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G = S3xC3xC27order 486 = 2·35

Direct product of C3xC27 and S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3xC3xC27, C32:3C54, C33.6C18, C3:(C3xC54), C9.7(S3xC9), (C3xC27):13C6, (S3xC9).3C9, (C32xC27):1C2, (C3xC9).10C18, C9.6(S3xC32), (S3xC9).2C32, (S3xC32).2C9, C32.21(S3xC9), (C32xC9).27C6, C32.16(C3xC18), C3.7(S3xC3xC9), (S3xC3xC9).5C3, (C3xS3).4(C3xC9), (C3xC9).37(C3xC6), (C3xC9).63(C3xS3), SmallGroup(486,112)

Series: Derived Chief Lower central Upper central

C1C3 — S3xC3xC27
C1C3C32C3xC9C3xC27C32xC27 — S3xC3xC27
C3 — S3xC3xC27
C1C3xC27

Generators and relations for S3xC3xC27
 G = < a,b,c,d | a3=b27=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 78 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C9, C32, C32, C32, C18, C3xS3, C3xS3, C3xC6, C27, C27, C3xC9, C3xC9, C3xC9, C33, C54, S3xC9, S3xC9, C3xC18, S3xC32, C3xC27, C3xC27, C3xC27, C32xC9, S3xC27, C3xC54, S3xC3xC9, C32xC27, S3xC3xC27
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3xS3, C3xC6, C27, C3xC9, C54, S3xC9, C3xC18, S3xC32, C3xC27, S3xC27, C3xC54, S3xC3xC9, S3xC3xC27

Smallest permutation representation of S3xC3xC27
On 162 points
Generators in S162
(1 61 32)(2 62 33)(3 63 34)(4 64 35)(5 65 36)(6 66 37)(7 67 38)(8 68 39)(9 69 40)(10 70 41)(11 71 42)(12 72 43)(13 73 44)(14 74 45)(15 75 46)(16 76 47)(17 77 48)(18 78 49)(19 79 50)(20 80 51)(21 81 52)(22 55 53)(23 56 54)(24 57 28)(25 58 29)(26 59 30)(27 60 31)(82 146 134)(83 147 135)(84 148 109)(85 149 110)(86 150 111)(87 151 112)(88 152 113)(89 153 114)(90 154 115)(91 155 116)(92 156 117)(93 157 118)(94 158 119)(95 159 120)(96 160 121)(97 161 122)(98 162 123)(99 136 124)(100 137 125)(101 138 126)(102 139 127)(103 140 128)(104 141 129)(105 142 130)(106 143 131)(107 144 132)(108 145 133)
(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 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)(55 64 73)(56 65 74)(57 66 75)(58 67 76)(59 68 77)(60 69 78)(61 70 79)(62 71 80)(63 72 81)(82 100 91)(83 101 92)(84 102 93)(85 103 94)(86 104 95)(87 105 96)(88 106 97)(89 107 98)(90 108 99)(109 127 118)(110 128 119)(111 129 120)(112 130 121)(113 131 122)(114 132 123)(115 133 124)(116 134 125)(117 135 126)(136 154 145)(137 155 146)(138 156 147)(139 157 148)(140 158 149)(141 159 150)(142 160 151)(143 161 152)(144 162 153)
(1 84)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 82)(27 83)(28 132)(29 133)(30 134)(31 135)(32 109)(33 110)(34 111)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 120)(44 121)(45 122)(46 123)(47 124)(48 125)(49 126)(50 127)(51 128)(52 129)(53 130)(54 131)(55 142)(56 143)(57 144)(58 145)(59 146)(60 147)(61 148)(62 149)(63 150)(64 151)(65 152)(66 153)(67 154)(68 155)(69 156)(70 157)(71 158)(72 159)(73 160)(74 161)(75 162)(76 136)(77 137)(78 138)(79 139)(80 140)(81 141)

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

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

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

243 conjugacy classes

class 1  2 3A···3H3I···3Q6A···6H9A···9R9S···9AJ18A···18R27A···27BB27BC···27DD54A···54BB
order123···33···36···69···99···918···1827···2727···2754···54
size131···12···23···31···12···23···31···12···23···3

243 irreducible representations

dim111111111111222222
type+++
imageC1C2C3C3C6C6C9C9C18C18C27C54S3C3xS3C3xS3S3xC9S3xC9S3xC27
kernelS3xC3xC27C32xC27S3xC27S3xC3xC9C3xC27C32xC9S3xC9S3xC32C3xC9C33C3xS3C32C3xC27C27C3xC9C9C32C3
# reps116262126126545416212654

Matrix representation of S3xC3xC27 in GL3(F109) generated by

100
0630
0063
,
2500
0780
0078
,
100
0450
0063
,
10800
001
010
G:=sub<GL(3,GF(109))| [1,0,0,0,63,0,0,0,63],[25,0,0,0,78,0,0,0,78],[1,0,0,0,45,0,0,0,63],[108,0,0,0,0,1,0,1,0] >;

S3xC3xC27 in GAP, Magma, Sage, TeX

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

G:=Group("S3xC3xC27");
// GroupNames label

G:=SmallGroup(486,112);
// by ID

G=gap.SmallGroup(486,112);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,68,93,11669]);
// 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,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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