direct product, metabelian, soluble, monomial, A-group
Aliases: C3×C9.A4, C62.1C9, (C2×C6)⋊C27, C9.4(C3×A4), (C3×C9).5A4, (C2×C18).2C9, (C6×C18).1C3, C22⋊2(C3×C27), C9.3(C3.A4), (C2×C18).4C32, C32.2(C3.A4), (C2×C6).5(C3×C9), C3.1(C3×C3.A4), SmallGroup(324,44)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — C3×C9.A4 |
Generators and relations for C3×C9.A4
G = < a,b,c,d,e | a3=b9=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 C3×C9.A4
►On 162 points
Generators in S162
(1 28 66)(2 29 67)(3 30 68)(4 31 69)(5 32 70)(6 33 71)(7 34 72)(8 35 73)(9 36 74)(10 37 75)(11 38 76)(12 39 77)(13 40 78)(14 41 79)(15 42 80)(16 43 81)(17 44 55)(18 45 56)(19 46 57)(20 47 58)(21 48 59)(22 49 60)(23 50 61)(24 51 62)(25 52 63)(26 53 64)(27 54 65)(82 127 148)(83 128 149)(84 129 150)(85 130 151)(86 131 152)(87 132 153)(88 133 154)(89 134 155)(90 135 156)(91 109 157)(92 110 158)(93 111 159)(94 112 160)(95 113 161)(96 114 162)(97 115 136)(98 116 137)(99 117 138)(100 118 139)(101 119 140)(102 120 141)(103 121 142)(104 122 143)(105 123 144)(106 124 145)(107 125 146)(108 126 147)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106)(83 86 89 92 95 98 101 104 107)(84 87 90 93 96 99 102 105 108)(109 112 115 118 121 124 127 130 133)(110 113 116 119 122 125 128 131 134)(111 114 117 120 123 126 129 132 135)(136 139 142 145 148 151 154 157 160)(137 140 143 146 149 152 155 158 161)(138 141 144 147 150 153 156 159 162)
(2 110)(3 111)(5 113)(6 114)(8 116)(9 117)(11 119)(12 120)(14 122)(15 123)(17 125)(18 126)(20 128)(21 129)(23 131)(24 132)(26 134)(27 135)(29 158)(30 159)(32 161)(33 162)(35 137)(36 138)(38 140)(39 141)(41 143)(42 144)(44 146)(45 147)(47 149)(48 150)(50 152)(51 153)(53 155)(54 156)(55 107)(56 108)(58 83)(59 84)(61 86)(62 87)(64 89)(65 90)(67 92)(68 93)(70 95)(71 96)(73 98)(74 99)(76 101)(77 102)(79 104)(80 105)
(1 109)(3 111)(4 112)(6 114)(7 115)(9 117)(10 118)(12 120)(13 121)(15 123)(16 124)(18 126)(19 127)(21 129)(22 130)(24 132)(25 133)(27 135)(28 157)(30 159)(31 160)(33 162)(34 136)(36 138)(37 139)(39 141)(40 142)(42 144)(43 145)(45 147)(46 148)(48 150)(49 151)(51 153)(52 154)(54 156)(56 108)(57 82)(59 84)(60 85)(62 87)(63 88)(65 90)(66 91)(68 93)(69 94)(71 96)(72 97)(74 99)(75 100)(77 102)(78 103)(80 105)(81 106)
(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)
G:=sub<Sym(162)| (1,28,66)(2,29,67)(3,30,68)(4,31,69)(5,32,70)(6,33,71)(7,34,72)(8,35,73)(9,36,74)(10,37,75)(11,38,76)(12,39,77)(13,40,78)(14,41,79)(15,42,80)(16,43,81)(17,44,55)(18,45,56)(19,46,57)(20,47,58)(21,48,59)(22,49,60)(23,50,61)(24,51,62)(25,52,63)(26,53,64)(27,54,65)(82,127,148)(83,128,149)(84,129,150)(85,130,151)(86,131,152)(87,132,153)(88,133,154)(89,134,155)(90,135,156)(91,109,157)(92,110,158)(93,111,159)(94,112,160)(95,113,161)(96,114,162)(97,115,136)(98,116,137)(99,117,138)(100,118,139)(101,119,140)(102,120,141)(103,121,142)(104,122,143)(105,123,144)(106,124,145)(107,125,146)(108,126,147), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (