | 24 | 24 | 1 | 24 |
Matrix representation of S3×C52 ►in GL3(𝔽31) generated by
| 16 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 1 | 0 | 0 |
| 0 | 30 | 30 |
| 0 | 1 | 0 |
| 30 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(31))| [16,0,0,0,8,0,0,0,8],[8,0,0,0,8,0,0,0,8],[1,0,0,0,30,1,0,30,0],[30,0,0,0,0,1,0,1,0] >;
S3×C52 in GAP, Magma, Sage, TeX
S_3\times C_5^2
% in TeX
G:=Group("S3xC5^2"); // GroupNames label
G:=SmallGroup(150,10);
// by ID
G=gap.SmallGroup(150,10);
# by ID
G:=PCGroup([4,-2,-5,-5,-3,1603]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export