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## G = C15×C3⋊S3order 270 = 2·33·5

### Direct product of C15 and C3⋊S3

Aliases: C15×C3⋊S3, C332C10, C324C30, C3⋊(S3×C15), C153(C3×S3), (C3×C15)⋊8S3, (C3×C15)⋊9C6, C323(C5×S3), (C32×C15)⋊6C2, SmallGroup(270,26)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C15×C3⋊S3
G = < a,b,c,d | a15=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 144 in 64 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, C5, S3, C6, C32, C32, C32, C10, C15, C15, C15, C3×S3, C3⋊S3, C33, C5×S3, C30, C3×C15, C3×C15, C3×C15, C3×C3⋊S3, S3×C15, C5×C3⋊S3, C32×C15, C15×C3⋊S3
Quotients: C1, C2, C3, C5, S3, C6, C10, C15, C3×S3, C3⋊S3, C5×S3, C30, C3×C3⋊S3, S3×C15, C5×C3⋊S3, C15×C3⋊S3

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

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

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

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

90 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I ··· 15BD 30A ··· 30H 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 9 1 1 2 ··· 2 1 1 1 1 9 9 9 9 9 9 1 ··· 1 2 ··· 2 9 ··· 9

90 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 C15×C3⋊S3 C32×C15 C5×C3⋊S3 C3×C3⋊S3 C3×C15 C33 C3⋊S3 C32 C3×C15 C15 C32 C3 # reps 1 1 2 4 2 4 8 8 4 8 16 32

Matrix representation of C15×C3⋊S3 in GL4(𝔽31) generated by

 16 0 0 0 0 16 0 0 0 0 25 0 0 0 0 25
,
 25 27 0 0 0 5 0 0 0 0 1 0 0 0 0 1
,
 5 4 0 0 0 25 0 0 0 0 25 0 0 0 23 5
,
 27 24 0 0 11 4 0 0 0 0 14 4 0 0 21 17
G:=sub<GL(4,GF(31))| [16,0,0,0,0,16,0,0,0,0,25,0,0,0,0,25],[25,0,0,0,27,5,0,0,0,0,1,0,0,0,0,1],[5,0,0,0,4,25,0,0,0,0,25,23,0,0,0,5],[27,11,0,0,24,4,0,0,0,0,14,21,0,0,4,17] >;

C15×C3⋊S3 in GAP, Magma, Sage, TeX

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

G:=Group("C15xC3:S3");
// GroupNames label

G:=SmallGroup(270,26);
// by ID

G=gap.SmallGroup(270,26);
# by ID

G:=PCGroup([5,-2,-3,-5,-3,-3,1203,4504]);
// Polycyclic

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