direct product, metabelian, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C9⋊3- 1+2, C18⋊33- 1+2, C9⋊C9⋊6C6, C32⋊C9.17C6, C6.6(C9○He3), (C3×C6).24C33, (C32×C9).24C6, (C3×C18).7C32, C33.41(C3×C6), (C32×C18).12C3, C9⋊6(C2×3- 1+2), (C32×C6).29C32, C32.28(C32×C6), C3.6(C6×3- 1+2), C6.6(C3×3- 1+2), (C3×C6).33- 1+2, (C6×3- 1+2).4C3, (C3×3- 1+2).7C6, C32.3(C2×3- 1+2), (C2×C9⋊C9)⋊3C3, (C3×C9).6(C3×C6), C3.6(C2×C9○He3), (C2×C32⋊C9).8C3, SmallGroup(486,200)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9⋊3- 1+2
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, cbc-1=b7, bd=db, dcd-1=c4 >
Subgroups: 216 in 124 conjugacy classes, 78 normal (18 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C32, C32, C32, C18, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C3×C18, C3×C18, C3×C18, C2×3- 1+2, C32×C6, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C2×C32⋊C9, C2×C9⋊C9, C32×C18, C6×3- 1+2, C9⋊3- 1+2, C2×C9⋊3- 1+2
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C9○He3, C6×3- 1+2, C2×C9○He3, C9⋊3- 1+2, C2×C9⋊3- 1+2
►On 162 points
Generators in S162
(1 108)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 106)(9 107)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 82)(18 83)(19 93)(20 94)(21 95)(22 96)(23 97)(24 98)(25 99)(26 91)(27 92)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 133)(53 134)(54 135)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(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 79 68 41 95 52 28 85 55)(2 74 66 42 99 50 29 89 62)(3 78 64 43 94 48 30 84 60)(4 73 71 44 98 46 31 88 58)(5 77 69 45 93 53 32 83 56)(6 81 67 37 97 51 33 87 63)(7 76 65 38 92 49 34 82 61)(8 80 72 39 96 47 35 86 59)(9 75 70 40 91 54 36 90 57)(10 141 101 159 145 124 20 129 111)(11 136 108 160 149 122 21 133 109)(12 140 106 161 153 120 22 128 116)(13 144 104 162 148 118 23 132 114)(14 139 102 154 152 125 24 127 112)(15 143 100 155 147 123 25 131 110)(16 138 107 156 151 121 26 135 117)(17 142 105 157 146 119 27 130 115)(18 137 103 158 150 126 19 134 113)
(10 20 159)(11 21 160)(12 22 161)(13 23 162)(14 24 154)(15 25 155)(16 26 156)(17 27 157)(18 19 158)(46 58 71)(47 59 72)(48 60 64)(49 61 65)(50 62 66)(51 63 67)(52 55 68)(53 56 69)(54 57 70)(73 88 98)(74 89 99)(75 90 91)(76 82 92)(77 83 93)(78 84 94)(79 85 95)(80 86 96)(81 87 97)(127 139 152)(128 140 153)(129 141 145)(130 142 146)(131 143 147)(132 144 148)(133 136 149)(134 137 150)(135 138 151)
G:=sub<Sym(162)| (1,108)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(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), (1,79,68,41,95,52,28,85,55)(2,74,66,42,99,50,29,89,62)(3,78,64,43,94,48,30,84,60)(4,73,71,44,98,46,31,88,58)(5,77,69,45,93,53,32,83,56)(6,81,67,37,97,51,33,87,63)(7,76,65,38,92,49,34,82,61)(8,80,72,39,96,47,35,86,59)(9,75,70,40,91,54,36,90,57)(10,141,101,159,145,124,20,129,111)(11,136,108,160,149,122,21,133,109)(12,140,106,161,153,120,22,128,116)(13,144,104,162,148,118,23,132,114)(14,139,102,154,152,125,24,127,112)(15,143,100,155,147,123,25,131,110)(16,138,107,156,151,121,26,135,117)(17,142,105,157,146,119,27,130,115)(18,137,103,158,150,126,19,134,113), (10,20,159)(11,21,160)(12,22,161)(13,23,162)(14,24,154)(15,25,155)(16,26,156)(17,27,157)(18,19,158)(46,58,71)(47,59,72)(48,60,64)(49,61,65)(50,62,66)(51,63,67)(52,55,68)(53,56,69)(54,57,70)(73,88,98)(74,89,99)(75,90,91)(76,82,92)(77,83,93)(78,84,94)(79,85,95)(80,86,96)(81,87,97)(127,139,152)(128,140,153)(129,141,145)(130,142,146)(131,143,147)(132,144,148)(133,136,149)(134,137,150)(135,138,151)>;
