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## G = C2×C9⋊D9order 324 = 22·34

### Direct product of C2 and C9⋊D9

Aliases: C2×C9⋊D9, C18⋊D9, C92D18, C923C22, (C9×C18)⋊2C2, (C3×C9).8D6, C6.3(C9⋊S3), (C3×C18).22S3, C3.1(C2×C9⋊S3), C32.9(C2×C3⋊S3), (C3×C6).17(C3⋊S3), SmallGroup(324,74)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C92 — C2×C9⋊D9
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C9⋊D9 — C2×C9⋊D9
 Lower central C92 — C2×C9⋊D9
 Upper central C1 — C2

Generators and relations for C2×C9⋊D9
G = < a,b,c,d | a2=b9=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1003 in 115 conjugacy classes, 49 normal (7 characteristic)
C1, C2, C2, C3, C22, S3, C6, C9, C32, D6, D9, C18, C3⋊S3, C3×C6, C3×C9, D18, C2×C3⋊S3, C9⋊S3, C3×C18, C92, C2×C9⋊S3, C9⋊D9, C9×C18, C2×C9⋊D9
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, C2×C3⋊S3, C9⋊S3, C2×C9⋊S3, C9⋊D9, C2×C9⋊D9

Smallest permutation representation of C2×C9⋊D9
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 90)(20 82)(21 83)(22 84)(23 85)(24 86)(25 87)(26 88)(27 89)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 133)(44 134)(45 135)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 100)(54 101)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 151)(62 152)(63 153)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 118)(72 119)(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 40 126 103 80 20 31 14 55)(2 41 118 104 81 21 32 15 56)(3 42 119 105 73 22 33 16 57)(4 43 120 106 74 23 34 17 58)(5 44 121 107 75 24 35 18 59)(6 45 122 108 76 25 36 10 60)(7 37 123 100 77 26 28 11 61)(8 38 124 101 78 27 29 12 62)(9 39 125 102 79 19 30 13 63)(46 160 90 111 94 153 144 129 69)(47 161 82 112 95 145 136 130 70)(48 162 83 113 96 146 137 131 71)(49 154 84 114 97 147 138 132 72)(50 155 85 115 98 148 139 133 64)(51 156 86 116 99 149 140 134 65)(52 157 87 117 91 150 141 135 66)(53 158 88 109 92 151 142 127 67)(54 159 89 110 93 152 143 128 68)
(1 55)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 45)(19 104)(20 103)(21 102)(22 101)(23 100)(24 108)(25 107)(26 106)(27 105)(28 120)(29 119)(30 118)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(46 83)(47 82)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(64 109)(65 117)(66 116)(67 115)(68 114)(69 113)(70 112)(71 111)(72 110)(73 78)(74 77)(75 76)(79 81)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 135)(136 145)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(154 159)(155 158)(156 157)(160 162)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 9A ··· 9AJ 18A ··· 18AJ order 1 2 2 2 3 3 3 3 6 6 6 6 9 ··· 9 18 ··· 18 size 1 1 81 81 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 S3 D6 D9 D18 kernel C2×C9⋊D9 C9⋊D9 C9×C18 C3×C18 C3×C9 C18 C9 # reps 1 2 1 4 4 36 36

Matrix representation of C2×C9⋊D9 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 0 1 0 0 18 18 0 0 0 0 5 7 0 0 12 17
,
 2 14 0 0 5 7 0 0 0 0 5 7 0 0 12 17
,
 2 7 0 0 5 17 0 0 0 0 12 17 0 0 5 7
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,18,0,0,1,18,0,0,0,0,5,12,0,0,7,17],[2,5,0,0,14,7,0,0,0,0,5,12,0,0,7,17],[2,5,0,0,7,17,0,0,0,0,12,5,0,0,17,7] >;

C2×C9⋊D9 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes D_9
% in TeX

G:=Group("C2xC9:D9");
// GroupNames label

G:=SmallGroup(324,74);
// by ID

G=gap.SmallGroup(324,74);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,794,338,3171,453,2164,7781]);
// Polycyclic

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

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