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## G = C2×C92⋊3C3order 486 = 2·35

### Direct product of C2 and C92⋊3C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×C92⋊3C3
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — C92⋊3C3 — C2×C92⋊3C3
 Lower central C1 — C3 — C2×C92⋊3C3
 Upper central C1 — C3×C18 — C2×C92⋊3C3

Generators and relations for C2×C923C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc3, cd=dc >

Subgroups: 198 in 138 conjugacy classes, 108 normal (14 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, C33, C3×C18, C3×C18, C32×C6, C92, C32⋊C9, C9⋊C9, C32×C9, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C32×C18, C923C3, C2×C923C3
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, C33, C3×C18, C32×C6, C32×C9, C9○He3, C32×C18, C2×C9○He3, C923C3, C2×C923C3

Smallest permutation representation of C2×C923C3
On 162 points
Generators in S162
(1 131)(2 132)(3 133)(4 134)(5 135)(6 127)(7 128)(8 129)(9 130)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 88)(29 89)(30 90)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 140)(38 141)(39 142)(40 143)(41 144)(42 136)(43 137)(44 138)(45 139)(46 151)(47 152)(48 153)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 115)(56 116)(57 117)(58 109)(59 110)(60 111)(61 112)(62 113)(63 114)(64 121)(65 122)(66 123)(67 124)(68 125)(69 126)(70 118)(71 119)(72 120)(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)
(1 116 88 80 45 65 13 21 50)(2 117 89 81 37 66 14 22 51)(3 109 90 73 38 67 15 23 52)(4 110 82 74 39 68 16 24 53)(5 111 83 75 40 69 17 25 54)(6 112 84 76 41 70 18 26 46)(7 113 85 77 42 71 10 27 47)(8 114 86 78 43 72 11 19 48)(9 115 87 79 44 64 12 20 49)(28 161 139 122 94 102 146 131 56)(29 162 140 123 95 103 147 132 57)(30 154 141 124 96 104 148 133 58)(31 155 142 125 97 105 149 134 59)(32 156 143 126 98 106 150 135 60)(33 157 144 118 99 107 151 127 61)(34 158 136 119 91 108 152 128 62)(35 159 137 120 92 100 153 129 63)(36 160 138 121 93 101 145 130 55)
(1 7 4)(2 11 75)(3 79 18)(5 14 78)(6 73 12)(8 17 81)(9 76 15)(10 16 13)(19 40 117)(20 112 38)(21 27 24)(22 43 111)(23 115 41)(25 37 114)(26 109 44)(28 34 31)(29 153 126)(30 121 151)(32 147 120)(33 124 145)(35 150 123)(36 118 148)(39 45 42)(46 90 64)(47 53 50)(48 69 89)(49 84 67)(51 72 83)(52 87 70)(54 66 86)(55 144 104)(56 62 59)(57 100 143)(58 138 107)(60 103 137)(61 141 101)(63 106 140)(65 71 68)(74 80 77)(82 88 85)(91 97 94)(92 156 132)(93 127 154)(95 159 135)(96 130 157)(98 162 129)(99 133 160)(102 108 105)(110 116 113)(119 125 122)(128 134 131)(136 142 139)(146 152 149)(155 161 158)

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

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

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

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 3 3 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 C9 C18 C9○He3 C2×C9○He3 kernel C2×C92⋊3C3 C92⋊3C3 C9×C18 C2×C32⋊C9 C2×C9⋊C9 C32×C18 C92 C32⋊C9 C9⋊C9 C32×C9 C3×C18 C3×C9 C6 C3 # reps 1 1 6 6 12 2 6 6 12 2 54 54 18 18

Matrix representation of C2×C923C3 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 6 0 0 0 0 7 3 10 0 14 10 3 0 6 7 2
,
 11 0 0 0 0 5 0 0 0 0 5 0 0 0 0 5
,
 7 0 0 0 0 1 1 11 0 0 11 16 0 0 0 7
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[6,0,0,0,0,7,14,6,0,3,10,7,0,10,3,2],[11,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[7,0,0,0,0,1,0,0,0,1,11,0,0,11,16,7] >;

C2×C923C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_3C_3
% in TeX

G:=Group("C2xC9^2:3C3");
// GroupNames label

G:=SmallGroup(486,193);
// by ID

G=gap.SmallGroup(486,193);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,979,548,93]);
// 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,b*c=c*b,d*b*d^-1=b*c^3,c*d=d*c>;
// generators/relations

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