direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C5×C10, C3⋊C102, C30⋊3C10, C6⋊(C5×C10), (C5×C30)⋊5C2, C15⋊4(C2×C10), (C5×C15)⋊10C22, SmallGroup(300,46)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×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
►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