direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: S3×C25, C3⋊C50, C75⋊3C2, C15.C10, C5.(C5×S3), (C5×S3).C5, SmallGroup(150,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C25 |
Generators and relations for S3×C25
G = < a,b,c | a25=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C25
►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 66 30)(2 67 31)(3 68 32)(4 69 33)(5 70 34)(6 71 35)(7 72 36)(8 73 37)(9 74 38)(10 75 39)(11 51 40)(12 52 41)(13 53 42)(14 54 43)(15 55 44)(16 56 45)(17 57 46)(18 58 47)(19 59 48)(20 60 49)(21 61 50)(22 62 26)(23 63 27)(24 64 28)(25 65 29)
(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)
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), (1,66,30)(2,67,31)(3,68,32)(4,69,33)(5,70,34)(6,71,35)(7,72,36)(8,73,37)(9,74,38)(10,75,39)(11,51,40)(12,52,41)(13,53,42)(14,54,43)(15,55,44)(16,56,45)(17,57,46)(18,58,47)(19,59,48)(20,60,49)(21,61,50)(22,62,26)(23,63,27)(24,64,28)(25,65,29), (26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)>;
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), (1,66,30)(2,67,31)(3,68,32)(4,69,33)(5,70,34)(6,71,35)(7,72,36)(8,73,37)(9,74,38)(10,75,39)(11,51,40)(12,52,41)(13,53,42)(14,54,43)(15,55,44)(16,56,45)(17,57,46)(18,58,47)(19,59,48)(20,60,49)(21,61,50)(22,62,26)(23,63,27)(24,64,28)(25,65,29), (26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61) );
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)], [(1,66,30),(2,67,31),(3,68,32),(4,69,33),(5,70,34),(6,71,35),(7,72,36),(8,73,37),(9,74,38),(10,75,39),(11,51,40),(12,52,41),(13,53,42),(14,54,43),(15,55,44),(16,56,45),(17,57,46),(18,58,47),(19,59,48),(20,60,49),(21,61,50),(22,62,26),(23,63,27),(24,64,28),(25,65,29)], [(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61)]])
75 conjugacy classes
| class | 1 | 2 | 3 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 25A | ··· | 25T | 50A | ··· | 50T | 75A | ··· | 75T |
| order | 1 | 2 | 3 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 50 | ··· | 50 | 75 | ··· | 75 |
| size | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | ||||||
| image | C1 | C2 | C5 | C10 | C25 | C50 | S3 | C5×S3 | S3×C25 |
| kernel | S3×C25 | C75 | C5×S3 | C15 | S3 | C3 | C25 | C5 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 20 | 1 | 4 | 20 |
Matrix representation of S3×C25 ►in GL2(𝔽151) generated by
| 125 | 0 |
| 0 | 125 |
| 150 | 150 |
| 1 | 0 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(151))| [125,0,0,125],[150,1,150,0],[0,1,1,0] >;
S3×C25 in GAP, Magma, Sage, TeX
S_3\times C_{25} % in TeX
G:=Group("S3xC25"); // GroupNames label
G:=SmallGroup(150,1);
// by ID
G=gap.SmallGroup(150,1);
# by ID
G:=PCGroup([4,-2,-5,-5,-3,45,1603]);
// Polycyclic
G:=Group<a,b,c|a^25=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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