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G = C2×C27⋊S3order 324 = 22·34

Direct product of C2 and C27⋊S3

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

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

C1C3×C27 — C2×C27⋊S3
C1C3C9C3×C9C3×C27C27⋊S3 — C2×C27⋊S3
C3×C27 — C2×C27⋊S3
C1C2

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 [×2], C3, C3 [×3], C22, S3 [×8], C6, C6 [×3], C9, C9 [×2], C32, D6 [×4], D9 [×6], C18, C18 [×2], C3⋊S3 [×2], C3×C6, C27 [×3], C3×C9, D18 [×3], C2×C3⋊S3, D27 [×6], C54 [×3], C9⋊S3 [×2], C3×C18, C3×C27, D54 [×3], C2×C9⋊S3, C27⋊S3 [×2], C3×C54, C2×C27⋊S3
Quotients: C1, C2 [×3], C22, S3 [×4], D6 [×4], D9 [×3], C3⋊S3, D18 [×3], C2×C3⋊S3, D27 [×3], C9⋊S3, D54 [×3], C2×C9⋊S3, C27⋊S3, C2×C27⋊S3

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

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B3C3D6A6B6C6D9A···9I18A···18I27A···27AA54A···54AA
order1222333366669···918···1827···2754···54
size118181222222222···22···22···22···2

84 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3S3D6D6D9D9D18D18D27D54
kernelC2×C27⋊S3C27⋊S3C3×C54C54C3×C18C27C3×C9C18C3×C6C9C32C6C3
# reps121313163632727

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

1000
0100
001080
000108
,
71700
929900
001646
006379
,
0100
10810800
0010
0001
,
1000
10810800
003282
005077
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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