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G = C2×C92⋊9C3order 486 = 2·35

Direct product of C2 and C92⋊9C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C92⋊9C3
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊9C3 — C2×C92⋊9C3
 Lower central C1 — C32 — C2×C92⋊9C3
 Upper central C1 — C3×C6 — C2×C92⋊9C3

Generators and relations for C2×C929C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b7, dcd-1=c7 >

Subgroups: 252 in 140 conjugacy classes, 90 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C9, C32, C32, C18, C18, C3×C6, C3×C6, C3×C9, 3- 1+2, C33, C3×C18, C2×3- 1+2, C32×C6, C92, C9⋊C9, C3×3- 1+2, C9×C18, C2×C9⋊C9, C6×3- 1+2, C929C3, C2×C929C3
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C6×3- 1+2, C929C3, C2×C929C3

Smallest permutation representation of C2×C929C3
On 162 points
Generators in S162
(1 136)(2 137)(3 138)(4 139)(5 140)(6 141)(7 142)(8 143)(9 144)(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 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 82)(36 83)(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)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 109)(63 110)(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)
(1 116 89 80 70 53 26 43 16)(2 117 90 81 71 54 27 44 17)(3 109 82 73 72 46 19 45 18)(4 110 83 74 64 47 20 37 10)(5 111 84 75 65 48 21 38 11)(6 112 85 76 66 49 22 39 12)(7 113 86 77 67 50 23 40 13)(8 114 87 78 68 51 24 41 14)(9 115 88 79 69 52 25 42 15)(28 156 146 129 102 119 92 140 55)(29 157 147 130 103 120 93 141 56)(30 158 148 131 104 121 94 142 57)(31 159 149 132 105 122 95 143 58)(32 160 150 133 106 123 96 144 59)(33 161 151 134 107 124 97 136 60)(34 162 152 135 108 125 98 137 61)(35 154 153 127 100 126 99 138 62)(36 155 145 128 101 118 91 139 63)
(2 5 8)(3 9 6)(10 47 83)(11 51 90)(12 46 88)(13 50 86)(14 54 84)(15 49 82)(16 53 89)(17 48 87)(18 52 85)(19 25 22)(21 24 27)(28 95 135)(29 99 133)(30 94 131)(31 98 129)(32 93 127)(33 97 134)(34 92 132)(35 96 130)(36 91 128)(37 110 64)(38 114 71)(39 109 69)(40 113 67)(41 117 65)(42 112 72)(43 116 70)(44 111 68)(45 115 66)(55 149 125)(56 153 123)(57 148 121)(58 152 119)(59 147 126)(60 151 124)(61 146 122)(62 150 120)(63 145 118)(73 79 76)(75 78 81)(100 106 103)(102 105 108)(137 140 143)(138 144 141)(154 160 157)(156 159 162)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I 3J 6A ··· 6H 6I 6J 9A ··· 9X 9Y ··· 9AN 18A ··· 18X 18Y ··· 18AN order 1 2 3 ··· 3 3 3 6 ··· 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 9 9 1 ··· 1 9 9 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 3- 1+2 C2×3- 1+2 kernel C2×C92⋊9C3 C92⋊9C3 C9×C18 C2×C9⋊C9 C6×3- 1+2 C92 C9⋊C9 C3×3- 1+2 C18 C9 # reps 1 1 2 16 8 2 16 8 24 24

Matrix representation of C2×C929C3 in GL7(𝔽19)

 18 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 11 0 0 0 0 0 18 12 9 0 0 0 0 11 0 7 0 0 0 0 0 0 0 16 10 0 0 0 0 0 5 3 1 0 0 0 0 15 17 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 12 8 6 0 0 0 0 1 0 11 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 11 0 0 0 0 0 12 1 7 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 7 0 0 0 0 0 2 0 11

G:=sub<GL(7,GF(19))| [18,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,18,11,0,0,0,0,11,12,0,0,0,0,0,0,9,7,0,0,0,0,0,0,0,16,5,15,0,0,0,0,10,3,17,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,1,8,0,0,0,0,0,0,6,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[7,0,0,0,0,0,0,0,1,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,7,0,0,0,0,0,0,0,1,2,2,0,0,0,0,0,7,0,0,0,0,0,0,0,11] >;

C2×C929C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_9C_3
% in TeX

G:=Group("C2xC9^2:9C3");
// GroupNames label

G:=SmallGroup(486,206);
// by ID

G=gap.SmallGroup(486,206);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,176,2169,237]);
// 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^7,d*c*d^-1=c^7>;
// generators/relations

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