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## G = C9×C33⋊C2order 486 = 2·35

### Direct product of C9 and C33⋊C2

Aliases: C9×C33⋊C2, C339C18, C34.14C6, (C33×C9)⋊3C2, C326(S3×C9), (C32×C9)⋊28S3, C33.86(C3×S3), C3⋊(C9×C3⋊S3), (C3×C9)⋊9(C3⋊S3), C32.53(C3×C3⋊S3), C3.5(C3×C33⋊C2), (C3×C33⋊C2).3C3, SmallGroup(486,241)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C9×C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C34 — C33×C9 — C9×C33⋊C2
 Lower central C33 — C9×C33⋊C2
 Upper central C1 — C9

Generators and relations for C9×C33⋊C2
G = < a,b,c,d,e | a9=b3=c3=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b-1, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 948 in 324 conjugacy classes, 87 normal (9 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C33, C33, S3×C9, C3×C3⋊S3, C33⋊C2, C32×C9, C32×C9, C34, C9×C3⋊S3, C3×C33⋊C2, C33×C9, C9×C33⋊C2
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C3×C3⋊S3, C33⋊C2, C9×C3⋊S3, C3×C33⋊C2, C9×C33⋊C2

Smallest permutation representation of C9×C33⋊C2
On 162 points
Generators in S162
(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)
(1 107 134)(2 108 135)(3 100 127)(4 101 128)(5 102 129)(6 103 130)(7 104 131)(8 105 132)(9 106 133)(10 51 28)(11 52 29)(12 53 30)(13 54 31)(14 46 32)(15 47 33)(16 48 34)(17 49 35)(18 50 36)(19 157 60)(20 158 61)(21 159 62)(22 160 63)(23 161 55)(24 162 56)(25 154 57)(26 155 58)(27 156 59)(37 78 64)(38 79 65)(39 80 66)(40 81 67)(41 73 68)(42 74 69)(43 75 70)(44 76 71)(45 77 72)(82 139 95)(83 140 96)(84 141 97)(85 142 98)(86 143 99)(87 144 91)(88 136 92)(89 137 93)(90 138 94)(109 121 150)(110 122 151)(111 123 152)(112 124 153)(113 125 145)(114 126 146)(115 118 147)(116 119 148)(117 120 149)
(1 109 95)(2 110 96)(3 111 97)(4 112 98)(5 113 99)(6 114 91)(7 115 92)(8 116 93)(9 117 94)(10 156 67)(11 157 68)(12 158 69)(13 159 70)(14 160 71)(15 161 72)(16 162 64)(17 154 65)(18 155 66)(19 73 29)(20 74 30)(21 75 31)(22 76 32)(23 77 33)(24 78 34)(25 79 35)(26 80 36)(27 81 28)(37 48 56)(38 49 57)(39 50 58)(40 51 59)(41 52 60)(42 53 61)(43 54 62)(44 46 63)(45 47 55)(82 107 121)(83 108 122)(84 100 123)(85 101 124)(86 102 125)(87 103 126)(88 104 118)(89 105 119)(90 106 120)(127 152 141)(128 153 142)(129 145 143)(130 146 144)(131 147 136)(132 148 137)(133 149 138)(134 150 139)(135 151 140)
(1 85 147)(2 86 148)(3 87 149)(4 88 150)(5 89 151)(6 90 152)(7 82 153)(8 83 145)(9 84 146)(10 37 21)(11 38 22)(12 39 23)(13 40 24)(14 41 25)(15 42 26)(16 43 27)(17 44 19)(18 45 20)(28 64 62)(29 65 63)(30 66 55)(31 67 56)(32 68 57)(33 69 58)(34 70 59)(35 71 60)(36 72 61)(46 73 154)(47 74 155)(48 75 156)(49 76 157)(50 77 158)(51 78 159)(52 79 160)(53 80 161)(54 81 162)(91 120 127)(92 121 128)(93 122 129)(94 123 130)(95 124 131)(96 125 132)(97 126 133)(98 118 134)(99 119 135)(100 144 117)(101 136 109)(102 137 110)(103 138 111)(104 139 112)(105 140 113)(106 141 114)(107 142 115)(108 143 116)
(1 76)(2 77)(3 78)(4 79)(5 80)(6 81)(7 73)(8 74)(9 75)(10 144)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 115)(20 116)(21 117)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 122)(56 123)(57 124)(58 125)(59 126)(60 118)(61 119)(62 120)(63 121)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)(151 161)(152 162)(153 154)

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

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

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

135 conjugacy classes

 class 1 2 3A 3B 3C ··· 3AO 6A 6B 9A ··· 9F 9G ··· 9CF 18A ··· 18F order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 27 1 1 2 ··· 2 27 27 1 ··· 1 2 ··· 2 27 ··· 27

135 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 kernel C9×C33⋊C2 C33×C9 C3×C33⋊C2 C34 C33⋊C2 C33 C32×C9 C33 C32 # reps 1 1 2 2 6 6 13 26 78

Matrix representation of C9×C33⋊C2 in GL6(𝔽19)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 12 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 11 0 0 0 0 0 0 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 18 6 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(19))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,12,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,6,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C9×C33⋊C2 in GAP, Magma, Sage, TeX

C_9\times C_3^3\rtimes C_2
% in TeX

G:=Group("C9xC3^3:C2");
// GroupNames label

G:=SmallGroup(486,241);
// by ID

G=gap.SmallGroup(486,241);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,867,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^3=c^3=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e=b^-1,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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