direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3×C5⋊D15, C15⋊2D15, C152⋊2C2, C5⋊(C3×D15), (C5×C15)⋊5C6, (C5×C15)⋊6S3, (C3×C15)⋊2D5, C15⋊1(C3×D5), C52⋊5(C3×S3), C32⋊1(C5⋊D5), C3⋊(C3×C5⋊D5), SmallGroup(450,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C3×C5⋊D15 |
Generators and relations for C3×C5⋊D15
G = < a,b,c,d | a3=b5=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 496 in 72 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C3, C3, C5, S3, C6, C32, D5, C15, C15, C3×S3, C52, C3×D5, D15, C3×C15, C5⋊D5, C5×C15, C5×C15, C3×D15, C3×C5⋊D5, C5⋊D15, C152, C3×C5⋊D15
Quotients: C1, C2, C3, S3, C6, D5, C3×S3, C3×D5, D15, C5⋊D5, C3×D15, C3×C5⋊D5, C5⋊D15, C3×C5⋊D15
►On 150 points
Generators in S150
(1 11 6)(2 12 7)(3 13 8)(4 14 9)(5 15 10)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 36 41)(32 37 42)(33 38 43)(34 39 44)(35 40 45)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 66 71)(62 67 72)(63 68 73)(64 69 74)(65 70 75)(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)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)(121 131 126)(122 132 127)(123 133 128)(124 134 129)(125 135 130)(136 146 141)(137 147 142)(138 148 143)(139 149 144)(140 150 145)
(1 53 145 125 118)(2 54 146 126 119)(3 55 147 127 120)(4 56 148 128 106)(5 57 149 129 107)(6 58 150 130 108)(7 59 136 131 109)(8 60 137 132 110)(9 46 138 133 111)(10 47 139 134 112)(11 48 140 135 113)(12 49 141 121 114)(13 50 142 122 115)(14 51 143 123 116)(15 52 144 124 117)(16 44 67 98 83)(17 45 68 99 84)(18 31 69 100 85)(19 32 70 101 86)(20 33 71 102 87)(21 34 72 103 88)(22 35 73 104 89)(23 36 74 105 90)(24 37 75 91 76)(25 38 61 92 77)(26 39 62 93 78)(27 40 63 94 79)(28 41 64 95 80)(29 42 65 96 81)(30 43 66 97 82)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 75)(11 74)(12 73)(13 72)(14 71)(15 70)(16 147)(17 146)(18 145)(19 144)(20 143)(21 142)(22 141)(23 140)(24 139)(25 138)(26 137)(27 136)(28 150)(29 149)(30 148)(31 53)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(76 134)(77 133)(78 132)(79 131)(80 130)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 135)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)
G:=sub<Sym(150)| (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(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)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130)(136,146,141)(137,147,142)(138,148,143)(139,149,144)(140,150,145), (1,53,145,125,118)(2,54,146,126,119)(3,55,147,127,120)(4,56,148,128,106)(5,57,149,129,107)(6,58,150,130,108)(7,59,136,131,109)(8,60,137,132,110)(9,46,138,133,111)(10,47,139,134,112)(11,48,140,135,113)(12,49,141,121,114)(13,50,142,122,115)(14,51,143,123,116)(15,52,144,124,117)(16,44,67,98,83)(17,45,68,99,84)(18,31,69,100,85)(19,32,70,101,86)(20,33,71,102,87)(21,34,72,103,88)(22,35,73,104,89)(23,36,74,105,90)(24,37,75,91,76)(25,38,61,92,77)(26,39,62,93,78)(27,40,63,94,79)(28,41,64,95,80)(29,42,65,96,81)(30,43,66,97,82), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,147)(17,146)(18,145)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,138)(26,137)(27,136)(28,150)(29,149)(30,148)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(76,134)(77,133)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,135)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)>;
G:=Group( (1,11,6)(2,12,7)(3,13,8)(4,14,9)(5,15,10)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,36,41)(32,37,42)(33,38,43)(34,39,44)(35,40,45)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55)(61,66,71)(62,67,72)(63,68,73)(64,69,74)(65,70,75)(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)(106,116,111)(107,117,112)(108,118,113)(109,119,114)(110,120,115)(121,131,126)(122,132,127)(123,133,128)(124,134,129)(125,135,130)(136,146,141)(137,147,142)(138,148,143)(139,149,144)(140,150,145), (1,53,145,125,118)(2,54,146,126,119)(3,55,147,127,120)(4,56,148,128,106)(5,57,149,129,107)(