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## G = C3×C27⋊S3order 486 = 2·35

### Direct product of C3 and C27⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C27 — C3×C27⋊S3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C32×C27 — C3×C27⋊S3
 Lower central C3×C27 — C3×C27⋊S3
 Upper central C1 — C3

Generators and relations for C3×C27⋊S3
G = < a,b,c,d | a3=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 548 in 78 conjugacy classes, 30 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C27, C27, C3×C9, C3×C9, C3×C9, C33, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C27, C3×C27, C3×C27, C32×C9, C3×D27, C27⋊S3, C3×C9⋊S3, C32×C27, C3×C27⋊S3
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×D27, C27⋊S3, C3×C9⋊S3, C3×C27⋊S3

Smallest permutation representation of C3×C27⋊S3
On 162 points
Generators in S162
(1 96 80)(2 97 81)(3 98 55)(4 99 56)(5 100 57)(6 101 58)(7 102 59)(8 103 60)(9 104 61)(10 105 62)(11 106 63)(12 107 64)(13 108 65)(14 82 66)(15 83 67)(16 84 68)(17 85 69)(18 86 70)(19 87 71)(20 88 72)(21 89 73)(22 90 74)(23 91 75)(24 92 76)(25 93 77)(26 94 78)(27 95 79)(28 130 161)(29 131 162)(30 132 136)(31 133 137)(32 134 138)(33 135 139)(34 109 140)(35 110 141)(36 111 142)(37 112 143)(38 113 144)(39 114 145)(40 115 146)(41 116 147)(42 117 148)(43 118 149)(44 119 150)(45 120 151)(46 121 152)(47 122 153)(48 123 154)(49 124 155)(50 125 156)(51 126 157)(52 127 158)(53 128 159)(54 129 160)
(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 87 62)(2 88 63)(3 89 64)(4 90 65)(5 91 66)(6 92 67)(7 93 68)(8 94 69)(9 95 70)(10 96 71)(11 97 72)(12 98 73)(13 99 74)(14 100 75)(15 101 76)(16 102 77)(17 103 78)(18 104 79)(19 105 80)(20 106 81)(21 107 55)(22 108 56)(23 82 57)(24 83 58)(25 84 59)(26 85 60)(27 86 61)(28 152 112)(29 153 113)(30 154 114)(31 155 115)(32 156 116)(33 157 117)(34 158 118)(35 159 119)(36 160 120)(37 161 121)(38 162 122)(39 136 123)(40 137 124)(41 138 125)(42 139 126)(43 140 127)(44 141 128)(45 142 129)(46 143 130)(47 144 131)(48 145 132)(49 146 133)(50 147 134)(51 148 135)(52 149 109)(53 150 110)(54 151 111)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 30)(14 29)(15 28)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(55 146)(56 145)(57 144)(58 143)(59 142)(60 141)(61 140)(62 139)(63 138)(64 137)(65 136)(66 162)(67 161)(68 160)(69 159)(70 158)(71 157)(72 156)(73 155)(74 154)(75 153)(76 152)(77 151)(78 150)(79 149)(80 148)(81 147)(82 131)(83 130)(84 129)(85 128)(86 127)(87 126)(88 125)(89 124)(90 123)(91 122)(92 121)(93 120)(94 119)(95 118)(96 117)(97 116)(98 115)(99 114)(100 113)(101 112)(102 111)(103 110)(104 109)(105 135)(106 134)(107 133)(108 132)

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

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

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

126 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9AA 27A ··· 27CC order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 27 ··· 27 size 1 81 1 1 2 ··· 2 81 81 2 ··· 2 2 ··· 2

126 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C3 C6 S3 S3 C3×S3 D9 C3×S3 D9 C3×D9 D27 C3×D9 C3×D27 kernel C3×C27⋊S3 C32×C27 C27⋊S3 C3×C27 C3×C27 C32×C9 C27 C3×C9 C3×C9 C33 C9 C32 C32 C3 # reps 1 1 2 2 3 1 6 6 2 3 12 27 6 54

Matrix representation of C3×C27⋊S3 in GL4(𝔽109) generated by

 45 0 0 0 0 45 0 0 0 0 45 0 0 0 0 45
,
 38 0 0 0 0 66 0 0 0 0 7 0 0 0 0 78
,
 45 0 0 0 0 63 0 0 0 0 63 0 0 0 0 45
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(109))| [45,0,0,0,0,45,0,0,0,0,45,0,0,0,0,45],[38,0,0,0,0,66,0,0,0,0,7,0,0,0,0,78],[45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,45],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3×C27⋊S3 in GAP, Magma, Sage, TeX

C_3\times C_{27}\rtimes S_3
% in TeX

G:=Group("C3xC27:S3");
// GroupNames label

G:=SmallGroup(486,160);
// by ID

G=gap.SmallGroup(486,160);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,824,867,8104,208,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^27=c^3=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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