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## G = C3×C9.A4order 324 = 22·34

### Direct product of C3 and C9.A4

Aliases: C3×C9.A4, C62.1C9, (C2×C6)⋊C27, C9.4(C3×A4), (C3×C9).5A4, (C2×C18).2C9, (C6×C18).1C3, C222(C3×C27), C9.3(C3.A4), (C2×C18).4C32, C32.2(C3.A4), (C2×C6).5(C3×C9), C3.1(C3×C3.A4), SmallGroup(324,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C9.A4
 Chief series C1 — C22 — C2×C6 — C2×C18 — C9.A4 — C3×C9.A4
 Lower central C22 — C3×C9.A4
 Upper central C1 — C3×C9

Generators and relations for C3×C9.A4
G = < a,b,c,d,e | a3=b9=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Smallest permutation representation of C3×C9.A4
On 162 points
Generators in S162
(1 28 66)(2 29 67)(3 30 68)(4 31 69)(5 32 70)(6 33 71)(7 34 72)(8 35 73)(9 36 74)(10 37 75)(11 38 76)(12 39 77)(13 40 78)(14 41 79)(15 42 80)(16 43 81)(17 44 55)(18 45 56)(19 46 57)(20 47 58)(21 48 59)(22 49 60)(23 50 61)(24 51 62)(25 52 63)(26 53 64)(27 54 65)(82 127 148)(83 128 149)(84 129 150)(85 130 151)(86 131 152)(87 132 153)(88 133 154)(89 134 155)(90 135 156)(91 109 157)(92 110 158)(93 111 159)(94 112 160)(95 113 161)(96 114 162)(97 115 136)(98 116 137)(99 117 138)(100 118 139)(101 119 140)(102 120 141)(103 121 142)(104 122 143)(105 123 144)(106 124 145)(107 125 146)(108 126 147)
(1 4 7 10 13 16 19 22 25)(2 5 8 11 14 17 20 23 26)(3 6 9 12 15 18 21 24 27)(28 31 34 37 40 43 46 49 52)(29 32 35 38 41 44 47 50 53)(30 33 36 39 42 45 48 51 54)(55 58 61 64 67 70 73 76 79)(56 59 62 65 68 71 74 77 80)(57 60 63 66 69 72 75 78 81)(82 85 88 91 94 97 100 103 106)(83 86 89 92 95 98 101 104 107)(84 87 90 93 96 99 102 105 108)(109 112 115 118 121 124 127 130 133)(110 113 116 119 122 125 128 131 134)(111 114 117 120 123 126 129 132 135)(136 139 142 145 148 151 154 157 160)(137 140 143 146 149 152 155 158 161)(138 141 144 147 150 153 156 159 162)
(2 110)(3 111)(5 113)(6 114)(8 116)(9 117)(11 119)(12 120)(14 122)(15 123)(17 125)(18 126)(20 128)(21 129)(23 131)(24 132)(26 134)(27 135)(29 158)(30 159)(32 161)(33 162)(35 137)(36 138)(38 140)(39 141)(41 143)(42 144)(44 146)(45 147)(47 149)(48 150)(50 152)(51 153)(53 155)(54 156)(55 107)(56 108)(58 83)(59 84)(61 86)(62 87)(64 89)(65 90)(67 92)(68 93)(70 95)(71 96)(73 98)(74 99)(76 101)(77 102)(79 104)(80 105)
(1 109)(3 111)(4 112)(6 114)(7 115)(9 117)(10 118)(12 120)(13 121)(15 123)(16 124)(18 126)(19 127)(21 129)(22 130)(24 132)(25 133)(27 135)(28 157)(30 159)(31 160)(33 162)(34 136)(36 138)(37 139)(39 141)(40 142)(42 144)(43 145)(45 147)(46 148)(48 150)(49 151)(51 153)(52 154)(54 156)(56 108)(57 82)(59 84)(60 85)(62 87)(63 88)(65 90)(66 91)(68 93)(69 94)(71 96)(72 97)(74 99)(75 100)(77 102)(78 103)(80 105)(81 106)
(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)| (1,28,66)(2,29,67)(3,30,68)(4,31,69)(5,32,70)(6,33,71)(7,34,72)(8,35,73)(9,36,74)(10,37,75)(11,38,76)(12,39,77)(13,40,78)(14,41,79)(15,42,80)(16,43,81)(17,44,55)(18,45,56)(19,46,57)(20,47,58)(21,48,59)(22,49,60)(23,50,61)(24,51,62)(25,52,63)(26,53,64)(27,54,65)(82,127,148)(83,128,149)(84,129,150)(85,130,151)(86,131,152)(87,132,153)(88,133,154)(89,134,155)(90,135,156)(91,109,157)(92,110,158)(93,111,159)(94,112,160)(95,113,161)(96,114,162)(97,115,136)(98,116,137)(99,117,138)(100,118,139)(101,119,140)(102,120,141)(103,121,142)(104,122,143)(105,123,144)(106,124,145)(107,125,146)(108,126,147), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (2,110)(3,111)(5,113)(6,114)(8,116)(9,117)(11,119)(12,120)(14,122)(15,123)(17,125)(18,126)(20,128)(21,129)(23,131)(24,132)(26,134)(27,135)(29,158)(30,159)(32,161)(33,162)(35,137)(36,138)(38,140)(39,141)(41,143)(42,144)(44,146)(45,147)(47,149)(48,150)(50,152)(51,153)(53,155)(54,156)(55,107)(56,108)(58,83)(59,84)(61,86)(62,87)(64,89)(65,90)(67,92)(68,93)(70,95)(71,96)(73,98)(74,99)(76,101)(77,102)(79,104)(80,105), (1,109)(3,111)(4,112)(6,114)(7,115)(9,117)(10,118)(12,120)(13,121)(15,123)(16,124)(18,126)(19,127)(21,129)(22,130)(24,132)(25,133)(27,135)(28,157)(30,159)(31,160)(33,162)(34,136)(36,138)(37,139)(39,141)(40,142)(42,144)(43,145)(45,147)(46,148)(48,150)(49,151)(51,153)(52,154)(54,156)(56,108)(57,82)(59,84)(60,85)(62,87)(63,88)(65,90)(66,91)(68,93)(69,94)(71,96)(72,97)(74,99)(75,100)(77,102)(78,103)(80,105)(81,106), (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( (1,