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## G = C5×C3⋊D15order 450 = 2·32·52

### Direct product of C5 and C3⋊D15

Aliases: C5×C3⋊D15, C153D15, C1524C2, C3⋊(C5×D15), C151(C5×S3), (C5×C15)⋊4S3, (C3×C15)⋊5D5, C322(C5×D5), (C3×C15)⋊1C10, C522(C3⋊S3), C5⋊(C5×C3⋊S3), SmallGroup(450,32)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C5×C3⋊D15
 Chief series C1 — C5 — C15 — C3×C15 — C152 — C5×C3⋊D15
 Lower central C3×C15 — C5×C3⋊D15
 Upper central C1 — C5

Generators and relations for C5×C3⋊D15
G = < a,b,c,d | a5=b3=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 312 in 60 conjugacy classes, 26 normal (10 characteristic)
C1, C2, C3, C5, C5, S3, C32, D5, C10, C15, C15, C3⋊S3, C52, C5×S3, D15, C3×C15, C3×C15, C5×D5, C5×C15, C5×C3⋊S3, C3⋊D15, C5×D15, C152, C5×C3⋊D15
Quotients: C1, C2, C5, S3, D5, C10, C3⋊S3, C5×S3, D15, C5×D5, C5×C3⋊S3, C3⋊D15, C5×D15, C5×C3⋊D15

Smallest permutation representation of C5×C3⋊D15
On 90 points
Generators in S90
(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 33 17)(2 34 18)(3 35 19)(4 36 20)(5 37 21)(6 38 22)(7 39 23)(8 40 24)(9 41 25)(10 42 26)(11 43 27)(12 44 28)(13 45 29)(14 31 30)(15 32 16)(46 65 88)(47 66 89)(48 67 90)(49 68 76)(50 69 77)(51 70 78)(52 71 79)(53 72 80)(54 73 81)(55 74 82)(56 75 83)(57 61 84)(58 62 85)(59 63 86)(60 64 87)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 65)(17 64)(18 63)(19 62)(20 61)(21 75)(22 74)(23 73)(24 72)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 90)

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

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

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

120 conjugacy classes

 class 1 2 3A 3B 3C 3D 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 15A ··· 15CR order 1 2 3 3 3 3 5 5 5 5 5 ··· 5 10 10 10 10 15 ··· 15 size 1 45 2 2 2 2 1 1 1 1 2 ··· 2 45 45 45 45 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C5 C10 S3 D5 C5×S3 D15 C5×D5 C5×D15 kernel C5×C3⋊D15 C152 C3⋊D15 C3×C15 C5×C15 C3×C15 C15 C15 C32 C3 # reps 1 1 4 4 4 2 16 16 8 64

Matrix representation of C5×C3⋊D15 in GL4(𝔽31) generated by

 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 25 0 0 0 0 5 0 0 0 0 29 30 0 0 3 1
,
 20 0 0 0 0 14 0 0 0 0 1 0 0 0 0 1
,
 0 14 0 0 20 0 0 0 0 0 12 9 0 0 22 19
G:=sub<GL(4,GF(31))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[25,0,0,0,0,5,0,0,0,0,29,3,0,0,30,1],[20,0,0,0,0,14,0,0,0,0,1,0,0,0,0,1],[0,20,0,0,14,0,0,0,0,0,12,22,0,0,9,19] >;

C5×C3⋊D15 in GAP, Magma, Sage, TeX

C_5\times C_3\rtimes D_{15}
% in TeX

G:=Group("C5xC3:D15");
// GroupNames label

G:=SmallGroup(450,32);
// by ID

G=gap.SmallGroup(450,32);
# by ID

G:=PCGroup([5,-2,-5,-3,-3,-5,302,1203,9004]);
// Polycyclic

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

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