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G = S3×C75order 450 = 2·32·52

Direct product of C75 and S3

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

Aliases: S3×C75, C3⋊C150, C753C6, C321C50, C15.2C30, (C3×C75)⋊4C2, C5.(S3×C15), (S3×C15).C5, (C5×S3).C15, C15.5(C5×S3), (C3×C15).1C10, SmallGroup(450,6)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C75
C1C3C15C75C3×C75 — S3×C75
C3 — S3×C75
C1C75

Generators and relations for S3×C75
 G = < a,b,c | a75=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
2C3
3C6
3C10
2C15
3C30
3C50
2C75
3C150

Smallest permutation representation of S3×C75
On 150 points
Generators in S150
(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)
(1 26 51)(2 27 52)(3 28 53)(4 29 54)(5 30 55)(6 31 56)(7 32 57)(8 33 58)(9 34 59)(10 35 60)(11 36 61)(12 37 62)(13 38 63)(14 39 64)(15 40 65)(16 41 66)(17 42 67)(18 43 68)(19 44 69)(20 45 70)(21 46 71)(22 47 72)(23 48 73)(24 49 74)(25 50 75)(76 126 101)(77 127 102)(78 128 103)(79 129 104)(80 130 105)(81 131 106)(82 132 107)(83 133 108)(84 134 109)(85 135 110)(86 136 111)(87 137 112)(88 138 113)(89 139 114)(90 140 115)(91 141 116)(92 142 117)(93 143 118)(94 144 119)(95 145 120)(96 146 121)(97 147 122)(98 148 123)(99 149 124)(100 150 125)
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 85)(8 86)(9 87)(10 88)(11 89)(12 90)(13 91)(14 92)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(37 115)(38 116)(39 117)(40 118)(41 119)(42 120)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 76)(74 77)(75 78)

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

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

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

225 conjugacy classes

class 1  2 3A3B3C3D3E5A5B5C5D6A6B10A10B10C10D15A···15H15I···15T25A···25T30A···30H50A···50T75A···75AN75AO···75CV150A···150AN
order12333335555661010101015···1515···1525···2530···3050···5075···7575···75150···150
size131122211113333331···12···21···13···33···31···12···23···3

225 irreducible representations

dim111111111111222222
type+++
imageC1C2C3C5C6C10C15C25C30C50C75C150S3C3×S3C5×S3S3×C15S3×C25S3×C75
kernelS3×C75C3×C75S3×C25S3×C15C75C3×C15C5×S3C3×S3C15C32S3C3C75C25C15C5C3C1
# reps112424820820404012482040

Matrix representation of S3×C75 in GL2(𝔽151) generated by

1160
0116
,
3219
0118
,
12895
3123
G:=sub<GL(2,GF(151))| [116,0,0,116],[32,0,19,118],[128,31,95,23] >;

S3×C75 in GAP, Magma, Sage, TeX

S_3\times C_{75}
% in TeX

G:=Group("S3xC75");
// GroupNames label

G:=SmallGroup(450,6);
// by ID

G=gap.SmallGroup(450,6);
# by ID

G:=PCGroup([5,-2,-3,-5,-5,-3,87,7504]);
// Polycyclic

G:=Group<a,b,c|a^75=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×C75 in TeX

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