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## G = C3×C9⋊D9order 486 = 2·35

### Direct product of C3 and C9⋊D9

Aliases: C3×C9⋊D9, C9224C6, C93(C3×D9), (C3×C9)⋊10D9, (C3×C92)⋊4C2, (C32×C9).25S3, C32.16(C9⋊S3), C33.47(C3⋊S3), C3.1(C3×C9⋊S3), (C3×C9).57(C3×S3), C32.25(C3×C3⋊S3), SmallGroup(486,134)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — C3×C9⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C3×C92 — C3×C9⋊D9
 Lower central C92 — C3×C9⋊D9
 Upper central C1 — C3

Generators and relations for C3×C9⋊D9
G = < a,b,c,d | a3=b9=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 764 in 132 conjugacy classes, 48 normal (8 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C92, C92, C32×C9, C9⋊D9, C3×C9⋊S3, C3×C92, C3×C9⋊D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C9⋊D9, C3×C9⋊S3, C3×C9⋊D9

Smallest permutation representation of C3×C9⋊D9
On 162 points
Generators in S162
(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 52 49)(47 53 50)(48 54 51)(55 58 61)(56 59 62)(57 60 63)(64 70 67)(65 71 68)(66 72 69)(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)(100 103 106)(101 104 107)(102 105 108)(109 115 112)(110 116 113)(111 117 114)(118 121 124)(119 122 125)(120 123 126)(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 40 101 126 80 22 31 12 55)(2 41 102 118 81 23 32 13 56)(3 42 103 119 73 24 33 14 57)(4 43 104 120 74 25 34 15 58)(5 44 105 121 75 26 35 16 59)(6 45 106 122 76 27 36 17 60)(7 37 107 123 77 19 28 18 61)(8 38 108 124 78 20 29 10 62)(9 39 100 125 79 21 30 11 63)(46 65 82 140 145 112 93 135 156)(47 66 83 141 146 113 94 127 157)(48 67 84 142 147 114 95 128 158)(49 68 85 143 148 115 96 129 159)(50 69 86 144 149 116 97 130 160)(51 70 87 136 150 117 98 131 161)(52 71 88 137 151 109 99 132 162)(53 72 89 138 152 110 91 133 154)(54 64 90 139 153 111 92 134 155)
(1 89)(2 88)(3 87)(4 86)(5 85)(6 84)(7 83)(8 82)(9 90)(10 145)(11 153)(12 152)(13 151)(14 150)(15 149)(16 148)(17 147)(18 146)(19 94)(20 93)(21 92)(22 91)(23 99)(24 98)(25 97)(26 96)(27 95)(28 113)(29 112)(30 111)(31 110)(32 109)(33 117)(34 116)(35 115)(36 114)(37 66)(38 65)(39 64)(40 72)(41 71)(42 70)(43 69)(44 68)(45 67)(46 108)(47 107)(48 106)(49 105)(50 104)(51 103)(52 102)(53 101)(54 100)(55 138)(56 137)(57 136)(58 144)(59 143)(60 142)(61 141)(62 140)(63 139)(73 131)(74 130)(75 129)(76 128)(77 127)(78 135)(79 134)(80 133)(81 132)(118 162)(119 161)(120 160)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9DD 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 type + + + + image C1 C2 C3 C6 S3 D9 C3×S3 C3×D9 kernel C3×C9⋊D9 C3×C92 C9⋊D9 C92 C32×C9 C3×C9 C3×C9 C9 # reps 1 1 2 2 4 36 8 72

Matrix representation of C3×C9⋊D9 in GL4(𝔽19) generated by

 7 0 0 0 0 7 0 0 0 0 11 0 0 0 0 11
,
 4 0 0 0 0 5 0 0 0 0 17 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 6 0 0 0 0 16
,
 0 1 0 0 1 0 0 0 0 0 0 16 0 0 6 0
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,11,0,0,0,0,11],[4,0,0,0,0,5,0,0,0,0,17,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,6,0,0,0,0,16],[0,1,0,0,1,0,0,0,0,0,0,6,0,0,16,0] >;

C3×C9⋊D9 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes D_9
% in TeX

G:=Group("C3xC9:D9");
// GroupNames label

G:=SmallGroup(486,134);
// by ID

G=gap.SmallGroup(486,134);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,338,4755,453,3244,11669]);
// Polycyclic

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

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