direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C9⋊D9, C18⋊D9, C9⋊2D18, C92⋊3C22, (C9×C18)⋊2C2, (C3×C9).8D6, C6.3(C9⋊S3), (C3×C18).22S3, C3.1(C2×C9⋊S3), C32.9(C2×C3⋊S3), (C3×C6).17(C3⋊S3), SmallGroup(324,74)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C2×C9⋊D9 |
Generators and relations for C2×C9⋊D9
G = < a,b,c,d | a2=b9=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1003 in 115 conjugacy classes, 49 normal (7 characteristic)
C1, C2, C2, C3, C22, S3, C6, C9, C32, D6, D9, C18, C3⋊S3, C3×C6, C3×C9, D18, C2×C3⋊S3, C9⋊S3, C3×C18, C92, C2×C9⋊S3, C9⋊D9, C9×C18, C2×C9⋊D9
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, C2×C3⋊S3, C9⋊S3, C2×C9⋊S3, C9⋊D9, C2×C9⋊D9
►On 162 points
Generators in S162
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 90)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 100)(54 101)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 118)(72 119)(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 40 126 103 80 20 31 14 55)(2 41 118 104 81 21 32 15 56)(3 42 119 105 73 22 33 16 57)(4 43 120 106 74 23 34 17 58)(5 44 121 107 75 24 35 18 59)(6 45 122 108 76 25 36 10 60)(7 37 123 100 77 26 28 11 61)(8 38 124 101 78 27 29 12 62)(9 39 125 102 79 19 30 13 63)(46 160 90 111 94 153 144 129 69)(47 161 82 112 95 145 136 130 70)(48 162 83 113 96 146 137 131 71)(49 154 84 114 97 147 138 132 72)(50 155 85 115 98 148 139 133 64)(51 156 86 116 99 149 140 134 65)(52 157 87 117 91 150 141 135 66)(53 158 88 109 92 151 142 127 67)(54 159 89 110 93 152 143 128 68)
(1 55)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 104)(20 103)(21 102)(22 101)(23 100)(24 108)(25 107)(26 106)(27 105)(28 120)(29 119)(30 118)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(46 83)(47 82)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(64 109)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 78)(74 77)(75 76)(79 81)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 135)(136 145)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(154 159)(155 158)(156 157)(160 162)
G:=sub<Sym(162)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,90)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,100)(54,101)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,118)(72,119)(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), (1,40,126,103,80,20,31,14,55)(2,41,118,104,81,21,32,15,56)(3,42,119,105,73,22,33,16,57)(4,43,120,106,74,23,34,17,58)(5,44,121,107,75,24,35,18,59)(6,45,122,108,76,25,36,10,60)(7,37,123,100,77,26,28,11,61)(8,38,124,101,78,27,29,12,62)(9,39,125,102,79,19,30,13,63)(46,160,90,111,94,153,144,129,69)(47,161,82,112,95,145,136,130,70)(48,162,83,113,96,146,137,131,71)(49,154,84,114,97,147,138,132,72)(50,155,85,115,98,148,139,133,64)(51,156,86,116,99,149,140,134,65)(52,157,87,117,91,150,141,135,66)(53,158,88,109,92,151,142,127,67)(54,159,89,110,93,152,143,128,68), (1,55)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,104)(20,103)(21,102)(22,101)(23,100)(24,108)(25,107)(26,106)(27,105)(28,120)(29,119)(30,118)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(46,83)(47,82)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(64,109)(65,117)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,78)(74,77)(75,76)(79,81)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,135)(136,145)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(154,159)(155,158)(156,157)(160,162)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,141)(7,142)(8,143)(9,144)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,90)(20,82)(21,83)(22,84)(23,85)(24,86)(25,87)(26,88)(27,89)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,133)(44,134)(45,135)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,100)(54,101)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(61,151)(62,152)(63,153)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,118)(72,119)(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), (1,40,126,103,80,20,31,14,55)(2,41,118,104,81,21,32,15,56)(3,42,119,105,73,22,33,16,57)(4,43,120,106,74,23,34,17,58)(5,44,121,107