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## G = S3×C9⋊C9order 486 = 2·35

### Direct product of S3 and C9⋊C9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — S3×C9⋊C9
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C3×C9⋊C9 — S3×C9⋊C9
 Lower central C3 — C32 — S3×C9⋊C9
 Upper central C1 — C32 — C9⋊C9

Generators and relations for S3×C9⋊C9
G = < a,b,c,d | a3=b2=c9=d9=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c7 >

Subgroups: 218 in 96 conjugacy classes, 42 normal (15 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, C3×C9, C3×C9, C33, S3×C9, S3×C9, C3×C18, S3×C32, C9⋊C9, C9⋊C9, C32×C9, C32×C9, C2×C9⋊C9, S3×C3×C9, S3×C3×C9, C3×C9⋊C9, S3×C9⋊C9
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, 3- 1+2, S3×C9, C3×C18, C2×3- 1+2, S3×C32, C9⋊C9, C2×C9⋊C9, S3×C3×C9, S3×3- 1+2, S3×C9⋊C9

Smallest permutation representation of S3×C9⋊C9
On 162 points
Generators in S162
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 43 40)(38 44 41)(39 45 42)(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 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(91 94 97)(92 95 98)(93 96 99)(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)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 157 160)(155 158 161)(156 159 162)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 91)(10 119)(11 120)(12 121)(13 122)(14 123)(15 124)(16 125)(17 126)(18 118)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 86)(29 87)(30 88)(31 89)(32 90)(33 82)(34 83)(35 84)(36 85)(37 143)(38 144)(39 136)(40 137)(41 138)(42 139)(43 140)(44 141)(45 142)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(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 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 11 49 22 66 76 114 84)(2 44 18 50 26 64 77 109 82)(3 39 16 51 21 71 78 113 89)(4 43 14 52 25 69 79 117 87)(5 38 12 53 20 67 80 112 85)(6 42 10 54 24 65 81 116 83)(7 37 17 46 19 72 73 111 90)(8 41 15 47 23 70 74 115 88)(9 45 13 48 27 68 75 110 86)(28 91 142 122 129 108 149 156 58)(29 95 140 123 133 106 150 160 56)(30 99 138 124 128 104 151 155 63)(31 94 136 125 132 102 152 159 61)(32 98 143 126 127 100 153 154 59)(33 93 141 118 131 107 145 158 57)(34 97 139 119 135 105 146 162 55)(35 92 137 120 130 103 147 157 62)(36 96 144 121 134 101 148 161 60)

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

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

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

99 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3Q 6A ··· 6H 9A ··· 9X 9Y ··· 9AV 18A ··· 18X order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 size 1 3 1 ··· 1 2 ··· 2 3 ··· 3 3 ··· 3 6 ··· 6 9 ··· 9

99 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 3 3 6 type + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 S3×C9 3- 1+2 C2×3- 1+2 S3×3- 1+2 kernel S3×C9⋊C9 C3×C9⋊C9 S3×C3×C9 C32×C9 S3×C9 C3×C9 C9⋊C9 C3×C9 C9 C3×S3 C32 C3 # reps 1 1 8 8 18 18 1 8 18 6 6 6

Matrix representation of S3×C9⋊C9 in GL5(𝔽19)

 0 1 0 0 0 18 18 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 18 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 0 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 11 0 0 0 0 0 7 0 0 11 0 0
,
 4 0 0 0 0 0 4 0 0 0 0 0 0 7 0 0 0 0 0 7 0 0 1 0 0

G:=sub<GL(5,GF(19))| [0,18,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,11,0,0,0,0,0,7,0],[4,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,7,0,0,0,0,0,7,0] >;

S3×C9⋊C9 in GAP, Magma, Sage, TeX

S_3\times C_9\rtimes C_9
% in TeX

G:=Group("S3xC9:C9");
// GroupNames label

G:=SmallGroup(486,97);
// by ID

G=gap.SmallGroup(486,97);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,548,68,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^9=d^9=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^7>;
// generators/relations

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