direct product, metabelian, supersoluble, monomial
Aliases: S3×C9⋊C9, (S3×C9)⋊C9, C9⋊2(S3×C9), (C3×C9)⋊7C18, (C32×C9).11C6, C33.51(C3×C6), C32.15(C3×C18), (S3×C32).9C32, C32.52(S3×C32), C3.8(S3×3- 1+2), (C3×S3).23- 1+2, C32.10(C2×3- 1+2), C3⋊(C2×C9⋊C9), (C3×C9⋊C9)⋊3C2, C3.9(S3×C3×C9), (S3×C3×C9).1C3, (C3×S3).3(C3×C9), (C3×C9).44(C3×S3), SmallGroup(486,97)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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