28,66)(2,29,67)(3,30,68)(4,31,69)(5,32,70)(6,33,71)(7,34,72)(8,35,73)(9,36,74)(10,37,75)(11,38,76)(12,39,77)(13,40,78)(14,41,79)(15,42,80)(16,43,81)(17,44,55)(18,45,56)(19,46,57)(20,47,58)(21,48,59)(22,49,60)(23,50,61)(24,51,62)(25,52,63)(26,53,64)(27,54,65)(82,127,148)(83,128,149)(84,129,150)(85,130,151)(86,131,152)(87,132,153)(88,133,154)(89,134,155)(90,135,156)(91,109,157)(92,110,158)(93,111,159)(94,112,160)(95,113,161)(96,114,162)(97,115,136)(98,116,137)(99,117,138)(100,118,139)(101,119,140)(102,120,141)(103,121,142)(104,122,143)(105,123,144)(106,124,145)(107,125,146)(108,126,147), (1,4,7,10,13,16,19,22,25)(2,5,8,11,14,17,20,23,26)(3,6,9,12,15,18,21,24,27)(28,31,34,37,40,43,46,49,52)(29,32,35,38,41,44,47,50,53)(30,33,36,39,42,45,48,51,54)(55,58,61,64,67,70,73,76,79)(56,59,62,65,68,71,74,77,80)(57,60,63,66,69,72,75,78,81)(82,85,88,91,94,97,100,103,106)(83,86,89,92,95,98,101,104,107)(84,87,90,93,96,99,102,105,108)(109,112,115,118,121,124,127,130,133)(110,113,116,119,122,125,128,131,134)(111,114,117,120,123,126,129,132,135)(136,139,142,145,148,151,154,157,160)(137,140,143,146,149,152,155,158,161)(138,141,144,147,150,153,156,159,162), (2,110)(3,111)(5,113)(6,114)(8,116)(9,117)(11,119)(12,120)(14,122)(15,123)(17,125)(18,126)(20,128)(21,129)(23,131)(24,132)(26,134)(27,135)(29,158)(30,159)(32,161)(33,162)(35,137)(36,138)(38,140)(39,141)(41,143)(42,144)(44,146)(45,147)(47,149)(48,150)(50,152)(51,153)(53,155)(54,156)(55,107)(56,108)(58,83)(59,84)(61,86)(62,87)(64,89)(65,90)(67,92)(68,93)(70,95)(71,96)(73,98)(74,99)(76,101)(77,102)(79,104)(80,105), (1,109)(3,111)(4,112)(6,114)(7,115)(9,117)(10,118)(12,120)(13,121)(15,123)(16,124)(18,126)(19,127)(21,129)(22,130)(24,132)(25,133)(27,135)(28,157)(30,159)(31,160)(33,162)(34,136)(36,138)(37,139)(39,141)(40,142)(42,144)(43,145)(45,147)(46,148)(48,150)(49,151)(51,153)(52,154)(54,156)(56,108)(57,82)(59,84)(60,85)(62,87)(63,88)(65,90)(66,91)(68,93)(69,94)(71,96)(72,97)(74,99)(75,100)(77,102)(78,103)(80,105)(81,106), (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([[(1,28,66),(2,29,67),(3,30,68),(4,31,69),(5,32,70),(6,33,71),(7,34,72),(8,35,73),(9,36,74),(10,37,75),(11,38,76),(12,39,77),(13,40,78),(14,41,79),(15,42,80),(16,43,81),(17,44,55),(18,45,56),(19,46,57),(20,47,58),(21,48,59),(22,49,60),(23,50,61),(24,51,62),(25,52,63),(26,53,64),(27,54,65),(82,127,148),(83,128,149),(84,129,150),(85,130,151),(86,131,152),(87,132,153),(88,133,154),(89,134,155),(90,135,156),(91,109,157),(92,110,158),(93,111,159),(94,112,160),(95,113,161),(96,114,162),(97,115,136),(98,116,137),(99,117,138),(100,118,139),(101,119,140),(102,120,141),(103,121,142),(104,122,143),(105,123,144),(106,124,145),(107,125,146),(108,126,147)], [(1,4,7,10,13,16,19,22,25),(2,5,8,11,14,17,20,23,26),(3,6,9,12,15,18,21,24,27),(28,31,34,37,40,43,46,49,52),(29,32,35,38,41,44,47,50,53),(30,33,36,39,42,45,48,51,54),(55,58,61,64,67,70,73,76,79),(56,59,62,65,68,71,74,77,80),(57,60,63,66,69,72,75,78,81),(82,85,88,91,94,97,100,103,106),(83,86,89,92,95,98,101,104,107),(84,87,90,93,96,99,102,105,108),(109,112,115,118,121,124,127,130,133),(110,113,116,119,122,125,128,131,134),(111,114,117,120,123,126,129,132,135),(136,139,142,145,148,151,154,157,160),(137,140,143,146,149,152,155,158,161),(138,141,144,147,150,153,156,159,162)], [(2,110),(3,111),(5,113),(6,114),(8,116),(9,117),(11,119),(12,120),(14,122),(15,123),(17,125),(18,126),(20,128),(21,129),(23,131),(24,132),(26,134),(27,135),(29,158),(30,159),(32,161),(33,162),(35,137),(36,138),(38,140),(39,141),(41,143),(42,144),(44,146),(45,147),(47,149),(48,150),(50,152),(51,153),(53,155),(54,156),(55,107),(56,108),(58,83),(59,84),(61,86),(62,87),(64,89),(65,90),(67,92),(68,93),(70,95),(71,96),(73,98),(74,99),(76,101),(77,102),(79,104),(80,105)], [(1,109),(3,111),(4,112),(6,114),(7,115),(9,117),(10,118),(12,120),(13,121),(15,123),(16,124),(18,126),(19,127),(21,129),(22,130),(24,132),(25,133),(27,135),(28,157),(30,159),(31,160),(33,162),(34,136),(36,138),(37,139),(39,141),(40,142),(42,144),(43,145),(45,147),(46,148),(48,150),(49,151),(51,153),(52,154),(54,156),(56,108),(57,82),(59,84),(60,85),(62,87),(63,88),(65,90),(66,91),(68,93),(69,94),(71,96),(72,97),(74,99),(75,100),(77,102),(78,103),(80,105),(81,106)], [(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)]])

