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## G = D9×C33order 486 = 2·35

### Direct product of C33 and D9

Aliases: D9×C33, C34.12S3, (C33×C9)⋊4C2, C93(C32×C6), (C32×C9)⋊39C6, C3.1(S3×C33), C33.81(C3×S3), C32.44(S3×C32), (C3×C9)⋊22(C3×C6), SmallGroup(486,220)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D9×C33
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 760 in 324 conjugacy classes, 112 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3×C6, C3×C9, C3×C9, C33, C33, C33, C3×D9, S3×C32, C32×C6, C32×C9, C32×C9, C34, C32×D9, S3×C33, C33×C9, D9×C33
Quotients: C1, C2, C3, S3, C6, C32, D9, C3×S3, C3×C6, C33, C3×D9, S3×C32, C32×C6, C32×D9, S3×C33, D9×C33

Smallest permutation representation of D9×C33
On 162 points
Generators in S162
(1 62 29)(2 63 30)(3 55 31)(4 56 32)(5 57 33)(6 58 34)(7 59 35)(8 60 36)(9 61 28)(10 67 43)(11 68 44)(12 69 45)(13 70 37)(14 71 38)(15 72 39)(16 64 40)(17 65 41)(18 66 42)(19 76 52)(20 77 53)(21 78 54)(22 79 46)(23 80 47)(24 81 48)(25 73 49)(26 74 50)(27 75 51)(82 142 112)(83 143 113)(84 144 114)(85 136 115)(86 137 116)(87 138 117)(88 139 109)(89 140 110)(90 141 111)(91 151 121)(92 152 122)(93 153 123)(94 145 124)(95 146 125)(96 147 126)(97 148 118)(98 149 119)(99 150 120)(100 160 130)(101 161 131)(102 162 132)(103 154 133)(104 155 134)(105 156 135)(106 157 127)(107 158 128)(108 159 129)
(1 26 11)(2 27 12)(3 19 13)(4 20 14)(5 21 15)(6 22 16)(7 23 17)(8 24 18)(9 25 10)(28 49 43)(29 50 44)(30 51 45)(31 52 37)(32 53 38)(33 54 39)(34 46 40)(35 47 41)(36 48 42)(55 76 70)(56 77 71)(57 78 72)(58 79 64)(59 80 65)(60 81 66)(61 73 67)(62 74 68)(63 75 69)(82 106 94)(83 107 95)(84 108 96)(85 100 97)(86 101 98)(87 102 99)(88 103 91)(89 104 92)(90 105 93)(109 133 121)(110 134 122)(111 135 123)(112 127 124)(113 128 125)(114 129 126)(115 130 118)(116 131 119)(117 132 120)(136 160 148)(137 161 149)(138 162 150)(139 154 151)(140 155 152)(141 156 153)(142 157 145)(143 158 146)(144 159 147)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)(145 151 148)(146 152 149)(147 153 150)(154 160 157)(155 161 158)(156 162 159)
(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 84)(2 83)(3 82)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 99)(18 98)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 108)(27 107)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 117)(36 116)(37 124)(38 123)(39 122)(40 121)(41 120)(42 119)(43 118)(44 126)(45 125)(46 133)(47 132)(48 131)(49 130)(50 129)(51 128)(52 127)(53 135)(54 134)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 136)(62 144)(63 143)(64 151)(65 150)(66 149)(67 148)(68 147)(69 146)(70 145)(71 153)(72 152)(73 160)(74 159)(75 158)(76 157)(77 156)(78 155)(79 154)(80 162)(81 161)

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

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

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

162 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3BA 6A ··· 6Z 9A ··· 9CC order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 size 1 9 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2

162 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C3 C6 S3 D9 C3×S3 C3×D9 kernel D9×C33 C33×C9 C32×D9 C32×C9 C34 C33 C33 C32 # reps 1 1 26 26 1 3 26 78

Matrix representation of D9×C33 in GL4(𝔽19) generated by

 7 0 0 0 0 7 0 0 0 0 11 0 0 0 0 11
,
 11 0 0 0 0 1 0 0 0 0 11 0 0 0 0 11
,
 7 0 0 0 0 7 0 0 0 0 7 0 0 0 0 7
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 17
,
 18 0 0 0 0 18 0 0 0 0 0 17 0 0 9 0
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,11,0,0,0,0,11],[11,0,0,0,0,1,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,17],[18,0,0,0,0,18,0,0,0,0,0,9,0,0,17,0] >;

D9×C33 in GAP, Magma, Sage, TeX

D_9\times C_3^3
% in TeX

G:=Group("D9xC3^3");
// GroupNames label

G:=SmallGroup(486,220);
// by ID

G=gap.SmallGroup(486,220);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,8104,208,11669]);
// 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,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