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## G = C3×C32⋊4D9order 486 = 2·35

### Direct product of C3 and C32⋊4D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C3×C32⋊4D9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C33×C9 — C3×C32⋊4D9
 Lower central C32×C9 — C3×C32⋊4D9
 Upper central C1 — C3

Generators and relations for C3×C324D9
G = < a,b,c,d,e | a3=b3=c3=d9=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: 1736 in 348 conjugacy classes, 102 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C33, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C33⋊C2, C32×C9, C32×C9, C32×C9, C34, C3×C9⋊S3, C324D9, C3×C33⋊C2, C33×C9, C3×C324D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C33⋊C2, C3×C9⋊S3, C324D9, C3×C33⋊C2, C3×C324D9

Smallest permutation representation of C3×C324D9
On 162 points
Generators in S162
(1 109 25)(2 110 26)(3 111 27)(4 112 19)(5 113 20)(6 114 21)(7 115 22)(8 116 23)(9 117 24)(10 78 42)(11 79 43)(12 80 44)(13 81 45)(14 73 37)(15 74 38)(16 75 39)(17 76 40)(18 77 41)(28 120 133)(29 121 134)(30 122 135)(31 123 127)(32 124 128)(33 125 129)(34 126 130)(35 118 131)(36 119 132)(46 70 84)(47 71 85)(48 72 86)(49 64 87)(50 65 88)(51 66 89)(52 67 90)(53 68 82)(54 69 83)(55 92 106)(56 93 107)(57 94 108)(58 95 100)(59 96 101)(60 97 102)(61 98 103)(62 99 104)(63 91 105)(136 149 157)(137 150 158)(138 151 159)(139 152 160)(140 153 161)(141 145 162)(142 146 154)(143 147 155)(144 148 156)
(1 12 71)(2 13 72)(3 14 64)(4 15 65)(5 16 66)(6 17 67)(7 18 68)(8 10 69)(9 11 70)(19 38 50)(20 39 51)(21 40 52)(22 41 53)(23 42 54)(24 43 46)(25 44 47)(26 45 48)(27 37 49)(28 61 155)(29 62 156)(30 63 157)(31 55 158)(32 56 159)(33 57 160)(34 58 161)(35 59 162)(36 60 154)(73 87 111)(74 88 112)(75 89 113)(76 90 114)(77 82 115)(78 83 116)(79 84 117)(80 85 109)(81 86 110)(91 136 122)(92 137 123)(93 138 124)(94 139 125)(95 140 126)(96 141 118)(97 142 119)(98 143 120)(99 144 121)(100 153 130)(101 145 131)(102 146 132)(103 147 133)(104 148 134)(105 149 135)(106 150 127)(107 151 128)(108 152 129)
(1 77 50)(2 78 51)(3 79 52)(4 80 53)(5 81 54)(6 73 46)(7 74 47)(8 75 48)(9 76 49)(10 89 26)(11 90 27)(12 82 19)(13 83 20)(14 84 21)(15 85 22)(16 86 23)(17 87 24)(18 88 25)(28 100 137)(29 101 138)(30 102 139)(31 103 140)(32 104 141)(33 105 142)(34 106 143)(35 107 144)(36 108 136)(37 70 114)(38 71 115)(39 72 116)(40 64 117)(41 65 109)(42 66 110)(43 67 111)(44 68 112)(45 69 113)(55 147 126)(56 148 118)(57 149 119)(58 150 120)(59 151 121)(60 152 122)(61 153 123)(62 145 124)(63 146 125)(91 154 129)(92 155 130)(93 156 131)(94 157 132)(95 158 133)(96 159 134)(97 160 135)(98 161 127)(99 162 128)
(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)
(1 131)(2 130)(3 129)(4 128)(5 127)(6 135)(7 134)(8 133)(9 132)(10 147)(11 146)(12 145)(13 153)(14 152)(15 151)(16 150)(17 149)(18 148)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 126)(27 125)(28 116)(29 115)(30 114)(31 113)(32 112)(33 111)(34 110)(35 109)(36 117)(37 139)(38 138)(39 137)(40 136)(41 144)(42 143)(43 142)(44 141)(45 140)(46 97)(47 96)(48 95)(49 94)(50 93)(51 92)(52 91)(53 99)(54 98)(55 89)(56 88)(57 87)(58 86)(59 85)(60 84)(61 83)(62 82)(63 90)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 160)(74 159)(75 158)(76 157)(77 156)(78 155)(79 154)(80 162)(81 161)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C ··· 3AO 6A 6B 9A ··· 9CC order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 size 1 81 1 1 2 ··· 2 81 81 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 C3×D9 kernel C3×C32⋊4D9 C33×C9 C32⋊4D9 C32×C9 C32×C9 C34 C3×C9 C33 C33 C32 # reps 1 1 2 2 12 1 24 27 2 54

Matrix representation of C3×C324D9 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 0 7 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 11 0 0 0 0 0 0 7 0 0 0 0 0 0 7 0 0 0 0 0 0 11
,
 17 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 17 0 0 0 0 0 0 0 0 17 0 0 0 0 9 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[17,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,17,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C324D9 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_4D_9
% in TeX

G:=Group("C3xC3^2:4D9");
// GroupNames label

G:=SmallGroup(486,240);
// by ID

G=gap.SmallGroup(486,240);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,986,867,3244,11669]);
// Polycyclic

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