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## G = S3×C5×C10order 300 = 22·3·52

### Direct product of C5×C10 and S3

Aliases: S3×C5×C10, C3⋊C102, C303C10, C6⋊(C5×C10), (C5×C30)⋊5C2, C154(C2×C10), (C5×C15)⋊10C22, SmallGroup(300,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C5×C10
 Chief series C1 — C3 — C15 — C5×C15 — S3×C52 — S3×C5×C10
 Lower central C3 — S3×C5×C10
 Upper central C1 — C5×C10

Generators and relations for S3×C5×C10
G = < a,b,c,d | a5=b10=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 128 in 80 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, C10, C10, D6, C15, C2×C10, C52, C5×S3, C30, C5×C10, C5×C10, S3×C10, C5×C15, C102, S3×C52, C5×C30, S3×C5×C10
Quotients: C1, C2, C22, C5, S3, C10, D6, C2×C10, C52, C5×S3, C5×C10, S3×C10, C102, S3×C52, S3×C5×C10

Smallest permutation representation of S3×C5×C10
On 150 points
Generators in S150
(1 58 77 104 139)(2 59 78 105 140)(3 60 79 106 131)(4 51 80 107 132)(5 52 71 108 133)(6 53 72 109 134)(7 54 73 110 135)(8 55 74 101 136)(9 56 75 102 137)(10 57 76 103 138)(11 39 64 25 44)(12 40 65 26 45)(13 31 66 27 46)(14 32 67 28 47)(15 33 68 29 48)(16 34 69 30 49)(17 35 70 21 50)(18 36 61 22 41)(19 37 62 23 42)(20 38 63 24 43)(81 114 149 94 129)(82 115 150 95 130)(83 116 141 96 121)(84 117 142 97 122)(85 118 143 98 123)(86 119 144 99 124)(87 120 145 100 125)(88 111 146 91 126)(89 112 147 92 127)(90 113 148 93 128)
(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 43 123)(2 44 124)(3 45 125)(4 46 126)(5 47 127)(6 48 128)(7 49 129)(8 50 130)(9 41 121)(10 42 122)(11 86 59)(12 87 60)(13 88 51)(14 89 52)(15 90 53)(16 81 54)(17 82 55)(18 83 56)(19 84 57)(20 85 58)(21 95 136)(22 96 137)(23 97 138)(24 98 139)(25 99 140)(26 100 131)(27 91 132)(28 92 133)(29 93 134)(30 94 135)(31 111 80)(32 112 71)(33 113 72)(34 114 73)(35 115 74)(36 116 75)(37 117 76)(38 118 77)(39 119 78)(40 120 79)(61 141 102)(62 142 103)(63 143 104)(64 144 105)(65 145 106)(66 146 107)(67 147 108)(68 148 109)(69 149 110)(70 150 101)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 100)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 56)(52 57)(53 58)(54 59)(55 60)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 76)(72 77)(73 78)(74 79)(75 80)(101 106)(102 107)(103 108)(104 109)(105 110)(131 136)(132 137)(133 138)(134 139)(135 140)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 3 5A ··· 5X 6 10A ··· 10X 10Y ··· 10BT 15A ··· 15X 30A ··· 30X order 1 2 2 2 3 5 ··· 5 6 10 ··· 10 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 3 3 2 1 ··· 1 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 S3 D6 C5×S3 S3×C10 kernel S3×C5×C10 S3×C52 C5×C30 S3×C10 C5×S3 C30 C5×C10 C52 C10 C5 # reps 1 2 1 24 48 24 1 1 24 24

Matrix representation of S3×C5×C10 in GL3(𝔽31) generated by

 1 0 0 0 16 0 0 0 16
,
 23 0 0 0 27 0 0 0 27
,
 1 0 0 0 0 30 0 1 30
,
 1 0 0 0 0 30 0 30 0
G:=sub<GL(3,GF(31))| [1,0,0,0,16,0,0,0,16],[23,0,0,0,27,0,0,0,27],[1,0,0,0,0,1,0,30,30],[1,0,0,0,0,30,0,30,0] >;

S3×C5×C10 in GAP, Magma, Sage, TeX

S_3\times C_5\times C_{10}
% in TeX

G:=Group("S3xC5xC10");
// GroupNames label

G:=SmallGroup(300,46);
// by ID

G=gap.SmallGroup(300,46);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-3,5004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^10=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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