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## G = S3×C3×C15order 270 = 2·33·5

### Direct product of C3×C15 and S3

Aliases: S3×C3×C15, C331C10, C323C30, C3⋊(C3×C30), (C3×C15)⋊8C6, C153(C3×C6), (C32×C15)⋊5C2, SmallGroup(270,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C15
 Chief series C1 — C3 — C15 — C3×C15 — C32×C15 — S3×C3×C15
 Lower central C3 — S3×C3×C15
 Upper central C1 — C3×C15

Generators and relations for S3×C3×C15
G = < a,b,c,d | a3=b15=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 104 in 64 conjugacy classes, 36 normal (12 characteristic)
C1, C2, C3, C3, C3, C5, S3, C6, C32, C32, C32, C10, C15, C15, C15, C3×S3, C3×C6, C33, C5×S3, C30, C3×C15, C3×C15, C3×C15, S3×C32, S3×C15, C3×C30, C32×C15, S3×C3×C15
Quotients: C1, C2, C3, C5, S3, C6, C32, C10, C15, C3×S3, C3×C6, C5×S3, C30, C3×C15, S3×C32, S3×C15, C3×C30, S3×C3×C15

Smallest permutation representation of S3×C3×C15
On 90 points
Generators in S90
(1 42 70)(2 43 71)(3 44 72)(4 45 73)(5 31 74)(6 32 75)(7 33 61)(8 34 62)(9 35 63)(10 36 64)(11 37 65)(12 38 66)(13 39 67)(14 40 68)(15 41 69)(16 56 78)(17 57 79)(18 58 80)(19 59 81)(20 60 82)(21 46 83)(22 47 84)(23 48 85)(24 49 86)(25 50 87)(26 51 88)(27 52 89)(28 53 90)(29 54 76)(30 55 77)
(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 32 65)(2 33 66)(3 34 67)(4 35 68)(5 36 69)(6 37 70)(7 38 71)(8 39 72)(9 40 73)(10 41 74)(11 42 75)(12 43 61)(13 44 62)(14 45 63)(15 31 64)(16 88 46)(17 89 47)(18 90 48)(19 76 49)(20 77 50)(21 78 51)(22 79 52)(23 80 53)(24 81 54)(25 82 55)(26 83 56)(27 84 57)(28 85 58)(29 86 59)(30 87 60)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 38)(17 39)(18 40)(19 41)(20 42)(21 43)(22 44)(23 45)(24 31)(25 32)(26 33)(27 34)(28 35)(29 36)(30 37)(46 71)(47 72)(48 73)(49 74)(50 75)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)

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

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

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

135 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 5A 5B 5C 5D 6A ··· 6H 10A 10B 10C 10D 15A ··· 15AF 15AG ··· 15BP 30A ··· 30AF order 1 2 3 ··· 3 3 ··· 3 5 5 5 5 6 ··· 6 10 10 10 10 15 ··· 15 15 ··· 15 30 ··· 30 size 1 3 1 ··· 1 2 ··· 2 1 1 1 1 3 ··· 3 3 3 3 3 1 ··· 1 2 ··· 2 3 ··· 3

135 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 S3 C3×S3 C5×S3 S3×C15 kernel S3×C3×C15 C32×C15 S3×C15 S3×C32 C3×C15 C33 C3×S3 C32 C3×C15 C15 C32 C3 # reps 1 1 8 4 8 4 32 32 1 8 4 32

Matrix representation of S3×C3×C15 in GL3(𝔽31) generated by

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

S3×C3×C15 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{15}
% in TeX

G:=Group("S3xC3xC15");
// GroupNames label

G:=SmallGroup(270,24);
// by ID

G=gap.SmallGroup(270,24);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-3,4504]);
// Polycyclic

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