6,58,150,130,108)(7,59,136,131,109)(8,60,137,132,110)(9,46,138,133,111)(10,47,139,134,112)(11,48,140,135,113)(12,49,141,121,114)(13,50,142,122,115)(14,51,143,123,116)(15,52,144,124,117)(16,44,67,98,83)(17,45,68,99,84)(18,31,69,100,85)(19,32,70,101,86)(20,33,71,102,87)(21,34,72,103,88)(22,35,73,104,89)(23,36,74,105,90)(24,37,75,91,76)(25,38,61,92,77)(26,39,62,93,78)(27,40,63,94,79)(28,41,64,95,80)(29,42,65,96,81)(30,43,66,97,82), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,147)(17,146)(18,145)(19,144)(20,143)(21,142)(22,141)(23,140)(24,139)(25,138)(26,137)(27,136)(28,150)(29,149)(30,148)(31,53)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(76,134)(77,133)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,135)(91,112)(92,111)(93,110)(94,109)(95,108)(96,107)(97,106)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113) );
G=PermutationGroup([[(1,11,6),(2,12,7),(3,13,8),(4,14,9),(5,15,10),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,36,41),(32,37,42),(33,38,43),(34,39,44),(35,40,45),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55),(61,66,71),(62,67,72),(63,68,73),(64,69,74),(65,70,75),(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),(106,116,111),(107,117,112),(108,118,113),(109,119,114),(110,120,115),(121,131,126),(122,132,127),(123,133,128),(124,134,129),(125,135,130),(136,146,141),(137,147,142),(138,148,143),(139,149,144),(140,150,145)], [(1,53,145,125,118),(2,54,146,126,119),(3,55,147,127,120),(4,56,148,128,106),(5,57,149,129,107),(6,58,150,130,108),(7,59,136,131,109),(8,60,137,132,110),(9,46,138,133,111),(10,47,139,134,112),(11,48,140,135,113),(12,49,141,121,114),(13,50,142,122,115),(14,51,143,123,116),(15,52,144,124,117),(16,44,67,98,83),(17,45,68,99,84),(18,31,69,100,85),(19,32,70,101,86),(20,33,71,102,87),(21,34,72,103,88),(22,35,73,104,89),(23,36,74,105,90),(24,37,75,91,76),(25,38,61,92,77),(26,39,62,93,78),(27,40,63,94,79),(28,41,64,95,80),(29,42,65,96,81),(30,43,66,97,82)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,75),(11,74),(12,73),(13,72),(14,71),(15,70),(16,147),(17,146),(18,145),(19,144),(20,143),(21,142),(22,141),(23,140),(24,139),(25,138),(26,137),(27,136),(28,150),(29,149),(30,148),(31,53),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54),(76,134),(77,133),(78,132),(79,131),(80,130),(81,129),(82,128),(83,127),(84,126),(85,125),(86,124),(87,123),(88,122),(89,121),(90,135),(91,112),(92,111),(93,110),(94,109),(95,108),(96,107),(97,106),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113)]])
117 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | ··· | 5L | 6A | 6B | 15A | ··· | 15CR |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | ··· | 5 | 6 | 6 | 15 | ··· | 15 |
| size | 1 | 75 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 75 | 75 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | D5 | C3×S3 | C3×D5 | D15 | C3×D15 |
| kernel | C3×C5⋊D15 | C152 | C5⋊D15 | C5×C15 | C5×C15 | C3×C15 | C52 | C15 | C15 | C5 |
| # reps | 1 | 1 | 2 | 2 | 1 | 12 | 2 | 24 | 24 | 48 |
Matrix representation of C3×C5⋊D15 ►in GL4(𝔽31) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 25 |
| 16 | 0 | 0 | 0 |
| 17 | 2 | 0 | 0 |
| 0 | 0 | 2 | 22 |
| 0 | 0 | 0 | 16 |
| 18 | 0 | 0 | 0 |
| 1 | 19 | 0 | 0 |
| 0 | 0 | 14 | 5 |
| 0 | 0 | 0 | 20 |
| 26 | 26 | 0 | 0 |
| 11 | 5 | 0 | 0 |
| 0 | 0 | 2 | 22 |
| 0 | 0 | 21 | 29 |
G:=sub<GL(4,GF(31))| [5,0,0,0,0,5,0,0,0,0,25,0,0,0,0,25],[16,17,0,0,0,2,0,0,0,0,2,0,0,0,22,16],[18,1,0,0,0,19,0,0,0,0,14,0,0,0,5,20],[26,11,0,0,26,5,0,0,0,0,2,21,0,0,22,29] >;
C3×C5⋊D15 in GAP, Magma, Sage, TeX
C_3\times C_5\rtimes D_{15} % in TeX
G:=Group("C3xC5:D15"); // GroupNames label
G:=SmallGroup(450,30);
// by ID
G=gap.SmallGroup(450,30);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,182,1443,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^5=c^15=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations