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G = S3×C55order 330 = 2·3·5·11

Direct product of C55 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: S3×C55, C3⋊C110, C1657C2, C153C22, C337C10, SmallGroup(330,8)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C55
C1C3C33C165 — S3×C55
C3 — S3×C55
C1C55

Generators and relations for S3×C55
 G = < a,b,c | a55=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C10
3C22
3C110

Smallest permutation representation of S3×C55
On 165 points
Generators in S165
(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)(111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165)
(1 130 101)(2 131 102)(3 132 103)(4 133 104)(5 134 105)(6 135 106)(7 136 107)(8 137 108)(9 138 109)(10 139 110)(11 140 56)(12 141 57)(13 142 58)(14 143 59)(15 144 60)(16 145 61)(17 146 62)(18 147 63)(19 148 64)(20 149 65)(21 150 66)(22 151 67)(23 152 68)(24 153 69)(25 154 70)(26 155 71)(27 156 72)(28 157 73)(29 158 74)(30 159 75)(31 160 76)(32 161 77)(33 162 78)(34 163 79)(35 164 80)(36 165 81)(37 111 82)(38 112 83)(39 113 84)(40 114 85)(41 115 86)(42 116 87)(43 117 88)(44 118 89)(45 119 90)(46 120 91)(47 121 92)(48 122 93)(49 123 94)(50 124 95)(51 125 96)(52 126 97)(53 127 98)(54 128 99)(55 129 100)
(56 140)(57 141)(58 142)(59 143)(60 144)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 121)(93 122)(94 123)(95 124)(96 125)(97 126)(98 127)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)

G:=sub<Sym(165)| (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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,130,101)(2,131,102)(3,132,103)(4,133,104)(5,134,105)(6,135,106)(7,136,107)(8,137,108)(9,138,109)(10,139,110)(11,140,56)(12,141,57)(13,142,58)(14,143,59)(15,144,60)(16,145,61)(17,146,62)(18,147,63)(19,148,64)(20,149,65)(21,150,66)(22,151,67)(23,152,68)(24,153,69)(25,154,70)(26,155,71)(27,156,72)(28,157,73)(29,158,74)(30,159,75)(31,160,76)(32,161,77)(33,162,78)(34,163,79)(35,164,80)(36,165,81)(37,111,82)(38,112,83)(39,113,84)(40,114,85)(41,115,86)(42,116,87)(43,117,88)(44,118,89)(45,119,90)(46,120,91)(47,121,92)(48,122,93)(49,123,94)(50,124,95)(51,125,96)(52,126,97)(53,127,98)(54,128,99)(55,129,100), (56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)>;

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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165), (1,130,101)(2,131,102)(3,132,103)(4,133,104)(5,134,105)(6,135,106)(7,136,107)(8,137,108)(9,138,109)(10,139,110)(11,140,56)(12,141,57)(13,142,58)(14,143,59)(15,144,60)(16,145,61)(17,146,62)(18,147,63)(19,148,64)(20,149,65)(21,150,66)(22,151,67)(23,152,68)(24,153,69)(25,154,70)(26,155,71)(27,156,72)(28,157,73)(29,158,74)(30,159,75)(31,160,76)(32,161,77)(33,162,78)(34,163,79)(35,164,80)(36,165,81)(37,111,82)(38,112,83)(39,113,84)(40,114,85)(41,115,86)(42,116,87)(43,117,88)(44,118,89)(45,119,90)(46,120,91)(47,121,92)(48,122,93)(49,123,94)(50,124,95)(51,125,96)(52,126,97)(53,127,98)(54,128,99)(55,129,100), (56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,111)(83,112)(84,113)(85,114)(86,115)(87,116)(88,117)(89,118)(90,119)(91,120)(92,121)(93,122)(94,123)(95,124)(96,125)(97,126)(98,127)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139) );

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),(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165)], [(1,130,101),(2,131,102),(3,132,103),(4,133,104),(5,134,105),(6,135,106),(7,136,107),(8,137,108),(9,138,109),(10,139,110),(11,140,56),(12,141,57),(13,142,58),(14,143,59),(15,144,60),(16,145,61),(17,146,62),(18,147,63),(19,148,64),(20,149,65),(21,150,66),(22,151,67),(23,152,68),(24,153,69),(25,154,70),(26,155,71),(27,156,72),(28,157,73),(29,158,74),(30,159,75),(31,160,76),(32,161,77),(33,162,78),(34,163,79),(35,164,80),(36,165,81),(37,111,82),(38,112,83),(39,113,84),(40,114,85),(41,115,86),(42,116,87),(43,117,88),(44,118,89),(45,119,90),(46,120,91),(47,121,92),(48,122,93),(49,123,94),(50,124,95),(51,125,96),(52,126,97),(53,127,98),(54,128,99),(55,129,100)], [(56,140),(57,141),(58,142),(59,143),(60,144),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,153),(70,154),(71,155),(72,156),(73,157),(74,158),(75,159),(76,160),(77,161),(78,162),(79,163),(80,164),(81,165),(82,111),(83,112),(84,113),(85,114),(86,115),(87,116),(88,117),(89,118),(90,119),(91,120),(92,121),(93,122),(94,123),(95,124),(96,125),(97,126),(98,127),(99,128),(100,129),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139)])

165 conjugacy classes

class 1  2  3 5A5B5C5D10A10B10C10D11A···11J15A15B15C15D22A···22J33A···33J55A···55AN110A···110AN165A···165AN
order12355551010101011···111515151522···2233···3355···55110···110165···165
size132111133331···122223···32···21···13···32···2

165 irreducible representations

dim111111112222
type+++
imageC1C2C5C10C11C22C55C110S3C5×S3S3×C11S3×C55
kernelS3×C55C165S3×C11C33C5×S3C15S3C3C55C11C5C1
# reps114410104040141040

Matrix representation of S3×C55 in GL3(𝔽331) generated by

12400
01800
00180
,
100
00330
01330
,
33000
001
010
G:=sub<GL(3,GF(331))| [124,0,0,0,180,0,0,0,180],[1,0,0,0,0,1,0,330,330],[330,0,0,0,0,1,0,1,0] >;

S3×C55 in GAP, Magma, Sage, TeX

S_3\times C_{55}
% in TeX

G:=Group("S3xC55");
// GroupNames label

G:=SmallGroup(330,8);
// by ID

G=gap.SmallGroup(330,8);
# by ID

G:=PCGroup([4,-2,-5,-11,-3,3523]);
// Polycyclic

G:=Group<a,b,c|a^55=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C55 in TeX

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