non-abelian, supersoluble, monomial
Aliases: (C3×C9)⋊4D9, C3.2(C9⋊D9), (C32×C9).9S3, C3.C92⋊2C2, C33.46(C3⋊S3), C32.13(C9⋊S3), C3.1(C32⋊2D9), C32.12(He3⋊C2), SmallGroup(486,42)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.C92 — C3.2(C9⋊D9) |
Generators and relations for C3.2(C9⋊D9)
G = < a,b,c,d | a3=b9=c9=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b-1, dcd=c-1 >
Subgroups: 754 in 102 conjugacy classes, 30 normal (6 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C32×C9, C3×C9⋊S3, C3.C92, C3.2(C9⋊D9)
Quotients: C1, C2, S3, D9, C3⋊S3, C9⋊S3, He3⋊C2, C9⋊D9, C32⋊2D9, C3.2(C9⋊D9)
►On 162 points
Generators in S162
(1 122 29)(2 123 30)(3 124 31)(4 125 32)(5 126 33)(6 118 34)(7 119 35)(8 120 36)(9 121 28)(10 26 74)(11 27 75)(12 19 76)(13 20 77)(14 21 78)(15 22 79)(16 23 80)(17 24 81)(18 25 73)(37 58 107)(38 59 108)(39 60 100)(40 61 101)(41 62 102)(42 63 103)(43 55 104)(44 56 105)(45 57 106)(46 136 94)(47 137 95)(48 138 96)(49 139 97)(50 140 98)(51 141 99)(52 142 91)(53 143 92)(54 144 93)(64 127 85)(65 128 86)(66 129 87)(67 130 88)(68 131 89)(69 132 90)(70 133 82)(71 134 83)(72 135 84)(109 158 152)(110 159 153)(111 160 145)(112 161 146)(113 162 147)(114 154 148)(115 155 149)(116 156 150)(117 157 151)
(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 26 101 122 74 40 29 10 61)(2 11 41 123 27 62 30 75 102)(3 76 63 124 12 103 31 19 42)(4 20 104 125 77 43 32 13 55)(5 14 44 126 21 56 33 78 105)(6 79 57 118 15 106 34 22 45)(7 23 107 119 80 37 35 16 58)(8 17 38 120 24 59 36 81 108)(9 73 60 121 18 100 28 25 39)(46 88 155 94 130 115 136 67 149)(47 131 150 95 68 156 137 89 116)(48 69 117 96 90 151 138 132 157)(49 82 158 97 133 109 139 70 152)(50 134 153 98 71 159 140 83 110)(51 72 111 99 84 145 141 135 160)(52 85 161 91 127 112 142 64 146)(53 128 147 92 65 162 143 86 113)(54 66 114 93 87 148 144 129 154)
(1 145)(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 135)(11 134)(12 133)(13 132)(14 131)(15 130)(16 129)(17 128)(18 127)(19 82)(20 90)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 161)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 162)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(55 138)(56 137)(57 136)(58 144)(59 143)(60 142)(61 141)(62 140)(63 139)(64 73)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(91 100)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(109 124)(110 123)(111 122)(112 121)(113 120)(114 119)(115 118)(116 126)(117 125)
G:=sub<Sym(162)| (1,122,29)(2,123,30)(3,124,31)(4,125,32)(5,126,33)(6,118,34)(7,119,35)(8,120,36)(9,121,28)(10,26,74)(11,27,75)(12,19,76)(13,20,77)(14,21,78)(15,22,79)(16,23,80)(17,24,81)(18,25,73)(37,58,107)(38,59,108)(39,60,100)(40,61,101)(41,62,102)(42,63,103)(43,55,104)(44,56,105)(45,57,106)(46,136,94)(47,137,95)(48,138,96)(49,139,97)(50,140,98)(51,141,99)(52,142,91)(53,143,92)(54,144,93)(64,127,85)(65,128,86)(66,129,87)(67,130,88)(68,131,89)(69,132,90)(70,133,82)(71,134,83)(72,135,84)(109,158,152)(110,159,153)(111,160,145)(112,161,146)(113,162,147)(114,154,148)(115,155,149)(116,156,150)(117,157,151), (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), (1,26,101,122,74,40,29,10,61)(2,11,41,123,27,62,30,75,102)(3,76,63,124,12,103,31,19,42)(4,20,104,125,77,43,32,13,55)(5,14,44,126,21,56,33,78,105)(6,79,57,118,15,106,34,22,45)(7,23,107,119,80,37,35,16,58)(8,17,38,120,24,59,36,81,108)(9,73,60,121,18,100,28,25,39)(46,88,155,94,130,115,136,67,149)(47,131,150,95,68,156,137,89,116)(48,69,117,96,90,151,138,132,157)(49,82,158,97,133,109,139,70,152)(50,134,153,98,71,159,140,83,110)(51,72,111,99,84,145,141,135,160)(52,85,161,91,127,112,142,64,146)(53,128,147,92,65,162,143,86,113)(54,66,114,93,87,148,144,129,154), (1,145)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,82)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,161)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,162)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(55,138)(56,137)(57,136)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(64,73)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(91,100)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,126)(117,125)>;