2,110)(3,111)(5,113)(6,114)(8,116)(9,117)(11,119)(12,120)(14,122)(15,123)(17,125)(18,126)(20,128)(21,129)(23,131)(24,132)(26,134)(27,135)(29,158)(30,159)(32,161)(33,162)(35,137)(36,138)(38,140)(39,141)(41,143)(42,144)(44,146)(45,147)(47,149)(48,150)(50,152)(51,153)(53,155)(54,156)(55,107)(56,108)(58,83)(59,84)(61,86)(62,87)(64,89)(65,90)(67,92)(68,93)(70,95)(71,96)(73,98)(74,99)(76,101)(77,102)(79,104)(80,105), (1,109)(3,111)(4,112)(6,114)(7,115)(9,117)(10,118)(12,120)(13,121)(15,123)(16,124)(18,126)(19,127)(21,129)(22,130)(24,132)(25,133)(27,135)(28,157)(30,159)(31,160)(33,162)(34,136)(36,138)(37,139)(39,141)(40,142)(42,144)(43,145)(45,147)(46,148)(48,150)(49,151)(51,153)(52,154)(54,156)(56,108)(57,82)(59,84)(60,85)(62,87)(63,88)(65,90)(66,91)(68,93)(69,94)(71,96)(72,97)(74,99)(75,100)(77,102)(78,103)(80,105)(81,106), (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)>;
G:=Group( (1,28,66)(2,29,67)(3,30,68)(4,31,69)(5,32,70)(6,33,71)(7,34,72)(8,35,73)(9,36,74)(10,37,75)(11,38,76)(12,39,77)(13,40,78)(14,41,79)(15,42,80)(16,43,81)(17,44,55)(18,45,56)(19,46,57)(20,47,58)(21,48,59)(22,49,60)(23,50,61)(24,51,62)(25,52,63)(26,53,64)(27,54,65)(82,127,148)(83,128,149)(84,129,150)(85,130,151)(86,131,152)(87,132,153)(88,133,154)(89,134,155)(90,135,156)(91,109,157)(92,110,158)(93,111,159)(94,112,160)(95,113,161)(96,114,162)(97,115,136)(98,116,137)(99,117,138)(100,118,139)(101,119,140)(102,120,141)(103,121,142)(104,122,143)(105,123,144)(106,124,145)(107,125,146)(108,126,147), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (2,110)(3,111)(5,113)(6,114)(8,116)(9,117)(11,119)(12,120)(14,122)(15,123)(17,125)(18,126)(20,128)(21,129)(23,131)(24,132)(26,134)(27,135)(29,158)(30,159)(32,161)(33,162)(35,137)(36,138)(38,140)(39,141)(41,143)(42,144)(44,146)(45,147)(47,149)(48,150)(50,152)(51,153)(53,155)(54,156)(55,107)(56,108)(58,83)(59,84)(61,86)(62,87)(64,89)(65,90)(67,92)(68,93)(70,95)(71,96)(73,98)(74,99)(76,101)(77,102)(79,104)(80,105), (1,109)(3,111)(4,112)(6,114)(7,115)(9,117)(10,118)(12,120)(13,121)(15,123)(16,124)(18,126)(19,127)(21,129)(22,130)(24,132)(25,133)(27,135)(28,157)(30,159)(31,160)(33,162)(34,136)(36,138)(37,139)(39,141)(40,142)(42,144)(43,145)(45,147)(46,148)(48,150)(49,151)(51,153)(52,154)(54,156)(56,108)(57,82)(59,84)(60,85)(62,87)(63,88)(65,90)(66,91)(68,93)(69,94)(71,96)(72,97)(74,99)(75,100)(77,102)(78,103)(80,105)(81,106), (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) );
G=PermutationGroup([[(1,28,66),(2,29,67),(3,30,68),(4,31,69),(5,32,70),(6,33,71),(7,34,72),(8,35,73),(9,36,74),(10,37,75),(11,38,76),(12,39,77),(13,40,78),(14,41,79),(15,42,80),(16,43,81),(17,44,55),(18,45,56),(19,46,57),(20,47,58),(21,48,59),(22,49,60),(23,50,61),(24,51,62),(25,52,63),(26,53,64),(27,54,65),(82,127,148),(83,128,149),(84,129,150),(85,130,151),(86,131,152),(87,132,153),(88,133,154),(89,134,155),(90,135,156),(91,109,157),(92,110,158),(93,111,159),(94,112,160),(95,113,161),(96,114,162),(97,115,136),(98,116,137),(99,117