direct product, abelian, monomial, 3-elementary
Aliases: C32×C18, SmallGroup(162,47)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C18 |
| C1 — C32×C18 |
| C1 — C32×C18 |
Generators and relations for C32×C18
G = < a,b,c | a3=b3=c18=1, ab=ba, ac=ca, bc=cb >
Subgroups: 100, all normal (8 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C32×C18
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C32×C18
►Regular action on 162 points
Generators in S162
(1 154 38)(2 155 39)(3 156 40)(4 157 41)(5 158 42)(6 159 43)(7 160 44)(8 161 45)(9 162 46)(10 145 47)(11 146 48)(12 147 49)(13 148 50)(14 149 51)(15 150 52)(16 151 53)(17 152 54)(18 153 37)(19 121 138)(20 122 139)(21 123 140)(22 124 141)(23 125 142)(24 126 143)(25 109 144)(26 110 127)(27 111 128)(28 112 129)(29 113 130)(30 114 131)(31 115 132)(32 116 133)(33 117 134)(34 118 135)(35 119 136)(36 120 137)(55 94 80)(56 95 81)(57 96 82)(58 97 83)(59 98 84)(60 99 85)(61 100 86)(62 101 87)(63 102 88)(64 103 89)(65 104 90)(66 105 73)(67 106 74)(68 107 75)(69 108 76)(70 91 77)(71 92 78)(72 93 79)
(1 62 143)(2 63 144)(3 64 127)(4 65 128)(5 66 129)(6 67 130)(7 68 131)(8 69 132)(9 70 133)(10 71 134)(11 72 135)(12 55 136)(13 56 137)(14 57 138)(15 58 139)(16 59 140)(17 60 141)(18 61 142)(19 149 96)(20 150 97)(21 151 98)(22 152 99)(23 153 100)(24 154 101)(25 155 102)(26 156 103)(27 157 104)(28 158 105)(29 159 106)(30 160 107)(31 161 108)(32 162 91)(33 145 92)(34 146 93)(35 147 94)(36 148 95)(37 86 125)(38 87 126)(39 88 109)(40 89 110)(41 90 111)(42 73 112)(43 74 113)(44 75 114)(45 76 115)(46 77 116)(47 78 117)(48 79 118)(49 80 119)(50 81 120)(51 82 121)(52 83 122)(53 84 123)(54 85 124)
(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)| (1,154,38)(2,155,39)(3,156,40)(4,157,41)(5,158,42)(6,159,43)(7,160,44)(8,161,45)(9,162,46)(10,145,47)(11,146,48)(12,147,49)(13,148,50)(14,149,51)(15,150,52)(16,151,53)(17,152,54)(18,153,37)(19,121,138)(20,122,139)(21,123,140)(22,124,141)(23,125,142)(24,126,143)(25,109,144)(26,110,127)(27,111,128)(28,112,129)(29,113,130)(30,114,131)(31,115,132)(32,116,133)(33,117,134)(34,118,135)(35,119,136)(36,120,137)(55,94,80)(56,95,81)(57,96,82)(58,97,83)(59,98,84)(60,99,85)(61,100,86)(62,101,87)(63,102,88)(64,103,89)(65,104,90)(66,105,73)(67,106,74)(68,107,75)(69,108,76)(70,91,77)(71,92,78)(72,93,79), (1,62,143)(2,63,144)(3,64,127)(4,65,128)(5,66,129)(6,67,130)(7,68,131)(8,69,132)(9,70,133)(10,71,134)(11,72,135)(12,55,136)(13,56,137)(14,57,138)(15,58,139)(16,59,140)(17,60,141)(18,61,142)(19,149,96)(20,150,97)(21,151,98)(22,152,99)(23,153,100)(24,154,101)(25,155,102)(26,156,103)(27,157,104)(28,158,105)(29,159,106)(30,160,107)(31,161,108)(32,162,91)(33,145,92)(34,146,93)(35,147,94)(36,148,95)(37,86,125)(38,87,126)(39,88,109)(40,89,110)(41,90,111)(42,73,112)(43,74,113)(44,75,114)(45,76,115)(46,77,116)(47,78,117)(48,79,118)(49,80,119)(50,81,120)(51,82,121)(52,83,122)(53,84,123)(54,85,124), (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( (1,154,38)(2,155,39)(3,156,40)(4,157,41)(5,158,42)(6,159,43)(7,160,44)(8,161,45)(9,162,46)(10,145,47)(11,146,48)(12,147,49)(13,148,50)(14,149,51)(15,150,52)(16,151,53)(17,152,54)(18,153,37)(19,121,138)(20,122,139)(21,123,140)(22,124,141)(23,125,142)(24,126,143)(25,109,144)(26,110,127)(27,111,128)(28,112,129)(29,113,130)(30,114,131)(31,115,132)(32,116,133)(33,117,134)(34,118,135)(35,119,136)(36,120,137)(55,94,80)(56,95,81)(57,96,82)(58,97,83)(59,98,84)(60,99,85)(61,100,86)(62,101,87)(63,102,88)(64,103,89)(65,104,90)(66,105,73)(67,106,74)(68,107,75)(69,108,76)(70,91,77)(71,92,78)(72,93,79), (1,62,143)(2,63,144)(3,64,127)(4,65,128)(5,66,129)(6,67,130)(7,68,131)(8,69,132)(9,70,133)(10,71,134)(11,72,135)(12,55,136)(13,56,137)(14,57,138)(15,58,139)(16,59,140)(17,60,141)(18,61,142)(19,149,96)(20,150,97)(21,151,98)(22,152,99)(23,153,100)(24,154,101)(25,155,102)(26,156,103)(27,157,104)(28,158,105)(29,159,106)(30,160,107)(31,161,108)(32,162,91)(33,145,92)(34,146,93)(35,147,94)(36,148,95)(37,86,125)(38,87,126)(39,88,109)(40,89,110)(41,90,111)(42,73,112)(43,74,113)(44,75,114)(45,76,115)(46,77,116)(47,78,117)(48,79,118)(49,80,119)(50,81,120)(51,82,121)(52,83,122)(53,84,123)(54,85,124), (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([[(1,154,38),(2,155,39),(3,156,40),(4,157,41),(5,158,42),(6,159,43),(7,160,44),(8,161,45),(9,162,46),(10,145,47),(11,146,48),(12,147,49),(13,148,50),(14,149,51),(15,150,52),(16,151,53),(17,152,54),(18,153,37),(19,121,138),(20,122,139),(21,123,140),(22,124,141),(23,125,142),(24,126,143),(25,109,144),(26,110,127),(27,111,128),(28,112,129),(29,113,130),(30,114,131),(31,115,132),(32,116,133),(33,117,134),(34,118,135),(35,119,136),(36,120,137),(55,94,80),(56,95,81),(57,96,82),(58,97,83),(59,98,84),(60,99,85),(61,100,86),(62,101,87),(63,102,88),(64,103,89),(65,104,90),(66,105,73),(67,106,74),(68,107,75),(69,108,76),(70,91,77),(71,92,78),(72,93,79)], [(1,62,143),(2,63,144),(3,64,127),(4,65,128),(5,66,129),(6,67,130),(7,68,131),(8,69,132),(9,70,133),(10,71,134),(11,72,135),(12,55,136),(13,56,137),(14,57,138),(15,58,139),(16,59,140),(17,60,141),(18,61,142),(19,149,96),(20,150,97),(21,151,98),(22,152,99),(23,153,100),(24,154,101),(25,155,102),(26,156,103),(27,157,104),(28,158,105),(29,159,106),(30,160,107),(31,161,108),(32,162,91),(33,145,92),(34,146,93),(35,147,94),(36,148,95),(37,86,125),(38,87,126),(39,88,109),(40,89,110),(41,90,111),(42,73,112),(43,74,113),(44,75,114),(45,76,115),(46,77,116),(47,78,117),(48,79,118),(49,80,119),(50,81,120),(51,82,121),(52,83,122),(53,84,123),(54,85,124)], [(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)]])
C32×C18 is a maximal subgroup of
C32⋊5Dic9
162 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 6A | ··· | 6Z | 9A | ··· | 9BB | 18A | ··· | 18BB |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C18 |
| kernel | C32×C18 | C32×C9 | C3×C18 | C32×C6 | C3×C9 | C33 | C3×C6 | C32 |
| # reps | 1 | 1 | 24 | 2 | 24 | 2 | 54 | 54 |
Matrix representation of C32×C18 ►in GL3(𝔽19) generated by
| 7 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 14 |
G:=sub<GL(3,GF(19))| [7,0,0,0,11,0,0,0,11],[1,0,0,0,11,0,0,0,1],[1,0,0,0,3,0,0,0,14] >;
C32×C18 in GAP, Magma, Sage, TeX
C_3^2\times C_{18} % in TeX
G:=Group("C3^2xC18"); // GroupNames label
G:=SmallGroup(162,47);
// by ID
G=gap.SmallGroup(162,47);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,276]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^18=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations