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## G = C18×3- 1+2order 486 = 2·35

### Direct product of C18 and 3- 1+2

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C18×3- 1+2
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9×3- 1+2 — C18×3- 1+2
 Lower central C1 — C3 — C18×3- 1+2
 Upper central C1 — C3×C18 — C18×3- 1+2

Generators and relations for C18×3- 1+2
G = < a,b,c | a18=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Subgroups: 216 in 148 conjugacy classes, 114 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, C92, C32⋊C9, C9⋊C9, C32×C9, C3×3- 1+2, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C32×C18, C6×3- 1+2, C9×3- 1+2, C18×3- 1+2
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C32×C9, C3×3- 1+2, C9○He3, C32×C18, C6×3- 1+2, C2×C9○He3, C9×3- 1+2, C18×3- 1+2

Smallest permutation representation of C18×3- 1+2
On 162 points
Generators in S162
(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)
(1 121 27 51 162 86 59 92 129)(2 122 28 52 145 87 60 93 130)(3 123 29 53 146 88 61 94 131)(4 124 30 54 147 89 62 95 132)(5 125 31 37 148 90 63 96 133)(6 126 32 38 149 73 64 97 134)(7 109 33 39 150 74 65 98 135)(8 110 34 40 151 75 66 99 136)(9 111 35 41 152 76 67 100 137)(10 112 36 42 153 77 68 101 138)(11 113 19 43 154 78 69 102 139)(12 114 20 44 155 79 70 103 140)(13 115 21 45 156 80 71 104 141)(14 116 22 46 157 81 72 105 142)(15 117 23 47 158 82 55 106 143)(16 118 24 48 159 83 56 107 144)(17 119 25 49 160 84 57 108 127)(18 120 26 50 161 85 58 91 128)
(19 78 139)(20 79 140)(21 80 141)(22 81 142)(23 82 143)(24 83 144)(25 84 127)(26 85 128)(27 86 129)(28 87 130)(29 88 131)(30 89 132)(31 90 133)(32 73 134)(33 74 135)(34 75 136)(35 76 137)(36 77 138)(91 161 120)(92 162 121)(93 145 122)(94 146 123)(95 147 124)(96 148 125)(97 149 126)(98 150 109)(99 151 110)(100 152 111)(101 153 112)(102 154 113)(103 155 114)(104 156 115)(105 157 116)(106 158 117)(107 159 118)(108 160 119)

G:=sub<Sym(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), (1,121,27,51,162,86,59,92,129)(2,122,28,52,145,87,60,93,130)(3,123,29,53,146,88,61,94,131)(4,124,30,54,147,89,62,95,132)(5,125,31,37,148,90,63,96,133)(6,126,32,38,149,73,64,97,134)(7,109,33,39,150,74,65,98,135)(8,110,34,40,151,75,66,99,136)(9,111,35,41,152,76,67,100,137)(10,112,36,42,153,77,68,101,138)(11,113,19,43,154,78,69,102,139)(12,114,20,44,155,79,70,103,140)(13,115,21,45,156,80,71,104,141)(14,116,22,46,157,81,72,105,142)(15,117,23,47,158,82,55,106,143)(16,118,24,48,159,83,56,107,144)(17,119,25,49,160,84,57,108,127)(18,120,26,50,161,85,58,91,128), (19,78,139)(20,79,140)(21,80,141)(22,81,142)(23,82,143)(24,83,144)(25,84,127)(26,85,128)(27,86,129)(28,87,130)(29,88,131)(30,89,132)(31,90,133)(32,73,134)(33,74,135)(34,75,136)(35,76,137)(36,77,138)(91,161,120)(92,162,121)(93,145,122)(94,146,123)(95,147,124)(96,148,125)(97,149,126)(98,150,109)(99,151,110)(100,152,111)(101,153,112)(102,154,113)(103,155,114)(104,156,115)(105,157,116)(106,158,117)(107,159,118)(108,160,119)>;

G:=Group( (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), (1,121,27,51,162,86,59,92,129)(2,122,28,52,145,87,60,93,130)(3,123,29,53,146,88,61,94,131)(4,124,30,54,147,89,62,95,132)(5,125,31,37,148,90,63,96,133)(6,126,32,38,149,73,64,97,134)(7,109,33,39,150,74,65,98,135)(8,110,34,40,151,75,66,99,136)(9,111,35,41,152,76,67,100,137)(10,112,36,42,153,77,68,101,138)(11,113,19,43,154,78,69,102,139)(12,114,20,44,155,79,70,103,140)(13,115,21,45,156,80,71,104,141)(14,116,22,46,157,81,72,105,142)(15,117,23,47,158,82,55,106,143)(16,118,24,48,159,83,56,107,144)(17,119,25,49,160,84,57,108,127)(18,120,26,50,161,85,58,91,128), (19,78,139)(20,79,140)(21,80,141)(22,81,142)(23,82,143)(24,83,144)(25,84,127)(26,85,128)(27,86,129)(28,87,130)(29,88,131)(30,89,132)(31,90,133)(32,73,134)(33,74,135)(34,75,136)(35,76,137)(36,77,138)(91,161,120)(92,162,121)(93,145,122)(94,146,123)(95,147,124)(96,148,125)(97,149,126)(98,150,109)(99,151,110)(100,152,111)(101,153,112)(102,154,113)(103,155,114)(104,156,115)(105,157,116)(106,158,117)(107,159,118)(108,160,119) );

G=PermutationGroup([[(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)], [(1,121,27,51,162,86,59,92,129),(2,122,28,52,145,87,60,93,130),(3,123,29,53,146,88,61,94,131),(4,124,30,54,147,89,62,95,132),(5,125,31,37,148,90,63,96,133),(6,126,32,38,149,73,64,97,134),(7,109,33,39,150,74,65,98,135),(8,110,34,40,151,75,66,99,136),(9,111,35,41,152,76,67,100,137),(10,112,36,42,153,77,68,101,138),(11,113,19,43,154,78,69,102,139),(12,114,20,44,155,79,70,103,140),(13,115,21,45,156,80,71,104,141),(14,116,22,46,157,81,72,105,142),(15,117,23,47,158,82,55,106,143),(16,118,24,48,159,83,56,107,144),(17,119,25,49,160,84,57,108,127),(18,120,26,50,161,85,58,91,128)], [(19,78,139),(20,79,140),(21,80,141),(22,81,142),(23,82,143),(24,83,144),(25,84,127),(26,85,128),(27,86,129),(28,87,130),(29,88,131),(30,89,132),(31,90,133),(32,73,134),(33,74,135),(34,75,136),(35,76,137),(36,77,138),(91,161,120),(92,162,121),(93,145,122),(94,146,123),(95,147,124),(96,148,125),(97,149,126),(98,150,109),(99,151,110),(100,152,111),(101,153,112),(102,154,113),(103,155,114),(104,156,115),(105,157,116),(106,158,117),(107,159,118),(108,160,119)]])

198 conjugacy classes

 class 1 2 3A ··· 3H 3I ··· 3N 6A ··· 6H 6I ··· 6N 9A ··· 9R 9S ··· 9CF 18A ··· 18R 18S ··· 18CF 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 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C3 C3 C6 C6 C6 C6 C6 C9 C18 3- 1+2 C2×3- 1+2 C9○He3 C2×C9○He3 kernel C18×3- 1+2 C9×3- 1+2 C9×C18 C2×C32⋊C9 C2×C9⋊C9 C32×C18 C6×3- 1+2 C92 C32⋊C9 C9⋊C9 C32×C9 C3×3- 1+2 C2×3- 1+2 3- 1+2 C18 C9 C6 C3 # reps 1 1 6 4 12 2 2 6 4 12 2 2 54 54 6 6 12 12

Matrix representation of C18×3- 1+2 in GL4(𝔽19) generated by

 10 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 3 9 3 0 0 0 11 0 9 14 16
,
 11 0 0 0 0 1 14 2 0 0 11 0 0 0 0 7
G:=sub<GL(4,GF(19))| [10,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,3,0,9,0,9,0,14,0,3,11,16],[11,0,0,0,0,1,0,0,0,14,11,0,0,2,0,7] >;

C18×3- 1+2 in GAP, Magma, Sage, TeX

C_{18}\times 3_-^{1+2}
% in TeX

G:=Group("C18xES-(3,1)");
// GroupNames label

G:=SmallGroup(486,195);
// by ID

G=gap.SmallGroup(486,195);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,176,237]);
// Polycyclic

G:=Group<a,b,c|a^18=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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