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## G = D5×C3×C15order 450 = 2·32·52

### Direct product of C3×C15 and D5

Aliases: D5×C3×C15, C152C30, C1525C2, C5⋊(C3×C30), (C5×C15)⋊7C6, C523(C3×C6), (C3×C15)⋊3C10, SmallGroup(450,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C3×C15
 Chief series C1 — C5 — C52 — C5×C15 — C152 — D5×C3×C15
 Lower central C5 — D5×C3×C15
 Upper central C1 — C3×C15

Generators and relations for D5×C3×C15
G = < a,b,c,d | a3=b15=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 120 in 60 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C3, C5, C5, C6, C32, D5, C10, C15, C15, C3×C6, C52, C3×D5, C30, C3×C15, C3×C15, C5×D5, C5×C15, C32×D5, C3×C30, D5×C15, C152, D5×C3×C15
Quotients: C1, C2, C3, C5, C6, C32, D5, C10, C15, C3×C6, C3×D5, C30, C3×C15, C5×D5, C32×D5, C3×C30, D5×C15, D5×C3×C15

Smallest permutation representation of D5×C3×C15
On 90 points
Generators in S90
(1 32 20)(2 33 21)(3 34 22)(4 35 23)(5 36 24)(6 37 25)(7 38 26)(8 39 27)(9 40 28)(10 41 29)(11 42 30)(12 43 16)(13 44 17)(14 45 18)(15 31 19)(46 86 74)(47 87 75)(48 88 61)(49 89 62)(50 90 63)(51 76 64)(52 77 65)(53 78 66)(54 79 67)(55 80 68)(56 81 69)(57 82 70)(58 83 71)(59 84 72)(60 85 73)
(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)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90)
(1 7 13 4 10)(2 8 14 5 11)(3 9 15 6 12)(16 22 28 19 25)(17 23 29 20 26)(18 24 30 21 27)(31 37 43 34 40)(32 38 44 35 41)(33 39 45 36 42)(46 55 49 58 52)(47 56 50 59 53)(48 57 51 60 54)(61 70 64 73 67)(62 71 65 74 68)(63 72 66 75 69)(76 85 79 88 82)(77 86 80 89 83)(78 87 81 90 84)
(1 59)(2 60)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 89)(38 90)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)

G:=sub<Sym(90)| (1,32,20)(2,33,21)(3,34,22)(4,35,23)(5,36,24)(6,37,25)(7,38,26)(8,39,27)(9,40,28)(10,41,29)(11,42,30)(12,43,16)(13,44,17)(14,45,18)(15,31,19)(46,86,74)(47,87,75)(48,88,61)(49,89,62)(50,90,63)(51,76,64)(52,77,65)(53,78,66)(54,79,67)(55,80,68)(56,81,69)(57,82,70)(58,83,71)(59,84,72)(60,85,73), (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,7,13,4,10)(2,8,14,5,11)(3,9,15,6,12)(16,22,28,19,25)(17,23,29,20,26)(18,24,30,21,27)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,55,49,58,52)(47,56,50,59,53)(48,57,51,60,54)(61,70,64,73,67)(62,71,65,74,68)(63,72,66,75,69)(76,85,79,88,82)(77,86,80,89,83)(78,87,81,90,84), (1,59)(2,60)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82)>;

G:=Group( (1,32,20)(2,33,21)(3,34,22)(4,35,23)(5,36,24)(6,37,25)(7,38,26)(8,39,27)(9,40,28)(10,41,29)(11,42,30)(12,43,16)(13,44,17)(14,45,18)(15,31,19)(46,86,74)(47,87,75)(48,88,61)(49,89,62)(50,90,63)(51,76,64)(52,77,65)(53,78,66)(54,79,67)(55,80,68)(56,81,69)(57,82,70)(58,83,71)(59,84,72)(60,85,73), (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)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90), (1,7,13,4,10)(2,8,14,5,11)(3,9,15,6,12)(16,22,28,19,25)(17,23,29,20,26)(18,24,30,21,27)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,55,49,58,52)(47,56,50,59,53)(48,57,51,60,54)(61,70,64,73,67)(62,71,65,74,68)(63,72,66,75,69)(76,85,79,88,82)(77,86,80,89,83)(78,87,81,90,84), (1,59)(2,60)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,68)(17,69)(18,70)(19,71)(20,72)(21,73)(22,74)(23,75)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,83)(32,84)(33,85)(34,86)(35,87)(36,88)(37,89)(38,90)(39,76)(40,77)(41,78)(42,79)(43,80)(44,81)(45,82) );

G=PermutationGroup([[(1,32,20),(2,33,21),(3,34,22),(4,35,23),(5,36,24),(6,37,25),(7,38,26),(8,39,27),(9,40,28),(10,41,29),(11,42,30),(12,43,16),(13,44,17),(14,45,18),(15,31,19),(46,86,74),(47,87,75),(48,88,61),(49,89,62),(50,90,63),(51,76,64),(52,77,65),(53,78,66),(54,79,67),(55,80,68),(56,81,69),(57,82,70),(58,83,71),(59,84,72),(60,85,73)], [(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),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90)], [(1,7,13,4,10),(2,8,14,5,11),(3,9,15,6,12),(16,22,28,19,25),(17,23,29,20,26),(18,24,30,21,27),(31,37,43,34,40),(32,38,44,35,41),(33,39,45,36,42),(46,55,49,58,52),(47,56,50,59,53),(48,57,51,60,54),(61,70,64,73,67),(62,71,65,74,68),(63,72,66,75,69),(76,85,79,88,82),(77,86,80,89,83),(78,87,81,90,84)], [(1,59),(2,60),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,68),(17,69),(18,70),(19,71),(20,72),(21,73),(22,74),(23,75),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,83),(32,84),(33,85),(34,86),(35,87),(36,88),(37,89),(38,90),(39,76),(40,77),(41,78),(42,79),(43,80),(44,81),(45,82)]])

180 conjugacy classes

 class 1 2 3A ··· 3H 5A 5B 5C 5D 5E ··· 5N 6A ··· 6H 10A 10B 10C 10D 15A ··· 15AF 15AG ··· 15DH 30A ··· 30AF order 1 2 3 ··· 3 5 5 5 5 5 ··· 5 6 ··· 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 5 1 ··· 1 1 1 1 1 2 ··· 2 5 ··· 5 5 5 5 5 1 ··· 1 2 ··· 2 5 ··· 5

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C5 C6 C10 C15 C30 D5 C3×D5 C5×D5 D5×C15 kernel D5×C3×C15 C152 D5×C15 C32×D5 C5×C15 C3×C15 C3×D5 C15 C3×C15 C15 C32 C3 # reps 1 1 8 4 8 4 32 32 2 16 8 64

Matrix representation of D5×C3×C15 in GL3(𝔽31) generated by

 5 0 0 0 25 0 0 0 25
,
 20 0 0 0 18 0 0 0 18
,
 1 0 0 0 16 0 0 1 2
,
 1 0 0 0 20 30 0 27 11
G:=sub<GL(3,GF(31))| [5,0,0,0,25,0,0,0,25],[20,0,0,0,18,0,0,0,18],[1,0,0,0,16,1,0,0,2],[1,0,0,0,20,27,0,30,11] >;

D5×C3×C15 in GAP, Magma, Sage, TeX

D_5\times C_3\times C_{15}
% in TeX

G:=Group("D5xC3xC15");
// GroupNames label

G:=SmallGroup(450,26);
// by ID

G=gap.SmallGroup(450,26);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-5,9004]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^5=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

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