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## G = C32×D27order 486 = 2·35

### Direct product of C32 and D27

Aliases: C32×D27, C33.6D9, C273(C3×C6), (C3×C27)⋊14C6, (C32×C27)⋊3C2, C9.3(S3×C32), C3.2(C32×D9), (C32×C9).24S3, C32.15(C3×D9), (C3×C9).55(C3×S3), SmallGroup(486,111)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C27 — C32×D27
 Chief series C1 — C3 — C9 — C27 — C3×C27 — C32×C27 — C32×D27
 Lower central C27 — C32×D27
 Upper central C1 — C32

Generators and relations for C32×D27
G = < a,b,c,d | a3=b3=c27=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 312 in 72 conjugacy classes, 30 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C33, D27, C3×D9, S3×C32, C3×C27, C3×C27, C32×C9, C3×D27, C32×D9, C32×C27, C32×D27
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, D27, C3×D9, S3×C32, C3×D27, C32×D9, C32×D27

Smallest permutation representation of C32×D27
On 162 points
Generators in S162
(1 39 56)(2 40 57)(3 41 58)(4 42 59)(5 43 60)(6 44 61)(7 45 62)(8 46 63)(9 47 64)(10 48 65)(11 49 66)(12 50 67)(13 51 68)(14 52 69)(15 53 70)(16 54 71)(17 28 72)(18 29 73)(19 30 74)(20 31 75)(21 32 76)(22 33 77)(23 34 78)(24 35 79)(25 36 80)(26 37 81)(27 38 55)(82 117 159)(83 118 160)(84 119 161)(85 120 162)(86 121 136)(87 122 137)(88 123 138)(89 124 139)(90 125 140)(91 126 141)(92 127 142)(93 128 143)(94 129 144)(95 130 145)(96 131 146)(97 132 147)(98 133 148)(99 134 149)(100 135 150)(101 109 151)(102 110 152)(103 111 153)(104 112 154)(105 113 155)(106 114 156)(107 115 157)(108 116 158)
(1 74 48)(2 75 49)(3 76 50)(4 77 51)(5 78 52)(6 79 53)(7 80 54)(8 81 28)(9 55 29)(10 56 30)(11 57 31)(12 58 32)(13 59 33)(14 60 34)(15 61 35)(16 62 36)(17 63 37)(18 64 38)(19 65 39)(20 66 40)(21 67 41)(22 68 42)(23 69 43)(24 70 44)(25 71 45)(26 72 46)(27 73 47)(82 141 135)(83 142 109)(84 143 110)(85 144 111)(86 145 112)(87 146 113)(88 147 114)(89 148 115)(90 149 116)(91 150 117)(92 151 118)(93 152 119)(94 153 120)(95 154 121)(96 155 122)(97 156 123)(98 157 124)(99 158 125)(100 159 126)(101 160 127)(102 161 128)(103 162 129)(104 136 130)(105 137 131)(106 138 132)(107 139 133)(108 140 134)
(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)(136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)
(1 89)(2 88)(3 87)(4 86)(5 85)(6 84)(7 83)(8 82)(9 108)(10 107)(11 106)(12 105)(13 104)(14 103)(15 102)(16 101)(17 100)(18 99)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 90)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 140)(56 139)(57 138)(58 137)(59 136)(60 162)(61 161)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)(81 141)

G:=sub<Sym(162)| (1,39,56)(2,40,57)(3,41,58)(4,42,59)(5,43,60)(6,44,61)(7,45,62)(8,46,63)(9,47,64)(10,48,65)(11,49,66)(12,50,67)(13,51,68)(14,52,69)(15,53,70)(16,54,71)(17,28,72)(18,29,73)(19,30,74)(20,31,75)(21,32,76)(22,33,77)(23,34,78)(24,35,79)(25,36,80)(26,37,81)(27,38,55)(82,117,159)(83,118,160)(84,119,161)(85,120,162)(86,121,136)(87,122,137)(88,123,138)(89,124,139)(90,125,140)(91,126,141)(92,127,142)(93,128,143)(94,129,144)(95,130,145)(96,131,146)(97,132,147)(98,133,148)(99,134,149)(100,135,150)(101,109,151)(102,110,152)(103,111,153)(104,112,154)(105,113,155)(106,114,156)(107,115,157)(108,116,158), (1,74,48)(2,75,49)(3,76,50)(4,77,51)(5,78,52)(6,79,53)(7,80,54)(8,81,28)(9,55,29)(10,56,30)(11,57,31)(12,58,32)(13,59,33)(14,60,34)(15,61,35)(16,62,36)(17,63,37)(18,64,38)(19,65,39)(20,66,40)(21,67,41)(22,68,42)(23,69,43)(24,70,44)(25,71,45)(26,72,46)(27,73,47)(82,141,135)(83,142,109)(84,143,110)(85,144,111)(86,145,112)(87,146,113)(88,147,114)(89,148,115)(90,149,116)(91,150,117)(92,151,118)(93,152,119)(94,153,120)(95,154,121)(96,155,122)(97,156,123)(98,157,124)(99,158,125)(100,159,126)(101,160,127)(102,161,128)(103,162,129)(104,136,130)(105,137,131)(106,138,132)(107,139,133)(108,140,134), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,140)(56,139)(57,138)(58,137)(59,136)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)>;