108 conjugacy classes

 class 1 2 3A ··· 3H 6A ··· 6H 9A ··· 9R 18A ··· 18R 27A ··· 27BB order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 27 ··· 27 size 1 3 1 ··· 1 3 ··· 3 1 ··· 1 3 ··· 3 4 ··· 4

108 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 3 type + + image C1 C3 C3 C9 C9 C27 A4 C3.A4 C3×A4 C3.A4 C9.A4 kernel C3×C9.A4 C9.A4 C6×C18 C2×C18 C62 C2×C6 C3×C9 C9 C9 C32 C3 # reps 1 6 2 12 6 54 1 4 2 2 18

Matrix representation of C3×C9.A4 in GL4(𝔽109) generated by

 45 0 0 0 0 63 0 0 0 0 63 0 0 0 0 63
,
 1 0 0 0 0 38 0 0 0 0 38 0 0 0 0 38
,
 1 0 0 0 0 1 0 0 0 78 108 0 0 89 0 108
,
 1 0 0 0 0 108 0 0 0 0 108 0 0 20 0 1
,
 1 0 0 0 0 22 19 0 0 0 87 45 0 15 28 0
G:=sub<GL(4,GF(109))| [45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[1,0,0,0,0,38,0,0,0,0,38,0,0,0,0,38],[1,0,0,0,0,1,78,89,0,0,108,0,0,0,0,108],[1,0,0,0,0,108,0,20,0,0,108,0,0,0,0,1],[1,0,0,0,0,22,0,15,0,19,87,28,0,0,45,0] >;

C3×C9.A4 in GAP, Magma, Sage, TeX

C_3\times C_9.A_4
% in TeX

G:=Group("C3xC9.A4");
// GroupNames label

G:=SmallGroup(324,44);
// by ID

G=gap.SmallGroup(324,44);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,68,4864,8753]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^9=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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