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## G = C6×C32⋊C9order 486 = 2·35

### Direct product of C6 and C32⋊C9

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C6×C32⋊C9
 Chief series C1 — C3 — C32 — C33 — C34 — C3×C32⋊C9 — C6×C32⋊C9
 Lower central C1 — C3 — C6×C32⋊C9
 Upper central C1 — C32×C6 — C6×C32⋊C9

Generators and relations for C6×C32⋊C9
G = < a,b,c,d | a6=b3=c3=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, cd=dc >

Subgroups: 576 in 288 conjugacy classes, 144 normal (14 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C32, C32, C32, C18, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C33, C33, C33, C3×C18, C3×C18, C32×C6, C32×C6, C32×C6, C32⋊C9, C32×C9, C34, C2×C32⋊C9, C32×C18, C33×C6, C3×C32⋊C9, C6×C32⋊C9
Quotients: C1, C2, C3, C6, C9, C32, C18, C3×C6, C3×C9, He3, 3- 1+2, C33, C3×C18, C2×He3, C2×3- 1+2, C32×C6, C32⋊C9, C32×C9, C3×He3, C3×3- 1+2, C2×C32⋊C9, C32×C18, C6×He3, C6×3- 1+2, C3×C32⋊C9, C6×C32⋊C9

Smallest permutation representation of C6×C32⋊C9
On 162 points
Generators in S162
(1 155 55 109 65 145)(2 156 56 110 66 146)(3 157 57 111 67 147)(4 158 58 112 68 148)(5 159 59 113 69 149)(6 160 60 114 70 150)(7 161 61 115 71 151)(8 162 62 116 72 152)(9 154 63 117 64 153)(10 74 31 108 134 121)(11 75 32 100 135 122)(12 76 33 101 127 123)(13 77 34 102 128 124)(14 78 35 103 129 125)(15 79 36 104 130 126)(16 80 28 105 131 118)(17 81 29 106 132 119)(18 73 30 107 133 120)(19 44 49 82 95 140)(20 45 50 83 96 141)(21 37 51 84 97 142)(22 38 52 85 98 143)(23 39 53 86 99 144)(24 40 54 87 91 136)(25 41 46 88 92 137)(26 42 47 89 93 138)(27 43 48 90 94 139)
(1 20 16)(2 35 91)(3 70 63)(4 23 10)(5 29 94)(6 64 57)(7 26 13)(8 32 97)(9 67 60)(11 51 72)(12 130 30)(14 54 66)(15 133 33)(17 48 69)(18 127 36)(19 98 46)(21 62 135)(22 92 49)(24 56 129)(25 95 52)(27 59 132)(28 55 50)(31 58 53)(34 61 47)(37 116 122)(38 137 82)(39 74 158)(40 110 125)(41 140 85)(42 77 161)(43 113 119)(44 143 88)(45 80 155)(65 96 131)(68 99 134)(71 93 128)(73 123 104)(75 84 152)(76 126 107)(78 87 146)(79 120 101)(81 90 149)(83 105 109)(86 108 112)(89 102 115)(100 142 162)(103 136 156)(106 139 159)(111 160 153)(114 154 147)(117 157 150)(118 145 141)(121 148 144)(124 151 138)
(1 134 47)(2 135 48)(3 127 49)(4 128 50)(5 129 51)(6 130 52)(7 131 53)(8 132 54)(9 133 46)(10 93 55)(11 94 56)(12 95 57)(13 96 58)(14 97 59)(15 98 60)(16 99 61)(17 91 62)(18 92 63)(19 67 33)(20 68 34)(21 69 35)(22 70 36)(23 71 28)(24 72 29)(25 64 30)(26 65 31)(27 66 32)(37 149 103)(38 150 104)(39 151 105)(40 152 106)(41 153 107)(42 145 108)(43 146 100)(44 147 101)(45 148 102)(73 137 117)(74 138 109)(75 139 110)(76 140 111)(77 141 112)(78 142 113)(79 143 114)(80 144 115)(81 136 116)(82 157 123)(83 158 124)(84 159 125)(85 160 126)(86 161 118)(87 162 119)(88 154 120)(89 155 121)(90 156 122)
(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)| (1,155,55,109,65,145)(2,156,56,110,66,146)(3,157,57,111,67,147)(4,158,58,112,68,148)(5,159,59,113,69,149)(6,160,60,114,70,150)(7,161,61,115,71,151)(8,162,62,116,72,152)(9,154,63,117,64,153)(10,74,31,108,134,121)(11,75,32,100,135,122)(12,76,33,101,127,123)(13,77,34,102,128,124)(14,78,35,103,129,125)(15,79,36,104,130,126)(16,80,28,105,131,118)(17,81,29,106,132,119)(18,73,30,107,133,120)(19,44,49,82,95,140)(20,45,50,83,96,141)(21,37,51,84,97,142)(22,38,52,85,98,143)(23,39,53,86,99,144)(24,40,54,87,91,136)(25,41,46,88,92,137)(26,42,47,89,93,138)(27,43,48,90,94,139), (1,20,16)(2,35,91)(3,70,63)(4,23,10)(5,29,94)(6,64,57)(7,26,13)(8,32,97)(9,67,60)(11,51,72)(12,130,30)(14,54,66)(15,133,33)(17,48,69)(18,127,36)(19,98,46)(21,62,135)(22,92,49)(24,56,129)(25,95,52)(27,59,132)(28,55,50)(31,58,53)(34,61,47)(37,116,122)(38,137,82)(39,74,158)(40,110,125)(41,140,85)(42,77,161)(43,113,119)(44,143,88)(45,80,155)(65,96,131)(68,99,134)(71,93,128)(73,123,104)(75,84,152)(76,126,107)(78,87,146)(79,120,101)(81,90,149)(83,105,109)(86,108,112)(89,102,115)(100,142,162)(103,136,156)(106,139,159)(111,160,153)(114,154,147)(117,157,150)(118,145,141)(121,148,144)(124,151,138), (1,134,47)(2,135,48)(3,127,49)(4,128,50)(5,129,51)(6,130,52)(7,131,53)(8,132,54)(9,133,46)(10,93,55)(11,94,56)(12,95,57)(13,96,58)(14,97,59)(15,98,60)(16,99,61)(17,91,62)(18,92,63)(19,67,33)(20,68,34)(21