G:=Group( (1,108)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,106)(9,107)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,82)(18,83)(19,93)(20,94)(21,95)(22,96)(23,97)(24,98)(25,99)(26,91)(27,92)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,130)(50,131)(51,132)(52,133)(53,134)(54,135)(55,136)(56,137)(57,138)(58,139)(59,140)(60,141)(61,142)(62,143)(63,144)(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), (1,79,68,41,95,52,28,85,55)(2,74,66,42,99,50,29,89,62)(3,78,64,43,94,48,30,84,60)(4,73,71,44,98,46,31,88,58)(5,77,69,45,93,53,32,83,56)(6,81,67,37,97,51,33,87,63)(7,76,65,38,92,49,34,82,61)(8,80,72,39,96,47,35,86,59)(9,75,70,40,91,54,36,90,57)(10,141,101,159,145,124,20,129,111)(11,136,108,160,149,122,21,133,109)(12,140,106,161,153,120,22,128,116)(13,144,104,162,148,118,23,132,114)(14,139,102,154,152,125,24,127,112)(15,143,100,155,147,123,25,131,110)(16,138,107,156,151,121,26,135,117)(17,142,105,157,146,119,27,130,115)(18,137,103,158,150,126,19,134,113), (10,20,159)(11,21,160)(12,22,161)(13,23,162)(14,24,154)(15,25,155)(16,26,156)(17,27,157)(18,19,158)(46,58,71)(47,59,72)(48,60,64)(49,61,65)(50,62,66)(51,63,67)(52,55,68)(53,56,69)(54,57,70)(73,88,98)(74,89,99)(75,90,91)(76,82,92)(77,83,93)(78,84,94)(79,85,95)(80,86,96)(81,87,97)(127,139,152)(128,140,153)(129,141,145)(130,142,146)(131,143,147)(132,144,148)(133,136,149)(134,137,150)(135,138,151) );
G=PermutationGroup([[(1,108),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,106),(9,107),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,82),(18,83),(19,93),(20,94),(21,95),(22,96),(23,97),(24,98),(25,99),(26,91),(27,92),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,130),(50,131),(51,132),(52,133),(53,134),(54,135),(55,136),(56,137),(57,138),(58,139),(59,140),(60,141),(61,142),(62,143),(63,144),(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)], [(1,79,68,41,95,52,28,85,55),(2,74,66,42,99,50,29,89,62),(3,78,64,43,94,48,30,84,60),(4,73,71,44,98,46,31,88,58),(5,77,69,45,93,53,32,83,56),(6,81,67,37,97,51,33,87,63),(7,76,65,38,92,49,34,82,61),(8,80,72,39,96,47,35,86,59),(9,75,70,40,91,54,36,90,57),(10,141,101,159,145,124,20,129,111),(11,136,108,160,149,122,21,133,109),(12,140,106,161,153,120,22,128,116),(13,144,104,162,148,118,23,132,114),(14,139,102,154,152,125,24,127,112),(15,143,100,155,147,123,25,131,110),(16,138,107,156,151,121,26,135,117),(17,142,105,157,146,119,27,130,115),(18,137,103,158,150,126,19,134,113)], [(10,20,159),(11,21,160),(12,22,161),(13,23,162),(14,24,154),(15,25,155),(16,26,156),(17,27,157),(18,19,158),(46,58,71),(47,59,72),(48,60,64),(49,61,65),(50,62,66),(51,63,67),(52,55,68),(53,56,69),(54,57,70),(73,88,98),(74,89,99),(75,90,91),(76,82,92),(77,83,93),(78,84,94),(79,85,95),(80,86,96),(81,87,97),(127,139,152),(128,140,153),(129,141,145),(130,142,146),(131,143,147),(132,144,148),(133,136,149),(134,137,150),(135,138,151)]])
102 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 3I | ··· | 3N | 6A | ··· | 6H | 6I | ··· | 6N | 9A | ··· | 9R | 9S | ··· | 9AJ | 18A | ··· | 18R | 18S | ··· | 18AJ |
| order | 1 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 3 | ··· | 3 | 9 | ··· | 9 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C3 | C6 | C6 | C6 | C6 | 3- 1+2 | 3- 1+2 | C2×3- 1+2 | C2×3- 1+2 | C9○He3 | C2×C9○He3 |
| kernel | C2×C9⋊3- 1+2 | C9⋊3- 1+2 | C2×C32⋊C9 | C2×C9⋊C9 | C32×C18 | C6×3- 1+2 | C32⋊C9 | C9⋊C9 | C32×C9 | C3×3- 1+2 | C18 | C3×C6 | C9 | C32 | C6 | C3 |
| # reps | 1 | 1 | 4 | 18 | 2 | 2 | 4 | 18 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C2×C9⋊3- 1+2 ►in GL6(𝔽19)
| 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,17,0,0,0,0,0,0,16,0,0,0,0,0,0,5],[0,0,1,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,7,0],[1,0,0,0,0,0,0,11,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11] >;
C2×C9⋊3- 1+2 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes 3_-^{1+2} % in TeX
G:=Group("C2xC9:ES-(3,1)"); // GroupNames label
G:=SmallGroup(486,200);
// by ID
G=gap.SmallGroup(486,200);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,176,2169]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations