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## G = C2×C27⋊S3order 324 = 22·34

### Direct product of C2 and C27⋊S3

Aliases: C2×C27⋊S3, C54⋊S3, C6⋊D27, C272D6, C32D54, C18.5D9, C9.2D18, C32.4D18, (C3×C54)⋊3C2, (C3×C9).9D6, (C3×C6).8D9, C6.4(C9⋊S3), (C3×C27)⋊4C22, C18.2(C3⋊S3), (C3×C18).23S3, C9.(C2×C3⋊S3), C3.2(C2×C9⋊S3), SmallGroup(324,76)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C27 — C2×C27⋊S3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C27 — C27⋊S3 — C2×C27⋊S3
 Lower central C3×C27 — C2×C27⋊S3
 Upper central C1 — C2

Generators and relations for C2×C27⋊S3
G = < a,b,c,d | a2=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 742 in 70 conjugacy classes, 31 normal (13 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C9, C32, D6, D9, C18, C18, C3⋊S3, C3×C6, C27, C3×C9, D18, C2×C3⋊S3, D27, C54, C9⋊S3, C3×C18, C3×C27, D54, C2×C9⋊S3, C27⋊S3, C3×C54, C2×C27⋊S3
Quotients: C1, C2, C22, S3, D6, D9, C3⋊S3, D18, C2×C3⋊S3, D27, C9⋊S3, D54, C2×C9⋊S3, C27⋊S3, C2×C27⋊S3

Smallest permutation representation of C2×C27⋊S3
On 162 points
Generators in S162
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 109)(53 110)(54 111)(82 161)(83 162)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)
(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 44 98)(2 45 99)(3 46 100)(4 47 101)(5 48 102)(6 49 103)(7 50 104)(8 51 105)(9 52 106)(10 53 107)(11 54 108)(12 28 82)(13 29 83)(14 30 84)(15 31 85)(16 32 86)(17 33 87)(18 34 88)(19 35 89)(20 36 90)(21 37 91)(22 38 92)(23 39 93)(24 40 94)(25 41 95)(26 42 96)(27 43 97)(55 111 160)(56 112 161)(57 113 162)(58 114 136)(59 115 137)(60 116 138)(61 117 139)(62 118 140)(63 119 141)(64 120 142)(65 121 143)(66 122 144)(67 123 145)(68 124 146)(69 125 147)(70 126 148)(71 127 149)(72 128 150)(73 129 151)(74 130 152)(75 131 153)(76 132 154)(77 133 155)(78 134 156)(79 135 157)(80 109 158)(81 110 159)
(1 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 139)(29 138)(30 137)(31 136)(32 162)(33 161)(34 160)(35 159)(36 158)(37 157)(38 156)(39 155)(40 154)(41 153)(42 152)(43 151)(44 150)(45 149)(46 148)(47 147)(48 146)(49 145)(50 144)(51 143)(52 142)(53 141)(54 140)(82 117)(83 116)(84 115)(85 114)(86 113)(87 112)(88 111)(89 110)(90 109)(91 135)(92 134)(93 133)(94 132)(95 131)(96 130)(97 129)(98 128)(99 127)(100 126)(101 125)(102 124)(103 123)(104 122)(105 121)(106 120)(107 119)(108 118)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 6A 6B 6C 6D 9A ··· 9I 18A ··· 18I 27A ··· 27AA 54A ··· 54AA order 1 2 2 2 3 3 3 3 6 6 6 6 9 ··· 9 18 ··· 18 27 ··· 27 54 ··· 54 size 1 1 81 81 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 S3 S3 D6 D6 D9 D9 D18 D18 D27 D54 kernel C2×C27⋊S3 C27⋊S3 C3×C54 C54 C3×C18 C27 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 2 1 3 1 3 1 6 3 6 3 27 27

Matrix representation of C2×C27⋊S3 in GL4(𝔽109) generated by

 1 0 0 0 0 1 0 0 0 0 108 0 0 0 0 108
,
 7 17 0 0 92 99 0 0 0 0 16 46 0 0 63 79
,
 0 1 0 0 108 108 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 108 108 0 0 0 0 32 82 0 0 50 77
G:=sub<GL(4,GF(109))| [1,0,0,0,0,1,0,0,0,0,108,0,0,0,0,108],[7,92,0,0,17,99,0,0,0,0,16,63,0,0,46,79],[0,108,0,0,1,108,0,0,0,0,1,0,0,0,0,1],[1,108,0,0,0,108,0,0,0,0,32,50,0,0,82,77] >;

C2×C27⋊S3 in GAP, Magma, Sage, TeX

C_2\times C_{27}\rtimes S_3
% in TeX

G:=Group("C2xC27:S3");
// GroupNames label

G:=SmallGroup(324,76);
// by ID

G=gap.SmallGroup(324,76);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,794,824,579,5404,208,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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