direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C5×C15, C15⋊3C30, C152⋊7C2, C3⋊(C5×C30), (C5×C15)⋊9C6, (C3×C15)⋊4C10, C32⋊1(C5×C10), SmallGroup(450,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C5×C15 |
Generators and relations for S3×C5×C15
G = < a,b,c,d | a5=b15=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 112 in 72 conjugacy classes, 48 normal (12 characteristic)
C1, C2, C3, C3, C5, S3, C6, C32, C10, C15, C15, C3×S3, C52, C5×S3, C30, C3×C15, C5×C10, C5×C15, C5×C15, S3×C15, S3×C52, C5×C30, C152, S3×C5×C15
Quotients: C1, C2, C3, C5, S3, C6, C10, C15, C3×S3, C52, C5×S3, C30, C5×C10, C5×C15, S3×C15, S3×C52, C5×C30, S3×C5×C15
►On 150 points
Generators in S150
(1 73 55 40 22)(2 74 56 41 23)(3 75 57 42 24)(4 61 58 43 25)(5 62 59 44 26)(6 63 60 45 27)(7 64 46 31 28)(8 65 47 32 29)(9 66 48 33 30)(10 67 49 34 16)(11 68 50 35 17)(12 69 51 36 18)(13 70 52 37 19)(14 71 53 38 20)(15 72 54 39 21)(76 145 124 109 100)(77 146 125 110 101)(78 147 126 111 102)(79 148 127 112 103)(80 149 128 113 104)(81 150 129 114 105)(82 136 130 115 91)(83 137 131 116 92)(84 138 132 117 93)(85 139 133 118 94)(86 140 134 119 95)(87 141 135 120 96)(88 142 121 106 97)(89 143 122 107 98)(90 144 123 108 99)
(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 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 81 86)(77 82 87)(78 83 88)(79 84 89)(80 85 90)(91 96 101)(92 97 102)(93 98 103)(94 99 104)(95 100 105)(106 111 116)(107 112 117)(108 113 118)(109 114 119)(110 115 120)(121 126 131)(122 127 132)(123 128 133)(124 129 134)(125 130 135)(136 141 146)(137 142 147)(138 143 148)(139 144 149)(140 145 150)
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(41 116)(42 117)(43 118)(44 119)(45 120)(46 121)(47 122)(48 123)(49 124)(50 125)(51 126)(52 127)(53 128)(54 129)(55 130)(56 131)(57 132)(58 133)(59 134)(60 135)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 149)(72 150)(73 136)(74 137)(75 138)
G:=sub<Sym(150)| (1,73,55,40,22)(2,74,56,41,23)(3,75,57,42,24)(4,61,58,43,25)(5,62,59,44,26)(6,63,60,45,27)(7,64,46,31,28)(8,65,47,32,29)(9,66,48,33,30)(10,67,49,34,16)(11,68,50,35,17)(12,69,51,36,18)(13,70,52,37,19)(14,71,53,38,20)(15,72,54,39,21)(76,145,124,109,100)(77,146,125,110,101)(78,147,126,111,102)(79,148,127,112,103)(80,149,128,113,104)(81,150,129,114,105)(82,136,130,115,91)(83,137,131,116,92)(84,138,132,117,93)(85,139,133,118,94)(86,140,134,119,95)(87,141,135,120,96)(88,142,121,106,97)(89,143,122,107,98)(90,144,123,108,99), (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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120)(121,126,131)(122,127,132)(123,128,133)(124,129,134)(125,130,135)(136,141,146)(137,142,147)(138,143,148)(139,144,149)(140,145,150), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,136)(74,137)(75,138)>;
G:=Group( (1,73,55,40,22)(2,74,56,41,23)(3,75,57,42,24)(4,61,58,43,25)(5,62,59,44,26)(6,63,60,45,27)(7,64,46,31,28)(8,65,47,32,29)(9,66,48,33,30)(10,67,49,34,16)(11,68,50,35,17)(12,69,51,36,18)(13,70,52,37,19)(14,71,53,38,20)(15,72,54,39,21)(76,145,124,109,100)(77,146,125,110,101)(78,147,126,111,102)(79,148,127,112,103)(80,149,128,113,104)(81,150,129,114,105)(82,136,130,115,91)(83,137,131,116,92)(84,138,132,117,93)(85,139,133,118,94)(86,140,134,119,95)(87,141,135,120,96)(88,142,121,106,97)(89,143,122,107,98)(90,144,123,108,99), (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), (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,71,66)(62,72,67)(63,73,68)(64,74,69)(65,75,70)(76,81,86)(77,82,87)(78,83,88)(79,84,89)(80,85,90)(91,96,101)(92,97,102)(93,98,103)(94,99,104)(95,100,105