direct product, metabelian, nilpotent (class 3), monomial, 3-elementary
Aliases: C2×3- 1+2⋊C9, 3- 1+2⋊C18, C6.8C3≀C3, (C3×C6).25He3, C32⋊C9.16C6, C6.8(C32⋊C9), C32.2(C3×C18), (C32×C9).17C6, (C32×C18).5C3, C33.35(C3×C6), (C2×3- 1+2)⋊C9, C6.8(He3.C3), C32.23(C2×He3), C6.5(C3.He3), (C32×C6).23C32, (C3×C6).23- 1+2, (C6×3- 1+2).3C3, (C3×3- 1+2).6C6, C32.2(C2×3- 1+2), C3.3(C2×C3≀C3), (C3×C6).2(C3×C9), C3.8(C2×C32⋊C9), (C2×C32⋊C9).7C3, C3.3(C2×He3.C3), C3.2(C2×C3.He3), SmallGroup(486,78)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×3- 1+2⋊C9
G = < a,b,c,d | a2=b9=c3=d9=1, ab=ba, ac=ca, ad=da, cbc-1=b4, dbd-1=bc-1, dcd-1=b6c >
Subgroups: 198 in 82 conjugacy classes, 36 normal (26 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, 3- 1+2, 3- 1+2, C33, C3×C18, C2×3- 1+2, C2×3- 1+2, C32×C6, C32⋊C9, C32×C9, C3×3- 1+2, C2×C32⋊C9, C32×C18, C6×3- 1+2, 3- 1+2⋊C9, C2×3- 1+2⋊C9
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C3×C18, C2×He3, C2×3- 1+2, C32⋊C9, C3≀C3, He3.C3, C3.He3, C2×C32⋊C9, C2×C3≀C3, C2×He3.C3, C2×C3.He3, 3- 1+2⋊C9, C2×3- 1+2⋊C9
►On 162 points
Generators in S162
(1 111)(2 112)(3 113)(4 114)(5 115)(6 116)(7 117)(8 109)(9 110)(10 87)(11 88)(12 89)(13 90)(14 82)(15 83)(16 84)(17 85)(18 86)(19 98)(20 99)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 105)(29 106)(30 107)(31 108)(32 100)(33 101)(34 102)(35 103)(36 104)(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)
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 22 25)(21 27 24)(29 35 32)(30 33 36)(37 40 43)(39 45 42)(47 53 50)(48 51 54)(55 58 61)(57 63 60)(64 70 67)(65 68 71)(73 76 79)(75 81 78)(82 85 88)(84 90 87)(91 97 94)(92 95 98)(100 106 103)(101 104 107)(109 115 112)(110 113 116)(118 121 124)(120 126 123)(128 134 131)(129 132 135)(136 139 142)(138 144 141)(145 151 148)(146 149 152)(154 157 160)(156 162 159)
(1 78 51 28 91 71 44 84 55)(2 76 52 29 98 72 45 82 56)(3 80 47 30 93 67 37 86 60)(4 81 54 31 94 65 38 87 58)(5 79 46 32 92 66 39 85 59)(6 74 50 33 96 70 40 89 63)(7 75 48 34 97 68 41 90 61)(8 73 49 35 95 69 42 88 62)(9 77 53 36 99 64 43 83 57)(10 139 114 162 135 108 24 146 119)(11 143 109 154 130 103 25 150 123)(12 144 116 155 131 101 26 151 121)(13 142 117 156 129 102 27 149 122)(14 137 112 157 133 106 19 153 126)(15 138 110 158 134 104 20 145 124)(16 136 111 159 132 105 21 152 125)(17 140 115 160 127 100 22 147 120)(18 141 113 161 128 107 23 148 118)
G:=sub<Sym(162)| (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,109)(9,110)(10,87)(11,88)(12,89)(13,90)(14,82)(15,83)(16,84)(17,85)(18,86)(19,98)(20,99)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,105)(29,106)(30,107)(31,108)(32,100)(33,101)(34,102)(35,103)(36,104)(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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(64,70,67)(65,68,71)(73,76,79)(75,81,78)(82,85,88)(84,90,87)(91,97,94)(92,95,98)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(118,121,124)(120,126,123)(128,134,131)(129,132,135)(136,139,142)(138,144,141)(145,151,148)(146,149,152)(154,157,160)(156,162,159), (1,78,51,28,91,71,44,84,55)(2,76,52,29,98,72,45,82,56)(3,80,47,30,93,67,37,86,60)(4,81,54,31,94,65,38,87,58)(5,79,46,32,92,66,39,85,59)(6,74,50,33,96,70,40,89,63)(7,75,48,34,97,68,41,90,61)(8,73,49,35,95,69,42,88,62)(9,77,53,36,99,64,43,83,57)(10,139,114,162,135,108,24,146,119)(11,143,109,154,130,103,25,150,123)(12,144,116,155,131,101,26,151,121)(13,142,117,156,129,102,27,149,122)(14,137,112,157,133,106,19,153,126)(15,138,110,158,134,104,20,145,124)(16,136,111,159,132,105,21,152,125)(17,140,115,160,127,100,22,147,120)(18,141,113,161,128,107,23,148,118)>;
