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## G = D5×C3×C9order 270 = 2·33·5

### Direct product of C3×C9 and D5

Aliases: D5×C3×C9, C457C6, C152C18, C5⋊(C3×C18), (C3×C45)⋊6C2, (C3×C15).4C6, C15.1(C3×C6), C3.1(C32×D5), C32.3(C3×D5), (C32×D5).2C3, (C3×D5).1C32, SmallGroup(270,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C3×C9
 Chief series C1 — C5 — C15 — C45 — C3×C45 — D5×C3×C9
 Lower central C5 — D5×C3×C9
 Upper central C1 — C3×C9

Generators and relations for D5×C3×C9
G = < a,b,c,d | a3=b9=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of D5×C3×C9
On 135 points
Generators in S135
(1 12 97)(2 13 98)(3 14 99)(4 15 91)(5 16 92)(6 17 93)(7 18 94)(8 10 95)(9 11 96)(19 104 56)(20 105 57)(21 106 58)(22 107 59)(23 108 60)(24 100 61)(25 101 62)(26 102 63)(27 103 55)(28 113 65)(29 114 66)(30 115 67)(31 116 68)(32 117 69)(33 109 70)(34 110 71)(35 111 72)(36 112 64)(37 118 79)(38 119 80)(39 120 81)(40 121 73)(41 122 74)(42 123 75)(43 124 76)(44 125 77)(45 126 78)(46 129 90)(47 130 82)(48 131 83)(49 132 84)(50 133 85)(51 134 86)(52 135 87)(53 127 88)(54 128 89)
(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)(91 92 93 94 95 96 97 98 99)(100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117)(118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135)
(1 72 52 78 55)(2 64 53 79 56)(3 65 54 80 57)(4 66 46 81 58)(5 67 47 73 59)(6 68 48 74 60)(7 69 49 75 61)(8 70 50 76 62)(9 71 51 77 63)(10 33 133 43 25)(11 34 134 44 26)(12 35 135 45 27)(13 36 127 37 19)(14 28 128 38 20)(15 29 129 39 21)(16 30 130 40 22)(17 31 131 41 23)(18 32 132 42 24)(82 121 107 92 115)(83 122 108 93 116)(84 123 100 94 117)(85 124 101 95 109)(86 125 102 96 110)(87 126 103 97 111)(88 118 104 98 112)(89 119 105 99 113)(90 120 106 91 114)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 37)(64 79)(65 80)(66 81)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(91 106)(92 107)(93 108)(94 100)(95 101)(96 102)(97 103)(98 104)(99 105)(109 124)(110 125)(111 126)(112 118)(113 119)(114 120)(115 121)(116 122)(117 123)

G:=sub<Sym(135)| (1,12,97)(2,13,98)(3,14,99)(4,15,91)(5,16,92)(6,17,93)(7,18,94)(8,10,95)(9,11,96)(19,104,56)(20,105,57)(21,106,58)(22,107,59)(23,108,60)(24,100,61)(25,101,62)(26,102,63)(27,103,55)(28,113,65)(29,114,66)(30,115,67)(31,116,68)(32,117,69)(33,109,70)(34,110,71)(35,111,72)(36,112,64)(37,118,79)(38,119,80)(39,120,81)(40,121,73)(41,122,74)(42,123,75)(43,124,76)(44,125,77)(45,126,78)(46,129,90)(47,130,82)(48,131,83)(49,132,84)(50,133,85)(51,134,86)(52,135,87)(53,127,88)(54,128,89), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135), (1,72,52,78,55)(2,64,53,79,56)(3,65,54,80,57)(4,66,46,81,58)(5,67,47,73,59)(6,68,48,74,60)(7,69,49,75,61)(8,70,50,76,62)(9,71,51,77,63)(10,33,133,43,25)(11,34,134,44,26)(12,35,135,45,27)(13,36,127,37,19)(14,28,128,38,20)(15,29,129,39,21)(16,30,130,40,22)(17,31,131,41,23)(18,32,132,42,24)(82,121,107,92,115)(83,122,108,93,116)(84,123,100,94,117)(85,124,101,95,109)(86,125,102,96,110)(87,126,103,97,111)(88,118,104,98,112)(89,119,105,99,113)(90,120,106,91,114), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37)(64,79)(65,80)(66,81)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(91,106)(92,107)(93,108)(94,100)(95,101)(96,102)(97,103)(98,104)(99,105)(109,124)(110,125)(111,126)(112,118)(113,119)(114,120)(115,121)(116,122)(117,123)>;

G:=Group( (1,12,97)(2,13,98)(3,14,99)(4,15,91)(5,16,92)(6,17,93)(7,18,94)(8,10,95)(9,11,96)(19,104,56)(20,105,57)(21,106,58)(22,107,59)(23,108,60)(24,100,61)(25,101,62)(26,102,63)(27,103,55)(28,113,65)(29,114,66)(30,115,67)(31,116,68)(32,117,69)(33,109,70)(34,110,71)(35,111,72)(36,112,64)(37,118,79)(38,119,80)(39,120,81)(40,121,73)(41,122,74)(42,123,75)(43,124,76)(44,125,77)(45,126,78)(46,129,90)(47,130,82)(48,131,83)(49,132,84)(50,133,85)(51,134,86)(52,135,87)(53,127,88)(54,128,89), (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)(91,92,93,94,95,96,97,98,99)(100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117)(118,119,120,121,122,123,124,125,126)(127,128,129,130,131,132,133,134,135), (1,72,52,78,55)(2,64,53,79,56)(3,65,54,80,57)(4,66,46,81,58)(5,67,47,73,59)(6,68,48,74,60)(7,69,49,75,61)(8,70,50,76,62)(9,71,51,77,63)(10,33,133,43,25)(11,34,134,44,26)(12,35,135,45,27)(13,36,127,37,19)(14,28,128,38,20)(15,29,129,39,21)(16,30,130,40,22)(17,31,131,41,23)(18,32,132,42,24)(82,121,107,92,115)(83,122,108,93,116)(84,123,100,94,117)(85,124,101,95,109)(86,125,102,96,110)(87,126,103,97,111)(88,118,104,98,112)(89,119,105,99,113)(90,120,106,91,114), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,37)(64,79)(65,80)(66,81)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(91,106)(92,107)(93,108)(94,100)(95,101)(96,102)(97,103)(98,104)(99,105)(109,124)(110,125)(111,126)(112,118)(113,119)(114,120)(115,121)(116,122)(117,123) );

G=PermutationGroup([[(1,12,97),(2,13,98),(3,14,99),(4,15,91),(5,16,92),(6,17,93),(7,18,94),(8,10,95),(9,11,96),(19,104,56),(20,105,57),(21,106,58),(22,107,59),(23,108,60),(24,100,61),(25,101,62),(26,102,63),(27,103,55),(28,113,65),(29,114,66),(30,115,67),(31,116,68),(32,117,69),(33,109,70),(34,110,71),(35,111,72),(36,112,64),(37,118,79),(38,119,80),(39,120,81),(40,121,73),(41,122,74),(42,123,75),(43,124,76),(44,125,77),(45,126,78),(46,129,90),(47,130,82),(48,131,83),(49,132,84),(50,133,85),(51,134,86),(52,135,87),(53,127,88),(54,128,89)], [(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),(91,92,93,94,95,96,97,98,99),(100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117),(118,119,120,121,122,123,124,125,126),(127,128,129,130,131,132,133,134,135)], [(1,72,52,78,55),(2,64,53,79,56),(3,65,54,80,57),(4,66,46,81,58),(5,67,47,73,59),(6,68,48,74,60),(7,69,49,75,61),(8,70,50,76,62),(9,71,51,77,63),(10,33,133,43,25),(11,34,134,44,26),(12,35,135,45,27),(13,36,127,37,19),(14,28,128,38,20),(15,29,129,39,21),(16,30,130,40,22),(17,31,131,41,23),(18,32,132,42,24),(82,121,107,92,115),(83,122,108,93,116),(84,123,100,94,117),(85,124,101,95,109),(86,125,102,96,110),(87,126,103,97,111),(88,118,104,98,112),(89,119,105,99,113),(90,120,106,91,114)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,25),(11,26),(12,27),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,37),(64,79),(65,80),(66,81),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(91,106),(92,107),(93,108),(94,100),(95,101),(96,102),(97,103),(98,104),(99,105),(109,124),(110,125),(111,126),(112,118),(113,119),(114,120),(115,121),(116,122),(117,123)]])

108 conjugacy classes

 class 1 2 3A ··· 3H 5A 5B 6A ··· 6H 9A ··· 9R 15A ··· 15P 18A ··· 18R 45A ··· 45AJ order 1 2 3 ··· 3 5 5 6 ··· 6 9 ··· 9 15 ··· 15 18 ··· 18 45 ··· 45 size 1 5 1 ··· 1 2 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C3 C6 C6 C9 C18 D5 C3×D5 C3×D5 C9×D5 kernel D5×C3×C9 C3×C45 C9×D5 C32×D5 C45 C3×C15 C3×D5 C15 C3×C9 C9 C32 C3 # reps 1 1 6 2 6 2 18 18 2 12 4 36

Matrix representation of D5×C3×C9 in GL3(𝔽181) generated by

 132 0 0 0 48 0 0 0 48
,
 1 0 0 0 43 0 0 0 43
,
 1 0 0 0 0 1 0 180 13
,
 180 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(181))| [132,0,0,0,48,0,0,0,48],[1,0,0,0,43,0,0,0,43],[1,0,0,0,0,180,0,1,13],[180,0,0,0,0,1,0,1,0] >;

D5×C3×C9 in GAP, Magma, Sage, TeX

D_5\times C_3\times C_9
% in TeX

G:=Group("D5xC3xC9");
// GroupNames label

G:=SmallGroup(270,5);
// by ID

G=gap.SmallGroup(270,5);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-5,57,5404]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=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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