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## G = C2×C32⋊4D9order 324 = 22·34

### Direct product of C2 and C32⋊4D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C9 — C2×C32⋊4D9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32⋊4D9 — C2×C32⋊4D9
 Lower central C32×C9 — C2×C32⋊4D9
 Upper central C1 — C2

Generators and relations for C2×C324D9
G = < a,b,c,d,e | a2=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: 2110 in 250 conjugacy classes, 103 normal (9 characteristic)
C1, C2, C2 [×2], C3, C3 [×12], C22, S3 [×26], C6, C6 [×12], C9 [×9], C32 [×13], D6 [×13], D9 [×18], C18 [×9], C3⋊S3 [×26], C3×C6 [×13], C3×C9 [×12], C33, D18 [×9], C2×C3⋊S3 [×13], C9⋊S3 [×24], C3×C18 [×12], C33⋊C2 [×2], C32×C6, C32×C9, C2×C9⋊S3 [×12], C2×C33⋊C2, C324D9 [×2], C32×C18, C2×C324D9
Quotients: C1, C2 [×3], C22, S3 [×13], D6 [×13], D9 [×9], C3⋊S3 [×13], D18 [×9], C2×C3⋊S3 [×13], C9⋊S3 [×12], C33⋊C2, C2×C9⋊S3 [×12], C2×C33⋊C2, C324D9, C2×C324D9

Smallest permutation representation of C2×C324D9
On 162 points
Generators in S162
(1 134)(2 135)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 136)(11 137)(12 138)(13 139)(14 140)(15 141)(16 142)(17 143)(18 144)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 89)(29 90)(30 82)(31 83)(32 84)(33 85)(34 86)(35 87)(36 88)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 91)(44 92)(45 93)(46 126)(47 118)(48 119)(49 120)(50 121)(51 122)(52 123)(53 124)(54 125)(55 116)(56 117)(57 109)(58 110)(59 111)(60 112)(61 113)(62 114)(63 115)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)
(1 22 18)(2 23 10)(3 24 11)(4 25 12)(5 26 13)(6 27 14)(7 19 15)(8 20 16)(9 21 17)(28 158 58)(29 159 59)(30 160 60)(31 161 61)(32 162 62)(33 154 63)(34 155 55)(35 156 56)(36 157 57)(37 52 67)(38 53 68)(39 54 69)(40 46 70)(41 47 71)(42 48 72)(43 49 64)(44 50 65)(45 51 66)(73 115 85)(74 116 86)(75 117 87)(76 109 88)(77 110 89)(78 111 90)(79 112 82)(80 113 83)(81 114 84)(91 120 145)(92 121 146)(93 122 147)(94 123 148)(95 124 149)(96 125 150)(97 126 151)(98 118 152)(99 119 153)(100 141 131)(101 142 132)(102 143 133)(103 144 134)(104 136 135)(105 137 127)(106 138 128)(107 139 129)(108 140 130)
(1 77 41)(2 78 42)(3 79 43)(4 80 44)(5 81 45)(6 73 37)(7 74 38)(8 75 39)(9 76 40)(10 90 72)(11 82 64)(12 83 65)(13 84 66)(14 85 67)(15 86 68)(16 87 69)(17 88 70)(18 89 71)(19 116 53)(20 117 54)(21 109 46)(22 110 47)(23 111 48)(24 112 49)(25 113 50)(26 114 51)(27 115 52)(28 152 144)(29 153 136)(30 145 137)(31 146 138)(32 147 139)(33 148 140)(34 149 141)(35 150 142)(36 151 143)(55 124 100)(56 125 101)(57 126 102)(58 118 103)(59 119 104)(60 120 105)(61 121 106)(62 122 107)(63 123 108)(91 127 160)(92 128 161)(93 129 162)(94 130 154)(95 131 155)(96 132 156)(97 133 157)(98 134 158)(99 135 159)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 135)(9 134)(10 101)(11 100)(12 108)(13 107)(14 106)(15 105)(16 104)(17 103)(18 102)(19 137)(20 136)(21 144)(22 143)(23 142)(24 141)(25 140)(26 139)(27 138)(28 46)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 161)(38 160)(39 159)(40 158)(41 157)(42 156)(43 155)(44 154)(45 162)(55 64)(56 72)(57 71)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(73 92)(74 91)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 124)(83 123)(84 122)(85 121)(86 120)(87 119)(88 118)(89 126)(90 125)(109 152)(110 151)(111 150)(112 149)(113 148)(114 147)(115 146)(116 145)(117 153)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3M 6A ··· 6M 9A ··· 9AA 18A ··· 18AA order 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 81 81 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D18 kernel C2×C32⋊4D9 C32⋊4D9 C32×C18 C3×C18 C32×C6 C3×C9 C33 C3×C6 C32 # reps 1 2 1 12 1 12 1 27 27

Matrix representation of C2×C324D9 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 0 1 0 0 0 0 18 18 0 0 0 0 0 0 0 18 0 0 0 0 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 1 0 0 0 0 18 0 0 0 0 0 0 0 18 1 0 0 0 0 18 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 14 0 0 0 0 5 2 0 0 0 0 0 0 0 18 0 0 0 0 1 18
,
 1 0 0 0 0 0 18 18 0 0 0 0 0 0 14 17 0 0 0 0 12 5 0 0 0 0 0 0 18 1 0 0 0 0 0 1

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,1,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,0,0,0,0,1,0,0,0,0,0,0,0,18,18,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,5,0,0,0,0,14,2,0,0,0,0,0,0,0,1,0,0,0,0,18,18],[1,18,0,0,0,0,0,18,0,0,0,0,0,0,14,12,0,0,0,0,17,5,0,0,0,0,0,0,18,0,0,0,0,0,1,1] >;

C2×C324D9 in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes_4D_9
% in TeX

G:=Group("C2xC3^2:4D9");
// GroupNames label

G:=SmallGroup(324,149);
// by ID

G=gap.SmallGroup(324,149);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2090,986,579,2164,7781]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=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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