,69,35)(22,70,36)(23,71,28)(24,72,29)(25,64,30)(26,65,31)(27,66,32)(37,149,103)(38,150,104)(39,151,105)(40,152,106)(41,153,107)(42,145,108)(43,146,100)(44,147,101)(45,148,102)(73,137,117)(74,138,109)(75,139,110)(76,140,111)(77,141,112)(78,142,113)(79,143,114)(80,144,115)(81,136,116)(82,157,123)(83,158,124)(84,159,125)(85,160,126)(86,161,118)(87,162,119)(88,154,120)(89,155,121)(90,156,122), (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( (1,155,55,109,65,145)(2,156,56,110,66,146)(3,157,57,111,67,147)(4,158,58,112,68,148)(5,159,59,113,69,149)(6,160,60,114,70,150)(7,161,61,115,71,151)(8,162,62,116,72,152)(9,154,63,117,64,153)(10,74,31,108,134,121)(11,75,32,100,135,122)(12,76,33,101,127,123)(13,77,34,102,128,124)(14,78,35,103,129,125)(15,79,36,104,130,126)(16,80,28,105,131,118)(17,81,29,106,132,119)(18,73,30,107,133,120)(19,44,49,82,95,140)(20,45,50,83,96,141)(21,37,51,84,97,142)(22,38,52,85,98,143)(23,39,53,86,99,144)(24,40,54,87,91,136)(25,41,46,88,92,137)(26,42,47,89,93,138)(27,43,48,90,94,139), (1,20,16)(2,35,91)(3,70,63)(4,23,10)(5,29,94)(6,64,57)(7,26,13)(8,32,97)(9,67,60)(11,51,72)(12,130,30)(14,54,66)(15,133,33)(17,48,69)(18,127,36)(19,98,46)(21,62,135)(22,92,49)(24,56,129)(25,95,52)(27,59,132)(28,55,50)(31,58,53)(34,61,47)(37,116,122)(38,137,82)(39,74,158)(40,110,125)(41,140,85)(42,77,161)(43,113,119)(44,143,88)(45,80,155)(65,96,131)(68,99,134)(71,93,128)(73,123,104)(75,84,152)(76,126,107)(78,87,146)(79,120,101)(81,90,149)(83,105,109)(86,108,112)(89,102,115)(100,142,162)(103,136,156)(106,139,159)(111,160,153)(114,154,147)(117,157,150)(118,145,141)(121,148,144)(124,151,138), (1,134,47)(2,135,48)(3,127,49)(4,128,50)(5,129,51)(6,130,52)(7,131,53)(8,132,54)(9,133,46)(10,93,55)(11,94,56)(12,95,57)(13,96,58)(14,97,59)(15,98,60)(16,99,61)(17,91,62)(18,92,63)(19,67,33)(20,68,34)(21,69,35)(22,70,36)(23,71,28)(24,72,29)(25,64,30)(26,65,31)(27,66,32)(37,149,103)(38,150,104)(39,151,105)(40,152,106)(41,153,107)(42,145,108)(43,146,100)(44,147,101)(45,148,102)(73,137,117)(74,138,109)(75,139,110)(76,140,111)(77,141,112)(78,142,113)(79,143,114)(80,144,115)(81,136,116)(82,157,123)(83,158,124)(84,159,125)(85,160,126)(86,161,118)(87,162,119)(88,154,120)(89,155,121)(90,156,122), (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([[(1,155,55,109,65,145),(2,156,56,110,66,146),(3,157,57,111,67,147),(4,158,58,112,68,148),(5,159,59,113,69,149),(6,160,60,114,70,150),(7,161,61,115,71,151),(8,162,62,116,72,152),(9,154,63,117,64,153),(10,74,31,108,134,121),(11,75,32,100,135,122),(12,76,33,101,127,123),(13,77,34,102,128,124),(14,78,35,103,129,125),(15,79,36,104,130,126),(16,80,28,105,131,118),(17,81,29,106,132,119),(18,73,30,107,133,120),(19,44,49,82,95,140),(20,45,50,83,96,141),(21,37,51,84,97,142),(22,38,52,85,98,143),(23,39,53,86,99,144),(24,40,54,87,91,136),(25,41,46,88,92,137),(26,42,47,89,93,138),(27,43,48,90,94,139)], [(1,20,16),(2,35,91),(3,70,63),(4,23,10),(5,29,94),(6,64,57),(7,26,13),(8,32,97),(9,67,60),(11,51,72),(12,130,30),(14,54,66),(15,133,33),(17,48,69),(18,127,36),(19,98,46),(21,62,135),(22,92,49),(24,56,129),(25,95,52),(27,59,132),(28,55,50),(31,58,53),(34,61,47),(37,116,122),(38,137,82),(39,74,158),(40,110,125),(41,140,85),(42,77,161),(43,113,119),(44,143,88),(45,80,155),(65,96,131),(68,99,134),(71,93,128),(73,123,104),(75,84,152),(76,126,107),(78,87,146),(79,120,101),(81,90,149),(83,105,109),(86,108,112),(89,102,115),(100,142,162),(103,136,156),(106,139,159),(111,160,153),(114,154,147),(117,157,150),(118,145,141),(121,148,144),(124,151,138)], [(1,134,47),(2,135,48),(3,127,49),(4,128,50),(5,129,51),(6,130,52),(7,131,53),(8,132,54),(9,133,46),(10,93,55),(11,94,56),(12,95,57),(13,96,58),(14,97,59),(15,98,60),(16,99,61),(17,91,62),(18,92,63),(19,67,33),(20,68,34),(21,69,35),(22,70,36),(23,71,28),(24,72,29),(25,64,30),(26,65,31),(27,66,32),(37,149,103),(38,150,104),(39,151,105),(40,152,106),(41,153,107),(42,145,108),(43,146,100),(44,147,101),(45,148,102),(73,137,117),(74,138,109),(75,139,110),(76,140,111),(77,141,112),(78,142,113),(79,143,114),(80,144,115),(81,136,116),(82,157,123),(83,158,124),(84,159,125),(85,160,126),(86,161,118),(87,162,119),(88,154,120),(89,155,121),(90,156,122)], [(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)]])