)(106,111,116)(107,112,117)(108,113,118)(109,114,119)(110,115,120)(121,126,131)(122,127,132)(123,128,133)(124,129,134)(125,130,135)(136,141,146)(137,142,147)(138,143,148)(139,144,149)(140,145,150), (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,91)(23,92)(24,93)(25,94)(26,95)(27,96)(28,97)(29,98)(30,99)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(41,116)(42,117)(43,118)(44,119)(45,120)(46,121)(47,122)(48,123)(49,124)(50,125)(51,126)(52,127)(53,128)(54,129)(55,130)(56,131)(57,132)(58,133)(59,134)(60,135)(61,139)(62,140)(63,141)(64,142)(65,143)(66,144)(67,145)(68,146)(69,147)(70,148)(71,149)(72,150)(73,136)(74,137)(75,138) );
G=PermutationGroup([[(1,73,55,40,22),(2,74,56,41,23),(3,75,57,42,24),(4,61,58,43,25),(5,62,59,44,26),(6,63,60,45,27),(7,64,46,31,28),(8,65,47,32,29),(9,66,48,33,30),(10,67,49,34,16),(11,68,50,35,17),(12,69,51,36,18),(13,70,52,37,19),(14,71,53,38,20),(15,72,54,39,21),(76,145,124,109,100),(77,146,125,110,101),(78,147,126,111,102),(79,148,127,112,103),(80,149,128,113,104),(81,150,129,114,105),(82,136,130,115,91),(83,137,131,116,92),(84,138,132,117,93),(85,139,133,118,94),(86,140,134,119,95),(87,141,135,120,96),(88,142,121,106,97),(89,143,122,107,98),(90,144,123,108,99)], [(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)], [(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,71,66),(62,72,67),(63,73,68),(64,74,69),(65,75,70),(76,81,86),(77,82,87),(78,83,88),(79,84,89),(80,85,90),(91,96,101),(92,97,102),(93,98,103),(94,99,104),(95,100,105),(106,111,116),(107,112,117),(108,113,118),(109,114,119),(110,115,120),(121,126,131),(122,127,132),(123,128,133),(124,129,134),(125,130,135),(136,141,146),(137,142,147),(138,143,148),(139,144,149),(140,145,150)], [(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,76),(11,77),(12,78),(13,79),(14,80),(15,81),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,91),(23,92),(24,93),(25,94),(26,95),(27,96),(28,97),(29,98),(30,99),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(41,116),(42,117),(43,118),(44,119),(45,120),(46,121),(47,122),(48,123),(49,124),(50,125),(51,126),(52,127),(53,128),(54,129),(55,130),(56,131),(57,132),(58,133),(59,134),(60,135),(61,139),(62,140),(63,141),(64,142),(65,143),(66,144),(67,145),(68,146),(69,147),(70,148),(71,149),(72,150),(73,136),(74,137),(75,138)]])
225 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | ··· | 5X | 6A | 6B | 10A | ··· | 10X | 15A | ··· | 15AV | 15AW | ··· | 15DP | 30A | ··· | 30AV |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 10 | ··· | 10 | 15 | ··· | 15 | 15 | ··· | 15 | 30 | ··· | 30 |
| size | 1 | 3 | 1 | 1 | 2 | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C5 | C6 | C10 | C15 | C30 | S3 | C3×S3 | C5×S3 | S3×C15 |
| kernel | S3×C5×C15 | C152 | S3×C52 | S3×C15 | C5×C15 | C3×C15 | C5×S3 | C15 | C5×C15 | C52 | C15 | C5 |
| # reps | 1 | 1 | 2 | 24 | 2 | 24 | 48 | 48 | 1 | 2 | 24 | 48 |
Matrix representation of S3×C5×C15 ►in GL3(𝔽31) generated by
| 16 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 16 |
| 5 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 25 |
| 1 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 4 | 0 |
G:=sub<GL(3,GF(31))| [16,0,0,0,16,0,0,0,16],[5,0,0,0,18,0,0,0,18],[1,0,0,0,5,0,0,0,25],[1,0,0,0,0,4,0,8,0] >;
S3×C5×C15 in GAP, Magma, Sage, TeX
S_3\times C_5\times C_{15} % in TeX
G:=Group("S3xC5xC15"); // GroupNames label
G:=SmallGroup(450,28);
// by ID
G=gap.SmallGroup(450,28);
# by ID
G:=PCGroup([5,-2,-3,-5,-5,-3,7504]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^15=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