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## G = C32×D15order 270 = 2·33·5

### Direct product of C32 and D15

Aliases: C32×D15, C331D5, C5⋊(S3×C32), C3⋊(C32×D5), C152(C3×S3), (C3×C15)⋊6C6, (C3×C15)⋊7S3, C151(C3×C6), (C32×C15)⋊2C2, C323(C3×D5), SmallGroup(270,25)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C32×D15
 Chief series C1 — C5 — C15 — C3×C15 — C32×C15 — C32×D15
 Lower central C15 — C32×D15
 Upper central C1 — C32

Generators and relations for C32×D15
G = < a,b,c,d | a3=b3=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 200 in 64 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, C5, S3, C6, C32, C32, C32, D5, C15, C15, C15, C3×S3, C3×C6, C33, C3×D5, D15, C3×C15, C3×C15, C3×C15, S3×C32, C32×D5, C3×D15, C32×C15, C32×D15
Quotients: C1, C2, C3, S3, C6, C32, D5, C3×S3, C3×C6, C3×D5, D15, S3×C32, C32×D5, C3×D15, C32×D15

Smallest permutation representation of C32×D15
On 90 points
Generators in S90
(1 21 38)(2 22 39)(3 23 40)(4 24 41)(5 25 42)(6 26 43)(7 27 44)(8 28 45)(9 29 31)(10 30 32)(11 16 33)(12 17 34)(13 18 35)(14 19 36)(15 20 37)(46 68 88)(47 69 89)(48 70 90)(49 71 76)(50 72 77)(51 73 78)(52 74 79)(53 75 80)(54 61 81)(55 62 82)(56 63 83)(57 64 84)(58 65 85)(59 66 86)(60 67 87)
(1 43 16)(2 44 17)(3 45 18)(4 31 19)(5 32 20)(6 33 21)(7 34 22)(8 35 23)(9 36 24)(10 37 25)(11 38 26)(12 39 27)(13 40 28)(14 41 29)(15 42 30)(46 83 73)(47 84 74)(48 85 75)(49 86 61)(50 87 62)(51 88 63)(52 89 64)(53 90 65)(54 76 66)(55 77 67)(56 78 68)(57 79 69)(58 80 70)(59 81 71)(60 82 72)
(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 46)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 73)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 75)(30 74)(31 80)(32 79)(33 78)(34 77)(35 76)(36 90)(37 89)(38 88)(39 87)(40 86)(41 85)(42 84)(43 83)(44 82)(45 81)

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

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

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

81 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 5A 5B 6A ··· 6H 15A ··· 15AZ order 1 2 3 ··· 3 3 ··· 3 5 5 6 ··· 6 15 ··· 15 size 1 15 1 ··· 1 2 ··· 2 2 2 15 ··· 15 2 ··· 2

81 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 D5 C3×S3 C3×D5 D15 C3×D15 kernel C32×D15 C32×C15 C3×D15 C3×C15 C3×C15 C33 C15 C32 C32 C3 # reps 1 1 8 8 1 2 8 16 4 32

Matrix representation of C32×D15 in GL3(𝔽31) generated by

 25 0 0 0 25 0 0 0 25
,
 25 0 0 0 1 0 0 0 1
,
 1 0 0 0 18 0 0 0 19
,
 30 0 0 0 0 19 0 18 0
G:=sub<GL(3,GF(31))| [25,0,0,0,25,0,0,0,25],[25,0,0,0,1,0,0,0,1],[1,0,0,0,18,0,0,0,19],[30,0,0,0,0,18,0,19,0] >;

C32×D15 in GAP, Magma, Sage, TeX

C_3^2\times D_{15}
% in TeX

G:=Group("C3^2xD15");
// GroupNames label

G:=SmallGroup(270,25);
// by ID

G=gap.SmallGroup(270,25);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,723,5404]);
// Polycyclic

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