direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D25, C25⋊C6, C75⋊2C2, C15.2D5, C5.(C3×D5), SmallGroup(150,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C3×D25 |
Generators and relations for C3×D25
G = < a,b,c | a3=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D25
►On 75 points
Generators in S75
(1 52 31)(2 53 32)(3 54 33)(4 55 34)(5 56 35)(6 57 36)(7 58 37)(8 59 38)(9 60 39)(10 61 40)(11 62 41)(12 63 42)(13 64 43)(14 65 44)(15 66 45)(16 67 46)(17 68 47)(18 69 48)(19 70 49)(20 71 50)(21 72 26)(22 73 27)(23 74 28)(24 75 29)(25 51 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 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 35)(27 34)(28 33)(29 32)(30 31)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(51 52)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)
G:=sub<Sym(75)| (1,52,31)(2,53,32)(3,54,33)(4,55,34)(5,56,35)(6,57,36)(7,58,37)(8,59,38)(9,60,39)(10,61,40)(11,62,41)(12,63,42)(13,64,43)(14,65,44)(15,66,45)(16,67,46)(17,68,47)(18,69,48)(19,70,49)(20,71,50)(21,72,26)(22,73,27)(23,74,28)(24,75,29)(25,51,30), (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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(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)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,52)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)>;
G:=Group( (1,52,31)(2,53,32)(3,54,33)(4,55,34)(5,56,35)(6,57,36)(7,58,37)(8,59,38)(9,60,39)(10,61,40)(11,62,41)(12,63,42)(13,64,43)(14,65,44)(15,66,45)(16,67,46)(17,68,47)(18,69,48)(19,70,49)(20,71,50)(21,72,26)(22,73,27)(23,74,28)(24,75,29)(25,51,30), (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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(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)(26,35)(27,34)(28,33)(29,32)(30,31)(36,50)(37,49)(38,48)(39,47)(40,46)(41,45)(42,44)(51,52)(53,75)(54,74)(55,73)(56,72)(57,71)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65) );
G=PermutationGroup([[(1,52,31),(2,53,32),(3,54,33),(4,55,34),(5,56,35),(6,57,36),(7,58,37),(8,59,38),(9,60,39),(10,61,40),(11,62,41),(12,63,42),(13,64,43),(14,65,44),(15,66,45),(16,67,46),(17,68,47),(18,69,48),(19,70,49),(20,71,50),(21,72,26),(22,73,27),(23,74,28),(24,75,29),(25,51,30)], [(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),(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50),(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),(26,35),(27,34),(28,33),(29,32),(30,31),(36,50),(37,49),(38,48),(39,47),(40,46),(41,45),(42,44),(51,52),(53,75),(54,74),(55,73),(56,72),(57,71),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65)]])
C3×D25 is a maximal subgroup of
C75⋊C4
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 15A | 15B | 15C | 15D | 25A | ··· | 25J | 75A | ··· | 75T |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 25 | 1 | 1 | 2 | 2 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | D5 | C3×D5 | D25 | C3×D25 |
| kernel | C3×D25 | C75 | D25 | C25 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 10 | 20 |
Matrix representation of C3×D25 ►in GL2(𝔽151) generated by
| 32 | 0 |
| 0 | 32 |
| 71 | 36 |
| 115 | 120 |
| 71 | 36 |
| 11 | 80 |
G:=sub<GL(2,GF(151))| [32,0,0,32],[71,115,36,120],[71,11,36,80] >;
C3×D25 in GAP, Magma, Sage, TeX
C_3\times D_{25} % in TeX
G:=Group("C3xD25"); // GroupNames label
G:=SmallGroup(150,2);
// by ID
G=gap.SmallGroup(150,2);
# by ID
G:=PCGroup([4,-2,-3,-5,-5,650,250,1923]);
// Polycyclic
G:=Group<a,b,c|a^3=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export