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## G = S3×C75order 450 = 2·32·52

### Direct product of C75 and S3

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

 Derived series C1 — C3 — S3×C75
 Chief series C1 — C3 — C15 — C75 — C3×C75 — S3×C75
 Lower central C3 — S3×C75
 Upper central C1 — C75

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

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

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

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

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

225 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 5A 5B 5C 5D 6A 6B 10A 10B 10C 10D 15A ··· 15H 15I ··· 15T 25A ··· 25T 30A ··· 30H 50A ··· 50T 75A ··· 75AN 75AO ··· 75CV 150A ··· 150AN order 1 2 3 3 3 3 3 5 5 5 5 6 6 10 10 10 10 15 ··· 15 15 ··· 15 25 ··· 25 30 ··· 30 50 ··· 50 75 ··· 75 75 ··· 75 150 ··· 150 size 1 3 1 1 2 2 2 1 1 1 1 3 3 3 3 3 3 1 ··· 1 2 ··· 2 1 ··· 1 3 ··· 3 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

225 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + image C1 C2 C3 C5 C6 C10 C15 C25 C30 C50 C75 C150 S3 C3×S3 C5×S3 S3×C15 S3×C25 S3×C75 kernel S3×C75 C3×C75 S3×C25 S3×C15 C75 C3×C15 C5×S3 C3×S3 C15 C32 S3 C3 C75 C25 C15 C5 C3 C1 # reps 1 1 2 4 2 4 8 20 8 20 40 40 1 2 4 8 20 40

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

 116 0 0 116
,
 32 19 0 118
,
 128 95 31 23
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

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