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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, C3, C3, C22, S3, C6, C6, C9, C32, D6, D9, C18, C3⋊S3, C3×C6, C3×C9, C33, D18, C2×C3⋊S3, C9⋊S3, C3×C18, C33⋊C2, C32×C6, C32×C9, C2×C9⋊S3, C2×C33⋊C2, C324D9, C32×C18, C2×C324D9
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, C2×C3⋊S3, C9⋊S3, C33⋊C2, C2×C9⋊S3, C2×C33⋊C2, C324D9, C2×C324D9

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

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

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

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

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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