direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3xC5:D5, C15:2D5, C52:4C6, C5:(C3xD5), (C5xC15):4C2, SmallGroup(150,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C3xC5:D5 |
Generators and relations for C3xC5:D5
G = < a,b,c,d | a3=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Quotients: C1, C2, C3, C6, D5, C3xD5, C5:D5, C3xC5:D5
Smallest permutation representation of C3xC5:D5
►On 75 points
Generators in S75
(1 54 29)(2 55 30)(3 51 26)(4 52 27)(5 53 28)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 61 36)(12 62 37)(13 63 38)(14 64 39)(15 65 40)(16 66 41)(17 67 42)(18 68 43)(19 69 44)(20 70 45)(21 71 46)(22 72 47)(23 73 48)(24 74 49)(25 75 50)
(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 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)
(1 9)(2 8)(3 7)(4 6)(5 10)(11 22)(12 21)(13 25)(14 24)(15 23)(16 17)(18 20)(26 32)(27 31)(28 35)(29 34)(30 33)(36 47)(37 46)(38 50)(39 49)(40 48)(41 42)(43 45)(51 57)(52 56)(53 60)(54 59)(55 58)(61 72)(62 71)(63 75)(64 74)(65 73)(66 67)(68 70)
G:=sub<Sym(75)| (1,54,29)(2,55,30)(3,51,26)(4,52,27)(5,53,28)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,61,36)(12,62,37)(13,63,38)(14,64,39)(15,65,40)(16,66,41)(17,67,42)(18,68,43)(19,69,44)(20,70,45)(21,71,46)(22,72,47)(23,73,48)(24,74,49)(25,75,50), (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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)(26,32)(27,31)(28,35)(29,34)(30,33)(36,47)(37,46)(38,50)(39,49)(40,48)(41,42)(43,45)(51,57)(52,56)(53,60)(54,59)(55,58)(61,72)(62,71)(63,75)(64,74)(65,73)(66,67)(68,70)>;
G:=Group( (1,54,29)(2,55,30)(3,51,26)(4,52,27)(5,53,28)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,61,36)(12,62,37)(13,63,38)(14,64,39)(15,65,40)(16,66,41)(17,67,42)(18,68,43)(19,69,44)(20,70,45)(21,71,46)(22,72,47)(23,73,48)(24,74,49)(25,75,50), (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), (1,24,19,14,9)(2,25,20,15,10)(3,21,16,11,6)(4,22,17,12,7)(5,23,18,13,8)(26,46,41,36,31)(27,47,42,37,32)(28,48,43,38,33)(29,49,44,39,34)(30,50,45,40,35)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60), (1,9)(2,8)(3,7)(4,6)(5,10)(11,22)(12,21)(13,25)(14,24)(15,23)(16,17)(18,20)(26,32)(27,31)(28,35)(29,34)(30,33)(36,47)(37,46)(38,50)(39,49)(40,48)(41,42)(43,45)(51,57)(52,56)(53,60)(54,59)(55,58)(61,72)(62,71)(63,75)(64,74)(65,73)(66,67)(68,70) );
G=PermutationGroup([[(1,54,29),(2,55,30),(3,51,26),(4,52,27),(5,53,28),(6,56,31),(7,57,32),(8,58,33),(9,59,34),(10,60,35),(11,61,36),(12,62,37),(13,63,38),(14,64,39),(15,65,40),(16,66,41),(17,67,42),(18,68,43),(19,69,44),(20,70,45),(21,71,46),(22,72,47),(23,73,48),(24,74,49),(25,75,50)], [(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)], [(1,24,19,14,9),(2,25,20,15,10),(3,21,16,11,6),(4,22,17,12,7),(5,23,18,13,8),(26,46,41,36,31),(27,47,42,37,32),(28,48,43,38,33),(29,49,44,39,34),(30,50,45,40,35),(51,71,66,61,56),(52,72,67,62,57),(53,73,68,63,58),(54,74,69,64,59),(55,75,70,65,60)], [(1,9),(2,8),(3,7),(4,6),(5,10),(11,22),(12,21),(13,25),(14,24),(15,23),(16,17),(18,20),(26,32),(27,31),(28,35),(29,34),(30,33),(36,47),(37,46),(38,50),(39,49),(40,48),(41,42),(43,45),(51,57),(52,56),(53,60),(54,59),(55,58),(61,72),(62,71),(63,75),(64,74),(65,73),(66,67),(68,70)]])
C3xC5:D5 is a maximal subgroup of
C15:F5 C15:2F5 C3xD52 D15:D5 C52:C18
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | ··· | 5L | 6A | 6B | 15A | ··· | 15X |
| order | 1 | 2 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 15 | ··· | 15 |
| size | 1 | 25 | 1 | 1 | 2 | ··· | 2 | 25 | 25 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D5 | C3xD5 |
| kernel | C3xC5:D5 | C5xC15 | C5:D5 | C52 | C15 | C5 |
| # reps | 1 | 1 | 2 | 2 | 12 | 24 |
Matrix representation of C3xC5:D5 ►in GL4(F31) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 12 | 1 | 0 | 0 |
| 30 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 30 | 18 |
| 30 | 19 | 0 | 0 |
| 12 | 19 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 19 | 30 | 0 | 0 |
| 19 | 12 | 0 | 0 |
| 0 | 0 | 30 | 18 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(31))| [25,0,0,0,0,25,0,0,0,0,25,0,0,0,0,25],[12,30,0,0,1,0,0,0,0,0,0,30,0,0,1,18],[30,12,0,0,19,19,0,0,0,0,1,0,0,0,0,1],[19,19,0,0,30,12,0,0,0,0,30,0,0,0,18,1] >;
C3xC5:D5 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes D_5
% in TeX
G:=Group("C3xC5:D5"); // GroupNames label
G:=SmallGroup(150,9);
// by ID
G=gap.SmallGroup(150,9);
# by ID
G:=PCGroup([4,-2,-3,-5,-5,290,1923]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export