,138),(100,118,139),(101,119,140),(102,120,141),(103,121,142),(104,122,143),(105,123,144),(106,124,145),(107,125,146),(108,126,147)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106),(83,86,89,92,95,98,101,104,107),(84,87,90,93,96,99,102,105,108),(109,112,115,118,121,124,127,130,133),(110,113,116,119,122,125,128,131,134),(111,114,117,120,123,126,129,132,135),(136,139,142,145,148,151,154,157,160),(137,140,143,146,149,152,155,158,161),(138,141,144,147,150,153,156,159,162)], [(2,110),(3,111),(5,113),(6,114),(8,116),(9,117),(11,119),(12,120),(14,122),(15,123),(17,125),(18,126),(20,128),(21,129),(23,131),(24,132),(26,134),(27,135),(29,158),(30,159),(32,161),(33,162),(35,137),(36,138),(38,140),(39,141),(41,143),(42,144),(44,146),(45,147),(47,149),(48,150),(50,152),(51,153),(53,155),(54,156),(55,107),(56,108),(58,83),(59,84),(61,86),(62,87),(64,89),(65,90),(67,92),(68,93),(70,95),(71,96),(73,98),(74,99),(76,101),(77,102),(79,104),(80,105)], [(1,109),(3,111),(4,112),(6,114),(7,115),(9,117),(10,118),(12,120),(13,121),(15,123),(16,124),(18,126),(19,127),(21,129),(22,130),(24,132),(25,133),(27,135),(28,157),(30,159),(31,160),(33,162),(34,136),(36,138),(37,139),(39,141),(40,142),(42,144),(43,145),(45,147),(46,148),(48,150),(49,151),(51,153),(52,154),(54,156),(56,108),(57,82),(59,84),(60,85),(62,87),(63,88),(65,90),(66,91),(68,93),(69,94),(71,96),(72,97),(74,99),(75,100),(77,102),(78,103),(80,105),(81,106)], [(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)]])
108 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 9A | ··· | 9R | 18A | ··· | 18R | 27A | ··· | 27BB |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
| size | 1 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | |||||||||
| image | C1 | C3 | C3 | C9 | C9 | C27 | A4 | C3.A4 | C3×A4 | C3.A4 | C9.A4 |
| kernel | C3×C9.A4 | C9.A4 | C6×C18 | C2×C18 | C62 | C2×C6 | C3×C9 | C9 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 12 | 6 | 54 | 1 | 4 | 2 | 2 | 18 |
Matrix representation of C3×C9.A4 ►in GL4(𝔽109) generated by
| 45 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 63 |
| 1 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 0 | 38 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 78 | 108 | 0 |
| 0 | 89 | 0 | 108 |
| 1 | 0 | 0 | 0 |
| 0 | 108 | 0 | 0 |
| 0 | 0 | 108 | 0 |
| 0 | 20 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 22 | 19 | 0 |
| 0 | 0 | 87 | 45 |
| 0 | 15 | 28 | 0 |
G:=sub<GL(4,GF(109))| [45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,78,89,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,20,0,0,108,0,0,0,0,1],[1,0,0,0,0,22,0,15,0,19,87,28,0,0,45,0] >;
C3×C9.A4 in GAP, Magma, Sage, TeX
C_3\times C_9.A_4
% in TeX
G:=Group("C3xC9.A4"); // GroupNames label
G:=SmallGroup(324,44);
// by ID
G=gap.SmallGroup(324,44);
# by ID
G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,68,4864,8753]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^9=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