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## G = C3⋊S3×C27order 486 = 2·35

### Direct product of C27 and C3⋊S3

Aliases: C3⋊S3×C27, C324C54, C33.7C18, C3⋊(S3×C27), (C3×C27)⋊9S3, (C32×C27)⋊2C2, C32.20(S3×C9), (C32×C9).28C6, C9.6(C3×C3⋊S3), C3.5(C9×C3⋊S3), (C3×C3⋊S3).2C9, (C9×C3⋊S3).2C3, (C3×C9).51(C3×S3), SmallGroup(486,161)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C27
 Chief series C1 — C3 — C32 — C33 — C32×C9 — C32×C27 — C3⋊S3×C27
 Lower central C32 — C3⋊S3×C27
 Upper central C1 — C27

Generators and relations for C3⋊S3×C27
G = < a,b,c,d | a27=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 160 in 72 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, C18, C3×S3, C3⋊S3, C27, C27, C3×C9, C3×C9, C33, C54, S3×C9, C3×C3⋊S3, C3×C27, C3×C27, C32×C9, S3×C27, C9×C3⋊S3, C32×C27, C3⋊S3×C27
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, C27, C54, S3×C9, C3×C3⋊S3, S3×C27, C9×C3⋊S3, C3⋊S3×C27

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

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

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

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

162 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 6A 6B 9A ··· 9F 9G ··· 9AD 18A ··· 18F 27A ··· 27R 27S ··· 27CL 54A ··· 54R order 1 2 3 3 3 ··· 3 6 6 9 ··· 9 9 ··· 9 18 ··· 18 27 ··· 27 27 ··· 27 54 ··· 54 size 1 9 1 1 2 ··· 2 9 9 1 ··· 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C3 C6 C9 C18 C27 C54 S3 C3×S3 S3×C9 S3×C27 kernel C3⋊S3×C27 C32×C27 C9×C3⋊S3 C32×C9 C3×C3⋊S3 C33 C3⋊S3 C32 C3×C27 C3×C9 C32 C3 # reps 1 1 2 2 6 6 18 18 4 8 24 72

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

 7 0 0 0 0 7 0 0 0 0 49 0 0 0 0 49
,
 45 0 0 0 0 63 0 0 0 0 45 0 0 0 0 63
,
 45 0 0 0 0 63 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(109))| [7,0,0,0,0,7,0,0,0,0,49,0,0,0,0,49],[45,0,0,0,0,63,0,0,0,0,45,0,0,0,0,63],[45,0,0,0,0,63,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C3⋊S3×C27 in GAP, Magma, Sage, TeX

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

G:=Group("C3:S3xC27");
// GroupNames label

G:=SmallGroup(486,161);
// by ID

G=gap.SmallGroup(486,161);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,3244,11669]);
// Polycyclic

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