198 conjugacy classes

 class 1 2 3A ··· 3Z 3AA ··· 3AR 6A ··· 6Z 6AA ··· 6AR 9A ··· 9BB 18A ··· 18BB order 1 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 3 ··· 3 3 ··· 3

198 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 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 kernel C6×C32⋊C9 C3×C32⋊C9 C2×C32⋊C9 C32×C18 C33×C6 C32⋊C9 C32×C9 C34 C32×C6 C33 C3×C6 C3×C6 C32 C32 # reps 1 1 18 6 2 18 6 2 54 54 6 12 6 12

Matrix representation of C6×C32⋊C9 in GL5(𝔽19)

 12 0 0 0 0 0 12 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 7 0 0 0 0 0 1 0 0 0 0 0 1 9 2 0 0 0 11 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11 0 0 0 0 0 11
,
 9 0 0 0 0 0 7 0 0 0 0 0 17 1 6 0 0 0 0 1 0 0 13 13 2

G:=sub<GL(5,GF(19))| [12,0,0,0,0,0,12,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[7,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,9,11,0,0,0,2,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[9,0,0,0,0,0,7,0,0,0,0,0,17,0,13,0,0,1,0,13,0,0,6,1,2] >;

C6×C32⋊C9 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes C_9
% in TeX

G:=Group("C6xC3^2:C9");
// GroupNames label

G:=SmallGroup(486,191);
// by ID

G=gap.SmallGroup(486,191);
# by ID

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

G:=Group<a,b,c,d|a^6=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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