,75,24,35,18,59)(6,45,122,108,76,25,36,10,60)(7,37,123,100,77,26,28,11,61)(8,38,124,101,78,27,29,12,62)(9,39,125,102,79,19,30,13,63)(46,160,90,111,94,153,144,129,69)(47,161,82,112,95,145,136,130,70)(48,162,83,113,96,146,137,131,71)(49,154,84,114,97,147,138,132,72)(50,155,85,115,98,148,139,133,64)(51,156,86,116,99,149,140,134,65)(52,157,87,117,91,150,141,135,66)(53,158,88,109,92,151,142,127,67)(54,159,89,110,93,152,143,128,68), (1,55)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,56)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,45)(19,104)(20,103)(21,102)(22,101)(23,100)(24,108)(25,107)(26,106)(27,105)(28,120)(29,119)(30,118)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(46,83)(47,82)(48,90)(49,89)(50,88)(51,87)(52,86)(53,85)(54,84)(64,109)(65,117)(66,116)(67,115)(68,114)(69,113)(70,112)(71,111)(72,110)(73,78)(74,77)(75,76)(79,81)(91,134)(92,133)(93,132)(94,131)(95,130)(96,129)(97,128)(98,127)(99,135)(136,145)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)(144,146)(154,159)(155,158)(156,157)(160,162) );
G=PermutationGroup([[(1,136),(2,137),(3,138),(4,139),(5,140),(6,141),(7,142),(8,143),(9,144),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,90),(20,82),(21,83),(22,84),(23,85),(24,86),(25,87),(26,88),(27,89),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,133),(44,134),(45,135),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,100),(54,101),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(61,151),(62,152),(63,153),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,118),(72,119),(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)], [(1,40,126,103,80,20,31,14,55),(2,41,118,104,81,21,32,15,56),(3,42,119,105,73,22,33,16,57),(4,43,120,106,74,23,34,17,58),(5,44,121,107,75,24,35,18,59),(6,45,122,108,76,25,36,10,60),(7,37,123,100,77,26,28,11,61),(8,38,124,101,78,27,29,12,62),(9,39,125,102,79,19,30,13,63),(46,160,90,111,94,153,144,129,69),(47,161,82,112,95,145,136,130,70),(48,162,83,113,96,146,137,131,71),(49,154,84,114,97,147,138,132,72),(50,155,85,115,98,148,139,133,64),(51,156,86,116,99,149,140,134,65),(52,157,87,117,91,150,141,135,66),(53,158,88,109,92,151,142,127,67),(54,159,89,110,93,152,143,128,68)], [(1,55),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,56),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,45),(19,104),(20,103),(21,102),(22,101),(23,100),(24,108),(25,107),(26,106),(27,105),(28,120),(29,119),(30,118),(31,126),(32,125),(33,124),(34,123),(35,122),(36,121),(46,83),(47,82),(48,90),(49,89),(50,88),(51,87),(52,86),(53,85),(54,84),(64,109),(65,117),(66,116),(67,115),(68,114),(69,113),(70,112),(71,111),(72,110),(73,78),(74,77),(75,76),(79,81),(91,134),(92,133),(93,132),(94,131),(95,130),(96,129),(97,128),(98,127),(99,135),(136,145),(137,153),(138,152),(139,151),(140,150),(141,149),(142,148),(143,147),(144,146),(154,159),(155,158),(156,157),(160,162)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | ··· | 9AJ | 18A | ··· | 18AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 81 | 81 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D9 | D18 |
| kernel | C2×C9⋊D9 | C9⋊D9 | C9×C18 | C3×C18 | C3×C9 | C18 | C9 |
| # reps | 1 | 2 | 1 | 4 | 4 | 36 | 36 |
Matrix representation of C2×C9⋊D9 ►in GL4(𝔽19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 1 | 0 | 0 |
| 18 | 18 | 0 | 0 |
| 0 | 0 | 5 | 7 |
| 0 | 0 | 12 | 17 |
| 2 | 14 | 0 | 0 |
| 5 | 7 | 0 | 0 |
| 0 | 0 | 5 | 7 |
| 0 | 0 | 12 | 17 |
| 2 | 7 | 0 | 0 |
| 5 | 17 | 0 | 0 |
| 0 | 0 | 12 | 17 |
| 0 | 0 | 5 | 7 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,18,0,0,1,18,0,0,0,0,5,12,0,0,7,17],[2,5,0,0,14,7,0,0,0,0,5,12,0,0,7,17],[2,5,0,0,7,17,0,0,0,0,12,5,0,0,17,7] >;
C2×C9⋊D9 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes D_9
% in TeX
G:=Group("C2xC9:D9"); // GroupNames label
G:=SmallGroup(324,74);
// by ID
G=gap.SmallGroup(324,74);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,794,338,3171,453,2164,7781]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^9=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations