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## G = C2×C9⋊3- 1+2order 486 = 2·35

### Direct product of C2 and C9⋊3- 1+2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C9⋊3- 1+2
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C9⋊3- 1+2 — C2×C9⋊3- 1+2
 Lower central C1 — C32 — C2×C9⋊3- 1+2
 Upper central C1 — C3×C6 — C2×C9⋊3- 1+2

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

Smallest permutation representation of 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

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