direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×D25, D75⋊C2, C3⋊1D50, C25⋊1D6, C75⋊C22, C15.D10, C5.(S3×D5), (S3×C25)⋊C2, (C3×D25)⋊C2, (C5×S3).D5, SmallGroup(300,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C75 — S3×D25 |
Generators and relations for S3×D25
G = < a,b,c,d | a3=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of S3×D25
►On 75 points
Generators in S75
(1 46 70)(2 47 71)(3 48 72)(4 49 73)(5 50 74)(6 26 75)(7 27 51)(8 28 52)(9 29 53)(10 30 54)(11 31 55)(12 32 56)(13 33 57)(14 34 58)(15 35 59)(16 36 60)(17 37 61)(18 38 62)(19 39 63)(20 40 64)(21 41 65)(22 42 66)(23 43 67)(24 44 68)(25 45 69)
(26 75)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(49 73)(50 74)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 50)(42 49)(43 48)(44 47)(45 46)(51 63)(52 62)(53 61)(54 60)(55 59)(56 58)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)
G:=sub<Sym(75)| (1,46,70)(2,47,71)(3,48,72)(4,49,73)(5,50,74)(6,26,75)(7,27,51)(8,28,52)(9,29,53)(10,30,54)(11,31,55)(12,32,56)(13,33,57)(14,34,58)(15,35,59)(16,36,60)(17,37,61)(18,38,62)(19,39,63)(20,40,64)(21,41,65)(22,42,66)(23,43,67)(24,44,68)(25,45,69), (26,75)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)>;
G:=Group( (1,46,70)(2,47,71)(3,48,72)(4,49,73)(5,50,74)(6,26,75)(7,27,51)(8,28,52)(9,29,53)(10,30,54)(11,31,55)(12,32,56)(13,33,57)(14,34,58)(15,35,59)(16,36,60)(17,37,61)(18,38,62)(19,39,63)(20,40,64)(21,41,65)(22,42,66)(23,43,67)(24,44,68)(25,45,69), (26,75)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(49,73)(50,74), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(41,50)(42,49)(43,48)(44,47)(45,46)(51,63)(52,62)(53,61)(54,60)(55,59)(56,58)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70) );
G=PermutationGroup([[(1,46,70),(2,47,71),(3,48,72),(4,49,73),(5,50,74),(6,26,75),(7,27,51),(8,28,52),(9,29,53),(10,30,54),(11,31,55),(12,32,56),(13,33,57),(14,34,58),(15,35,59),(16,36,60),(17,37,61),(18,38,62),(19,39,63),(20,40,64),(21,41,65),(22,42,66),(23,43,67),(24,44,68),(25,45,69)], [(26,75),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(49,73),(50,74)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(41,50),(42,49),(43,48),(44,47),(45,46),(51,63),(52,62),(53,61),(54,60),(55,59),(56,58),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 6 | 10A | 10B | 15A | 15B | 25A | ··· | 25J | 50A | ··· | 50J | 75A | ··· | 75J |
| order | 1 | 2 | 2 | 2 | 3 | 5 | 5 | 6 | 10 | 10 | 15 | 15 | 25 | ··· | 25 | 50 | ··· | 50 | 75 | ··· | 75 |
| size | 1 | 3 | 25 | 75 | 2 | 2 | 2 | 50 | 6 | 6 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D10 | D25 | D50 | S3×D5 | S3×D25 |
| kernel | S3×D25 | S3×C25 | C3×D25 | D75 | D25 | C5×S3 | C25 | C15 | S3 | C3 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 10 | 10 | 2 | 10 |
Matrix representation of S3×D25 ►in GL4(𝔽151) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 121 |
| 0 | 0 | 136 | 149 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 150 | 30 |
| 0 | 0 | 0 | 1 |
| 88 | 136 | 0 | 0 |
| 133 | 70 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 136 | 40 | 0 | 0 |
| 85 | 15 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(151))| [1,0,0,0,0,1,0,0,0,0,1,136,0,0,121,149],[1,0,0,0,0,1,0,0,0,0,150,0,0,0,30,1],[88,133,0,0,136,70,0,0,0,0,1,0,0,0,0,1],[136,85,0,0,40,15,0,0,0,0,1,0,0,0,0,1] >;
S3×D25 in GAP, Magma, Sage, TeX
S_3\times D_{25} % in TeX
G:=Group("S3xD25"); // GroupNames label
G:=SmallGroup(300,7);
// by ID
G=gap.SmallGroup(300,7);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,67,2163,418,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^25=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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