G:=Group( (1,111)(2,112)(3,113)(4,114)(5,115)(6,116)(7,117)(8,109)(9,110)(10,87)(11,88)(12,89)(13,90)(14,82)(15,83)(16,84)(17,85)(18,86)(19,98)(20,99)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,105)(29,106)(30,107)(31,108)(32,100)(33,101)(34,102)(35,103)(36,104)(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), (2,8,5)(3,6,9)(10,16,13)(11,14,17)(19,22,25)(21,27,24)(29,35,32)(30,33,36)(37,40,43)(39,45,42)(47,53,50)(48,51,54)(55,58,61)(57,63,60)(64,70,67)(65,68,71)(73,76,79)(75,81,78)(82,85,88)(84,90,87)(91,97,94)(92,95,98)(100,106,103)(101,104,107)(109,115,112)(110,113,116)(118,121,124)(120,126,123)(128,134,131)(129,132,135)(136,139,142)(138,144,141)(145,151,148)(146,149,152)(154,157,160)(156,162,159), (1,78,51,28,91,71,44,84,55)(2,76,52,29,98,72,45,82,56)(3,80,47,30,93,67,37,86,60)(4,81,54,31,94,65,38,87,58)(5,79,46,32,92,66,39,85,59)(6,74,50,33,96,70,40,89,63)(7,75,48,34,97,68,41,90,61)(8,73,49,35,95,69,42,88,62)(9,77,53,36,99,64,43,83,57)(10,139,114,162,135,108,24,146,119)(11,143,109,154,130,103,25,150,123)(12,144,116,155,131,101,26,151,121)(13,142,117,156,129,102,27,149,122)(14,137,112,157,133,106,19,153,126)(15,138,110,158,134,104,20,145,124)(16,136,111,159,132,105,21,152,125)(17,140,115,160,127,100,22,147,120)(18,141,113,161,128,107,23,148,118) );
G=PermutationGroup([[(1,111),(2,112),(3,113),(4,114),(5,115),(6,116),(7,117),(8,109),(9,110),(10,87),(11,88),(12,89),(13,90),(14,82),(15,83),(16,84),(17,85),(18,86),(19,98),(20,99),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,105),(29,106),(30,107),(31,108),(32,100),(33,101),(34,102),(35,103),(36,104),(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)], [(2,8,5),(3,6,9),(10,16,13),(11,14,17),(19,22,25),(21,27,24),(29,35,32),(30,33,36),(37,40,43),(39,45,42),(47,53,50),(48,51,54),(55,58,61),(57,63,60),(64,70,67),(65,68,71),(73,76,79),(75,81,78),(82,85,88),(84,90,87),(91,97,94),(92,95,98),(100,106,103),(101,104,107),(109,115,112),(110,113,116),(118,121,124),(120,126,123),(128,134,131),(129,132,135),(136,139,142),(138,144,141),(145,151,148),(146,149,152),(154,157,160),(156,162,159)], [(1,78,51,28,91,71,44,84,55),(2,76,52,29,98,72,45,82,56),(3,80,47,30,93,67,37,86,60),(4,81,54,31,94,65,38,87,58),(5,79,46,32,92,66,39,85,59),(6,74,50,33,96,70,40,89,63),(7,75,48,34,97,68,41,90,61),(8,73,49,35,95,69,42,88,62),(9,77,53,36,99,64,43,83,57),(10,139,114,162,135,108,24,146,119),(11,143,109,154,130,103,25,150,123),(12,144,116,155,131,101,26,151,121),(13,142,117,156,129,102,27,149,122),(14,137,112,157,133,106,19,153,126),(15,138,110,158,134,104,20,145,124),(16,136,111,159,132,105,21,152,125),(17,140,115,160,127,100,22,147,120),(18,141,113,161,128,107,23,148,118)]])
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 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | C9 | C18 | He3 | 3- 1+2 | C2×He3 | C2×3- 1+2 | C3≀C3 | He3.C3 | C3.He3 | C2×C3≀C3 | C2×He3.C3 | C2×C3.He3 |
| kernel | C2×3- 1+2⋊C9 | 3- 1+2⋊C9 | C2×C32⋊C9 | C32×C18 | C6×3- 1+2 | C32⋊C9 | C32×C9 | C3×3- 1+2 | C2×3- 1+2 | 3- 1+2 | C3×C6 | C3×C6 | C32 | C32 | C6 | C6 | C6 | C3 | C3 | C3 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 2 | 18 | 18 | 2 | 4 | 2 | 4 | 6 | 6 | 6 | 6 | 6 | 6 |
Matrix representation of C2×3- 1+2⋊C9 ►in GL4(𝔽19) generated by
| 18 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 0 | 0 | 0 |
| 0 | 7 | 9 | 0 |
| 0 | 1 | 12 | 1 |
| 0 | 0 | 18 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 11 | 7 | 0 |
| 0 | 18 | 0 | 11 |
| 16 | 0 | 0 | 0 |
| 0 | 9 | 17 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 15 | 0 |
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[7,0,0,0,0,7,1,0,0,9,12,18,0,0,1,0],[1,0,0,0,0,1,11,18,0,0,7,0,0,0,0,11],[16,0,0,0,0,9,0,0,0,17,10,15,0,0,6,0] >;
C2×3- 1+2⋊C9 in GAP, Magma, Sage, TeX
C_2\times 3_-^{1+2}\rtimes C_9 % in TeX
G:=Group("C2xES-(3,1):C9"); // GroupNames label
G:=SmallGroup(486,78);
// by ID
G=gap.SmallGroup(486,78);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,224,500,2169,735]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^4,d*b*d^-1=b*c^-1,d*c*d^-1=b^6*c>;
// generators/relations