G:=Group( (1,39,56)(2,40,57)(3,41,58)(4,42,59)(5,43,60)(6,44,61)(7,45,62)(8,46,63)(9,47,64)(10,48,65)(11,49,66)(12,50,67)(13,51,68)(14,52,69)(15,53,70)(16,54,71)(17,28,72)(18,29,73)(19,30,74)(20,31,75)(21,32,76)(22,33,77)(23,34,78)(24,35,79)(25,36,80)(26,37,81)(27,38,55)(82,117,159)(83,118,160)(84,119,161)(85,120,162)(86,121,136)(87,122,137)(88,123,138)(89,124,139)(90,125,140)(91,126,141)(92,127,142)(93,128,143)(94,129,144)(95,130,145)(96,131,146)(97,132,147)(98,133,148)(99,134,149)(100,135,150)(101,109,151)(102,110,152)(103,111,153)(104,112,154)(105,113,155)(106,114,156)(107,115,157)(108,116,158), (1,74,48)(2,75,49)(3,76,50)(4,77,51)(5,78,52)(6,79,53)(7,80,54)(8,81,28)(9,55,29)(10,56,30)(11,57,31)(12,58,32)(13,59,33)(14,60,34)(15,61,35)(16,62,36)(17,63,37)(18,64,38)(19,65,39)(20,66,40)(21,67,41)(22,68,42)(23,69,43)(24,70,44)(25,71,45)(26,72,46)(27,73,47)(82,141,135)(83,142,109)(84,143,110)(85,144,111)(86,145,112)(87,146,113)(88,147,114)(89,148,115)(90,149,116)(91,150,117)(92,151,118)(93,152,119)(94,153,120)(95,154,121)(96,155,122)(97,156,123)(98,157,124)(99,158,125)(100,159,126)(101,160,127)(102,161,128)(103,162,129)(104,136,130)(105,137,131)(106,138,132)(107,139,133)(108,140,134), (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)(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162), (1,89)(2,88)(3,87)(4,86)(5,85)(6,84)(7,83)(8,82)(9,108)(10,107)(11,106)(12,105)(13,104)(14,103)(15,102)(16,101)(17,100)(18,99)(19,98)(20,97)(21,96)(22,95)(23,94)(24,93)(25,92)(26,91)(27,90)(28,135)(29,134)(30,133)(31,132)(32,131)(33,130)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,140)(56,139)(57,138)(58,137)(59,136)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141) );

G=PermutationGroup([[(1,39,56),(2,40,57),(3,41,58),(4,42,59),(5,43,60),(6,44,61),(7,45,62),(8,46,63),(9,47,64),(10,48,65),(11,49,66),(12,50,67),(13,51,68),(14,52,69),(15,53,70),(16,54,71),(17,28,72),(18,29,73),(19,30,74),(20,31,75),(21,32,76),(22,33,77),(23,34,78),(24,35,79),(25,36,80),(26,37,81),(27,38,55),(82,117,159),(83,118,160),(84,119,161),(85,120,162),(86,121,136),(87,122,137),(88,123,138),(89,124,139),(90,125,140),(91,126,141),(92,127,142),(93,128,143),(94,129,144),(95,130,145),(96,131,146),(97,132,147),(98,133,148),(99,134,149),(100,135,150),(101,109,151),(102,110,152),(103,111,153),(104,112,154),(105,113,155),(106,114,156),(107,115,157),(108,116,158)], [(1,74,48),(2,75,49),(3,76,50),(4,77,51),(5,78,52),(6,79,53),(7,80,54),(8,81,28),(9,55,29),(10,56,30),(11,57,31),(12,58,32),(13,59,33),(14,60,34),(15,61,35),(16,62,36),(17,63,37),(18,64,38),(19,65,39),(20,66,40),(21,67,41),(22,68,42),(23,69,43),(24,70,44),(25,71,45),(26,72,46),(27,73,47),(82,141,135),(83,142,109),(84,143,110),(85,144,111),(86,145,112),(87,146,113),(88,147,114),(89,148,115),(90,149,116),(91,150,117),(92,151,118),(93,152,119),(94,153,120),(95,154,121),(96,155,122),(97,156,123),(98,157,124),(99,158,125),(100,159,126),(101,160,127),(102,161,128),(103,162,129),(104,136,130),(105,137,131),(106,138,132),(107,139,133),(108,140,134)], [(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),(136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)], [(1,89),(2,88),(3,87),(4,86),(5,85),(6,84),(7,83),(8,82),(9,108),(10,107),(11,106),(12,105),(13,104),(14,103),(15,102),(16,101),(17,100),(18,99),(19,98),(20,97),(21,96),(22,95),(23,94),(24,93),(25,92),(26,91),(27,90),(28,135),(29,134),(30,133),(31,132),(32,131),(33,130),(34,129),(35,128),(36,127),(37,126),(38,125),(39,124),(40,123),(41,122),(42,121),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,140),(56,139),(57,138),(58,137),(59,136),(60,162),(61,161),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142),(81,141)]])

135 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9AA 27A ··· 27CC order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 27 ··· 27 size 1 27 1 ··· 1 2 ··· 2 27 ··· 27 2 ··· 2 2 ··· 2

135 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 C3×S3 D9 D27 C3×D9 C3×D27 kernel C32×D27 C32×C27 C3×D27 C3×C27 C32×C9 C3×C9 C33 C32 C32 C3 # reps 1 1 8 8 1 8 3 9 24 72

Matrix representation of C32×D27 in GL3(𝔽109) generated by

 63 0 0 0 1 0 0 0 1
,
 63 0 0 0 45 0 0 0 45
,
 1 0 0 0 81 0 0 106 35
,
 1 0 0 0 23 62 0 104 86
G:=sub<GL(3,GF(109))| [63,0,0,0,1,0,0,0,1],[63,0,0,0,45,0,0,0,45],[1,0,0,0,81,106,0,0,35],[1,0,0,0,23,104,0,62,86] >;

C32×D27 in GAP, Magma, Sage, TeX

C_3^2\times D_{27}
% in TeX

G:=Group("C3^2xD27");
// GroupNames label

G:=SmallGroup(486,111);
// by ID

G=gap.SmallGroup(486,111);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,2163,381,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^27=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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