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## G = C2×C9⋊C9order 162 = 2·34

### Direct product of C2 and C9⋊C9

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

Aliases: C2×C9⋊C9, C18⋊C9, C92C18, C6.23- 1+2, (C3×C9).2C6, C6.2(C3×C9), C3.2(C3×C18), (C3×C18).1C3, (C3×C6).7C32, C32.10(C3×C6), C3.2(C2×3- 1+2), SmallGroup(162,25)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C9⋊C9
G = < a,b,c | a2=b9=c9=1, ab=ba, ac=ca, cbc-1=b7 >

Smallest permutation representation of C2×C9⋊C9
Regular action on 162 points
Generators in S162
(1 114)(2 115)(3 116)(4 117)(5 109)(6 110)(7 111)(8 112)(9 113)(10 90)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 99)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 102)(29 103)(30 104)(31 105)(32 106)(33 107)(34 108)(35 100)(36 101)(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 78 51 31 93 66 39 87 60)(2 73 49 32 97 64 40 82 58)(3 77 47 33 92 71 41 86 56)(4 81 54 34 96 69 42 90 63)(5 76 52 35 91 67 43 85 61)(6 80 50 36 95 65 44 89 59)(7 75 48 28 99 72 45 84 57)(8 79 46 29 94 70 37 88 55)(9 74 53 30 98 68 38 83 62)(10 144 117 162 135 108 25 150 123)(11 139 115 154 130 106 26 145 121)(12 143 113 155 134 104 27 149 119)(13 138 111 156 129 102 19 153 126)(14 142 109 157 133 100 20 148 124)(15 137 116 158 128 107 21 152 122)(16 141 114 159 132 105 22 147 120)(17 136 112 160 127 103 23 151 118)(18 140 110 161 131 101 24 146 125)

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

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

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

C2×C9⋊C9 is a maximal subgroup of   C9⋊C36  C18×3- 1+2

66 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9X 18A ··· 18X order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 1 ··· 1 3 ··· 3 3 ··· 3

66 irreducible representations

 dim 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C6 C9 C18 3- 1+2 C2×3- 1+2 kernel C2×C9⋊C9 C9⋊C9 C3×C18 C3×C9 C18 C9 C6 C3 # reps 1 1 8 8 18 18 6 6

Matrix representation of C2×C9⋊C9 in GL4(𝔽19) generated by

 18 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 0 7 0 0 8 18 9 0 1 0 1
,
 5 0 0 0 0 11 8 15 0 2 8 1 0 15 18 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,0,8,1,0,7,18,0,0,0,9,1],[5,0,0,0,0,11,2,15,0,8,8,18,0,15,1,0] >;

C2×C9⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_9
% in TeX

G:=Group("C2xC9:C9");
// GroupNames label

G:=SmallGroup(162,25);
// by ID

G=gap.SmallGroup(162,25);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,276,187,57]);
// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^7>;
// generators/relations

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