G:=Group( (1,122,29)(2,123,30)(3,124,31)(4,125,32)(5,126,33)(6,118,34)(7,119,35)(8,120,36)(9,121,28)(10,26,74)(11,27,75)(12,19,76)(13,20,77)(14,21,78)(15,22,79)(16,23,80)(17,24,81)(18,25,73)(37,58,107)(38,59,108)(39,60,100)(40,61,101)(41,62,102)(42,63,103)(43,55,104)(44,56,105)(45,57,106)(46,136,94)(47,137,95)(48,138,96)(49,139,97)(50,140,98)(51,141,99)(52,142,91)(53,143,92)(54,144,93)(64,127,85)(65,128,86)(66,129,87)(67,130,88)(68,131,89)(69,132,90)(70,133,82)(71,134,83)(72,135,84)(109,158,152)(110,159,153)(111,160,145)(112,161,146)(113,162,147)(114,154,148)(115,155,149)(116,156,150)(117,157,151), (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), (1,26,101,122,74,40,29,10,61)(2,11,41,123,27,62,30,75,102)(3,76,63,124,12,103,31,19,42)(4,20,104,125,77,43,32,13,55)(5,14,44,126,21,56,33,78,105)(6,79,57,118,15,106,34,22,45)(7,23,107,119,80,37,35,16,58)(8,17,38,120,24,59,36,81,108)(9,73,60,121,18,100,28,25,39)(46,88,155,94,130,115,136,67,149)(47,131,150,95,68,156,137,89,116)(48,69,117,96,90,151,138,132,157)(49,82,158,97,133,109,139,70,152)(50,134,153,98,71,159,140,83,110)(51,72,111,99,84,145,141,135,160)(52,85,161,91,127,112,142,64,146)(53,128,147,92,65,162,143,86,113)(54,66,114,93,87,148,144,129,154), (1,145)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,128)(18,127)(19,82)(20,90)(21,89)(22,88)(23,87)(24,86)(25,85)(26,84)(27,83)(28,161)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,162)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(55,138)(56,137)(57,136)(58,144)(59,143)(60,142)(61,141)(62,140)(63,139)(64,73)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(91,100)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(109,124)(110,123)(111,122)(112,121)(113,120)(114,119)(115,118)(116,126)(117,125) );
G=PermutationGroup([[(1,122,29),(2,123,30),(3,124,31),(4,125,32),(5,126,33),(6,118,34),(7,119,35),(8,120,36),(9,121,28),(10,26,74),(11,27,75),(12,19,76),(13,20,77),(14,21,78),(15,22,79),(16,23,80),(17,24,81),(18,25,73),(37,58,107),(38,59,108),(39,60,100),(40,61,101),(41,62,102),(42,63,103),(43,55,104),(44,56,105),(45,57,106),(46,136,94),(47,137,95),(48,138,96),(49,139,97),(50,140,98),(51,141,99),(52,142,91),(53,143,92),(54,144,93),(64,127,85),(65,128,86),(66,129,87),(67,130,88),(68,131,89),(69,132,90),(70,133,82),(71,134,83),(72,135,84),(109,158,152),(110,159,153),(111,160,145),(112,161,146),(113,162,147),(114,154,148),(115,155,149),(116,156,150),(117,157,151)], [(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)], [(1,26,101,122,74,40,29,10,61),(2,11,41,123,27,62,30,75,102),(3,76,63,124,12,103,31,19,42),(4,20,104,125,77,43,32,13,55),(5,14,44,126,21,56,33,78,105),(6,79,57,118,15,106,34,22,45),(7,23,107,119,80,37,35,16,58),(8,17,38,120,24,59,36,81,108),(9,73,60,121,18,100,28,25,39),(46,88,155,94,130,115,136,67,149),(47,131,150,95,68,156,137,89,116),(48,69,117,96,90,151,138,132,157),(49,82,158,97,133,109,139,70,152),(50,134,153,98,71,159,140,83,110),(51,72,111,99,84,145,141,135,160),(52,85,161,91,127,112,142,64,146),(53,128,147,92,65,162,143,86,113),(54,66,114,93,87,148,144,129,154)], [(1,145),(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,135),(11,134),(12,133),(13,132),(14,131),(15,130),(16,129),(17,128),(18,127),(19,82),(20,90),(21,89),(22,88),(23,87),(24,86),(25,85),(26,84),(27,83),(28,161),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,154),(36,162),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(55,138),(56,137),(57,136),(58,144),(59,143),(60,142),(61,141),(62,140),(63,139),(64,73),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(91,100),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(109,124),(110,123),(111,122),(112,121),(113,120),(114,119),(115,118),(116,126),(117,125)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | ··· | 3N | 6A | 6B | 9A | ··· | 9AJ |
| order | 1 | 2 | 3 | 3 | 3 | ··· | 3 | 6 | 6 | 9 | ··· | 9 |
| size | 1 | 81 | 1 | 1 | 2 | ··· | 2 | 81 | 81 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 2 | 2 | 3 | 6 |
| type | + | + | + | + | ||
| image | C1 | C2 | S3 | D9 | He3⋊C2 | C32⋊2D9 |
| kernel | C3.2(C9⋊D9) | C3.C92 | C32×C9 | C3×C9 | C32 | C3 |
| # reps | 1 | 1 | 4 | 36 | 4 | 8 |
Matrix representation of C3.2(C9⋊D9) ►in GL7(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 | 0 |
| 0 | 0 | 12 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 0 | 0 | 0 |
| 0 | 0 | 14 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 5 | 0 | 0 | 0 |
| 0 | 0 | 17 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(7,GF(19))| [1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11],[11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,5,12,0,0,0,0,0,7,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,11],[17,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,7,14,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,11,0,0,0,0,0,0,0,7,0],[0,17,0,0,0,0,0,9,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,5,12,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,8,0,0,0,0,0,12,0] >;
C3.2(C9⋊D9) in GAP, Magma, Sage, TeX
C_3._2(C_9\rtimes D_9)
% in TeX
G:=Group("C3.2(C9:D9)"); // GroupNames label
G:=SmallGroup(486,42);
// by ID
G=gap.SmallGroup(486,42);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,223,1190,1520,338,867,3244]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^9=c^9=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations