direct product, metabelian, soluble, monomial, A-group
Aliases: C9×C3.A4, C22⋊C92, C62.10C32, (C2×C18)⋊1C9, C3.2(C9×A4), (C3×C9).6A4, (C6×C18).3C3, C32.15(C3×A4), (C2×C6).7(C3×C9), C3.2(C3×C3.A4), (C3×C3.A4).4C3, SmallGroup(324,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C9×C3.A4 |
Generators and relations for C9×C3.A4
G = < a,b,c,d,e | a9=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Smallest permutation representation of C9×C3.A4
►On 162 points
Generators in S162
(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 31 62)(2 32 63)(3 33 55)(4 34 56)(5 35 57)(6 36 58)(7 28 59)(8 29 60)(9 30 61)(10 79 52)(11 80 53)(12 81 54)(13 73 46)(14 74 47)(15 75 48)(16 76 49)(17 77 50)(18 78 51)(19 150 136)(20 151 137)(21 152 138)(22 153 139)(23 145 140)(24 146 141)(25 147 142)(26 148 143)(27 149 144)(37 158 125)(38 159 126)(39 160 118)(40 161 119)(41 162 120)(42 154 121)(43 155 122)(44 156 123)(45 157 124)(64 133 106)(65 134 107)(66 135 108)(67 127 100)(68 128 101)(69 129 102)(70 130 103)(71 131 104)(72 132 105)(82 113 96)(83 114 97)(84 115 98)(85 116 99)(86 117 91)(87 109 92)(88 110 93)(89 111 94)(90 112 95)
(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 161)(26 162)(27 154)(37 139)(38 140)(39 141)(40 142)(41 143)(42 144)(43 136)(44 137)(45 138)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 91)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 127)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 109)(107 110)(108 111)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 145)
(1 17)(2 18)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 73)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 55)(53 56)(54 57)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 91)(82 128)(83 129)(84 130)(85 131)(86 132)(87 133)(88 134)(89 135)(90 127)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 109)(107 110)(108 111)
(1 144 83 31 27 114 62 149 97)(2 136 84 32 19 115 63 150 98)(3 137 85 33 20 116 55 151 99)(4 138 86 34 21 117 56 152 91)(5 139 87 35 22 109 57 153 92)(6 140 88 36 23 110 58 145 93)(7 141 89 28 24 111 59 146 94)(8 142 90 29 25 112 60 147 95)(9 143 82 30 26 113 61 148 96)(10 44 131 79 156 104 52 123 71)(11 45 132 80 157 105 53 124 72)(12 37 133 81 158 106 54 125 64)(13 38 134 73 159 107 46 126 65)(14 39 135 74 160 108 47 118 66)(15 40 127 75 161 100 48 119 67)(16 41 128 76 162 101 49 120 68)(17 42 129 77 154 102 50 121 69)(18 43 130 78 155 103 51 122 70)
G:=sub<Sym(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), (1,31,62)(2,32,63)(3,33,55)(4,34,56)(5,35,57)(6,36,58)(7,28,59)(8,29,60)(9,30,61)(10,79,52)(11,80,53)(12,81,54)(13,73,46)(14,74,47)(15,75,48)(16,76,49)(17,77,50)(18,78,51)(19,150,136)(20,151,137)(21,152,138)(22,153,139)(23,145,140)(24,146,141)(25,147,142)(26,148,143)(27,149,144)(37,158,125)(38,159,126)(39,160,118)(40,161,119)(41,162,120)(42,154,121)(43,155,122)(44,156,123)(45,157,124)(64,133,106)(65,134,107)(66,135,108)(67,127,100)(68,128,101)(69,129,102)(70,130,103)(71,131,104)(72,132,105)(82,113,96)(83,114,97)(84,115,98)(85,116,99)(86,117,91)(87,109,92)(88,110,93)(89,111,94)(90,112,95), (19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,161)(26,162)(27,154)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,136)(44,137)(45,138)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,91)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,127)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,145), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,55)(53,56)(54,57)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,91)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,127)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111), (1,144,83,31,27,114,62,149,97)(2,136,84,32,19,115,63,150,98)(3,137,85,33,20,116,55,151,99)(4,138,86,34,21,117,56,152,91)(5,139,87,35,22,109,57,153,92)(6,140,88,36,23,110,58,145,93)(7,141,89,28,24,111,59,146,94)(8,142,90,29,25,112,60,147,95)(9,143,82,30,26,113,61,148,96)(10,44,131,79,156,104,52,123,71)(11,45,132,80,157,105,53,124,72)(12,37,133,81,158,106,54,125,64)(13,38,134,73,159,107,46,126,65)(14,39,135,74,160,108,47,118,66)(15,40,127,75,161,100,48,119,67)(16,41,128,76,162,101,49,120,68)(17,42,129,77,154,102,50,121,69)(18,43,130,78,155,103,51,122,70)>;
G:=Group( (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), (1,31,62)(2,32,63)(3,33,55)(4,34,56)(5,35,57)(6,36,58)(7,28,59)(8,29,60)(9,30,61)(10,79,52)(11,80,53)(12,81,54)(13,73,46)(14,74,47)(15,75,48)(16,76,49)(17,77,50)(18,78,51)(19,150,136)(20,151,137)(21,152,138)(22,153,139)(23,145,140)(24,146,141)(25,147,142)(26,148,143)(27,149,144)(37,158,125)(38,159,126)(39,160,118)(40,161,119)(41,162,120)(42,154,121)(43,155,122)(44,156,123)(45,157,124)(64,133,106)(65,134,107)(66,135,108)(67,127,100)(68,128,101)(69,129,102)(70,130,103)(71,131,104)(72,132,105)(82,113,96)(83,114,97)(84,115,98)(85,116,99)(86,117,91)(87,109,92)(88,110,93)(89,111,94)(90,112,95), (19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,161)(26,162)(27,154)(37,139)(38,140)(39,141)(40,142)(41,143)(42,144)(43,136)(44,137)(45,138)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,91)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,127)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,145), (1,17)(2,18)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(46,58)(47,59)(48,60)(49,61)(50,62)(51,63)(52,55)(53,56)(54,57)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,91)(82,128)(83,129)(84,130)(85,131)(86,132)(87,133)(88,134)(89,135)(90,127)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,109)(107,110)(108,111), (1,144,83,31,27,114,62,149,97)(2,136,84,32,19,115,63,150,98)(3,137,85,33,20,116,55,151,99)(4,138,86,34,21,117,56,152,91)(5,139,87,35,22,109,57,153,92)(6,140,88,36,23,110,58,145,93)(7,141,89,28,24,111,59,146,94)(8,142,90,29,25,112,60,147,95)(9,143,82,30,26,113,61,148,96)(10,44,131,79,156,104,52,123,71)(11,45,132,80,157,105,53,124,72)(12,37,133,81,158,106,54,125,64)(13,38,134,73,159,107,46,126,65)(14,39,135,74,160,108,47,118,66)(15,40,127,75,161,100,48,119,67)(16,41,128,76,162,101,49,120,68)(17,42,129,77,154,102,50,121,69)(18,43,130,78,155,103,51,122,70) );
G=PermutationGroup([[(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)], [(1,31,62),(2,32,63),(3,33,55),(4,34,56),(5,35,57),(6,36,58),(7,28,59),(8,29,60),(9,30,61),(10,79,52),(11,80,53),(12,81,54),(13,73,46),(14,74,47),(15,75,48),(16,76,49),(17,77,50),(18,78,51),(19,150,136),(20,151,137),(21,152,138),(22,153,139),(23,145,140),(24,146,141),(25,147,142),(26,148,143),(27,149,144),(37,158,125),(38,159,126),(39,160,118),(40,161,119),(41,162,120),(42,154,121),(43,155,122),(44,156,123),(45,157,124),(64,133,106),(65,134,107),(66,135,108),(67,127,100),(68,128,101),(69,129,102),(70,130,103),(71,131,104),(72,132,105),(82,113,96),(83,114,97),(84,115,98),(85,116,99),(86,117,91),(87,109,92),(88,110,93),(89,111,94),(90,112,95)], [(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,161),(26,162),(27,154),(37,139),(38,140),(39,141),(40,142),(41,143),(42,144),(43,136),(44,137),(45,138),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,91),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,127),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,109),(107,110),(108,111),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,145)], [(1,17),(2,18),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,73),(46,58),(47,59),(48,60),(49,61),(50,62),(51,63),(52,55),(53,56),(54,57),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,91),(82,128),(83,129),(84,130),(85,131),(86,132),(87,133),(88,134),(89,135),(90,127),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,109),(107,110),(108,111)], [(1,144,83,31,27,114,62,149,97),(2,136,84,32,19,115,63,150,98),(3,137,85,33,20,116,55,151,99),(4,138,86,34,21,117,56,152,91),(5,139,87,35,22,109,57,153,92),(6,140,88,36,23,110,58,145,93),(7,141,89,28,24,111,59,146,94),(8,142,90,29,25,112,60,147,95),(9,143,82,30,26,113,61,148,96),(10,44,131,79,156,104,52,123,71),(11,45,132,80,157,105,53,124,72),(12,37,133,81,158,106,54,125,64),(13,38,134,73,159,107,46,126,65),(14,39,135,74,160,108,47,118,66),(15,40,127,75,161,100,48,119,67),(16,41,128,76,162,101,49,120,68),(17,42,129,77,154,102,50,121,69),(18,43,130,78,155,103,51,122,70)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R | 9S | ··· | 9BT | 18A | ··· | 18R |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 4 | ··· | 4 | 3 | ··· | 3 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||
| image | C1 | C3 | C3 | C9 | C9 | A4 | C3.A4 | C3×A4 | C9×A4 |
| kernel | C9×C3.A4 | C3×C3.A4 | C6×C18 | C3.A4 | C2×C18 | C3×C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 54 | 18 | 1 | 6 | 2 | 18 |
Matrix representation of C9×C3.A4 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 0 | 4 | 0 | 0 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[11,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[1,0,0,0,0,1,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,0,0,0,18,0,0,0,0,1],[5,0,0,0,0,0,0,4,0,6,0,0,0,0,6,0] >;
C9×C3.A4 in GAP, Magma, Sage, TeX
C_9\times C_3.A_4
% in TeX
G:=Group("C9xC3.A4"); // GroupNames label
G:=SmallGroup(324,46);
// by ID
G=gap.SmallGroup(324,46);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,115,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^2=1,e^3=b,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,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export