direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C3×D75, C75⋊1C6, C75⋊2S3, C32⋊1D25, C15.4D15, C25⋊(C3×S3), C3⋊(C3×D25), (C3×C75)⋊2C2, C5.(C3×D15), (C3×C15).2D5, C15.1(C3×D5), SmallGroup(450,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C75 — C3×D75 |
Generators and relations for C3×D75
G = < a,b,c | a3=b75=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D75
►On 150 points
Generators in S150
(1 51 26)(2 52 27)(3 53 28)(4 54 29)(5 55 30)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 61 36)(12 62 37)(13 63 38)(14 64 39)(15 65 40)(16 66 41)(17 67 42)(18 68 43)(19 69 44)(20 70 45)(21 71 46)(22 72 47)(23 73 48)(24 74 49)(25 75 50)(76 101 126)(77 102 127)(78 103 128)(79 104 129)(80 105 130)(81 106 131)(82 107 132)(83 108 133)(84 109 134)(85 110 135)(86 111 136)(87 112 137)(88 113 138)(89 114 139)(90 115 140)(91 116 141)(92 117 142)(93 118 143)(94 119 144)(95 120 145)(96 121 146)(97 122 147)(98 123 148)(99 124 149)(100 125 150)
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 120)(32 119)(33 118)(34 117)(35 116)(36 115)(37 114)(38 113)(39 112)(40 111)(41 110)(42 109)(43 108)(44 107)(45 106)(46 105)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)(53 98)(54 97)(55 96)(56 95)(57 94)(58 93)(59 92)(60 91)(61 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)
G:=sub<Sym(150)| (1,51,26)(2,52,27)(3,53,28)(4,54,29)(5,55,30)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,61,36)(12,62,37)(13,63,38)(14,64,39)(15,65,40)(16,66,41)(17,67,42)(18,68,43)(19,69,44)(20,70,45)(21,71,46)(22,72,47)(23,73,48)(24,74,49)(25,75,50)(76,101,126)(77,102,127)(78,103,128)(79,104,129)(80,105,130)(81,106,131)(82,107,132)(83,108,133)(84,109,134)(85,110,135)(86,111,136)(87,112,137)(88,113,138)(89,114,139)(90,115,140)(91,116,141)(92,117,142)(93,118,143)(94,119,144)(95,120,145)(96,121,146)(97,122,147)(98,123,148)(99,124,149)(100,125,150), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)>;
G:=Group( (1,51,26)(2,52,27)(3,53,28)(4,54,29)(5,55,30)(6,56,31)(7,57,32)(8,58,33)(9,59,34)(10,60,35)(11,61,36)(12,62,37)(13,63,38)(14,64,39)(15,65,40)(16,66,41)(17,67,42)(18,68,43)(19,69,44)(20,70,45)(21,71,46)(22,72,47)(23,73,48)(24,74,49)(25,75,50)(76,101,126)(77,102,127)(78,103,128)(79,104,129)(80,105,130)(81,106,131)(82,107,132)(83,108,133)(84,109,134)(85,110,135)(86,111,136)(87,112,137)(88,113,138)(89,114,139)(90,115,140)(91,116,141)(92,117,142)(93,118,143)(94,119,144)(95,120,145)(96,121,146)(97,122,147)(98,123,148)(99,124,149)(100,125,150), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76) );
G=PermutationGroup([[(1,51,26),(2,52,27),(3,53,28),(4,54,29),(5,55,30),(6,56,31),(7,57,32),(8,58,33),(9,59,34),(10,60,35),(11,61,36),(12,62,37),(13,63,38),(14,64,39),(15,65,40),(16,66,41),(17,67,42),(18,68,43),(19,69,44),(20,70,45),(21,71,46),(22,72,47),(23,73,48),(24,74,49),(25,75,50),(76,101,126),(77,102,127),(78,103,128),(79,104,129),(80,105,130),(81,106,131),(82,107,132),(83,108,133),(84,109,134),(85,110,135),(86,111,136),(87,112,137),(88,113,138),(89,114,139),(90,115,140),(91,116,141),(92,117,142),(93,118,143),(94,119,144),(95,120,145),(96,121,146),(97,122,147),(98,123,148),(99,124,149),(100,125,150)], [(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)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,120),(32,119),(33,118),(34,117),(35,116),(36,115),(37,114),(38,113),(39,112),(40,111),(41,110),(42,109),(43,108),(44,107),(45,106),(46,105),(47,104),(48,103),(49,102),(50,101),(51,100),(52,99),(53,98),(54,97),(55,96),(56,95),(57,94),(58,93),(59,92),(60,91),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76)]])
117 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 5A | 5B | 6A | 6B | 15A | ··· | 15P | 25A | ··· | 25J | 75A | ··· | 75CB |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 5 | 5 | 6 | 6 | 15 | ··· | 15 | 25 | ··· | 25 | 75 | ··· | 75 |
| size | 1 | 75 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 75 | 75 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | S3 | D5 | C3×S3 | C3×D5 | D15 | D25 | C3×D15 | C3×D25 | D75 | C3×D75 |
| kernel | C3×D75 | C3×C75 | D75 | C75 | C75 | C3×C15 | C25 | C15 | C15 | C32 | C5 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 10 | 8 | 20 | 20 | 40 |
Matrix representation of C3×D75 ►in GL2(𝔽151) generated by
| 32 | 0 |
| 0 | 32 |
| 11 | 0 |
| 0 | 55 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(151))| [32,0,0,32],[11,0,0,55],[0,1,1,0] >;
C3×D75 in GAP, Magma, Sage, TeX
C_3\times D_{75} % in TeX
G:=Group("C3xD75"); // GroupNames label
G:=SmallGroup(450,7);
// by ID
G=gap.SmallGroup(450,7);
# by ID
G:=PCGroup([5,-2,-3,-3,-5,-5,182,3243,418,9004]);
// Polycyclic
G:=Group<a,b,c|a^3=b^75=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export