direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D25, C50⋊C6, C150⋊2C2, C75⋊3C22, C30.5D5, C15.3D10, C25⋊(C2×C6), C5.(C6×D5), C10.2(C3×D5), SmallGroup(300,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C6×D25 |
Generators and relations for C6×D25
G = < a,b,c | a6=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D25
►On 150 points
Generators in S150
(1 101 64 95 34 130)(2 102 65 96 35 131)(3 103 66 97 36 132)(4 104 67 98 37 133)(5 105 68 99 38 134)(6 106 69 100 39 135)(7 107 70 76 40 136)(8 108 71 77 41 137)(9 109 72 78 42 138)(10 110 73 79 43 139)(11 111 74 80 44 140)(12 112 75 81 45 141)(13 113 51 82 46 142)(14 114 52 83 47 143)(15 115 53 84 48 144)(16 116 54 85 49 145)(17 117 55 86 50 146)(18 118 56 87 26 147)(19 119 57 88 27 148)(20 120 58 89 28 149)(21 121 59 90 29 150)(22 122 60 91 30 126)(23 123 61 92 31 127)(24 124 62 93 32 128)(25 125 63 94 33 129)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(42 50)(43 49)(44 48)(45 47)(52 75)(53 74)(54 73)(55 72)(56 71)(57 70)(58 69)(59 68)(60 67)(61 66)(62 65)(63 64)(76 88)(77 87)(78 86)(79 85)(80 84)(81 83)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 125)(102 124)(103 123)(104 122)(105 121)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(126 133)(127 132)(128 131)(129 130)(134 150)(135 149)(136 148)(137 147)(138 146)(139 145)(140 144)(141 143)
G:=sub<Sym(150)| (1,101,64,95,34,130)(2,102,65,96,35,131)(3,103,66,97,36,132)(4,104,67,98,37,133)(5,105,68,99,38,134)(6,106,69,100,39,135)(7,107,70,76,40,136)(8,108,71,77,41,137)(9,109,72,78,42,138)(10,110,73,79,43,139)(11,111,74,80,44,140)(12,112,75,81,45,141)(13,113,51,82,46,142)(14,114,52,83,47,143)(15,115,53,84,48,144)(16,116,54,85,49,145)(17,117,55,86,50,146)(18,118,56,87,26,147)(19,119,57,88,27,148)(20,120,58,89,28,149)(21,121,59,90,29,150)(22,122,60,91,30,126)(23,123,61,92,31,127)(24,124,62,93,32,128)(25,125,63,94,33,129), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(42,50)(43,49)(44,48)(45,47)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,125)(102,124)(103,123)(104,122)(105,121)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(126,133)(127,132)(128,131)(129,130)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143)>;
G:=Group( (1,101,64,95,34,130)(2,102,65,96,35,131)(3,103,66,97,36,132)(4,104,67,98,37,133)(5,105,68,99,38,134)(6,106,69,100,39,135)(7,107,70,76,40,136)(8,108,71,77,41,137)(9,109,72,78,42,138)(10,110,73,79,43,139)(11,111,74,80,44,140)(12,112,75,81,45,141)(13,113,51,82,46,142)(14,114,52,83,47,143)(15,115,53,84,48,144)(16,116,54,85,49,145)(17,117,55,86,50,146)(18,118,56,87,26,147)(19,119,57,88,27,148)(20,120,58,89,28,149)(21,121,59,90,29,150)(22,122,60,91,30,126)(23,123,61,92,31,127)(24,124,62,93,32,128)(25,125,63,94,33,129), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(42,50)(43,49)(44,48)(45,47)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)(76,88)(77,87)(78,86)(79,85)(80,84)(81,83)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)(101,125)(102,124)(103,123)(104,122)(105,121)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(126,133)(127,132)(128,131)(129,130)(134,150)(135,149)(136,148)(137,147)(138,146)(139,145)(140,144)(141,143) );
G=PermutationGroup([[(1,101,64,95,34,130),(2,102,65,96,35,131),(3,103,66,97,36,132),(4,104,67,98,37,133),(5,105,68,99,38,134),(6,106,69,100,39,135),(7,107,70,76,40,136),(8,108,71,77,41,137),(9,109,72,78,42,138),(10,110,73,79,43,139),(11,111,74,80,44,140),(12,112,75,81,45,141),(13,113,51,82,46,142),(14,114,52,83,47,143),(15,115,53,84,48,144),(16,116,54,85,49,145),(17,117,55,86,50,146),(18,118,56,87,26,147),(19,119,57,88,27,148),(20,120,58,89,28,149),(21,121,59,90,29,150),(22,122,60,91,30,126),(23,123,61,92,31,127),(24,124,62,93,32,128),(25,125,63,94,33,129)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(26,41),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(42,50),(43,49),(44,48),(45,47),(52,75),(53,74),(54,73),(55,72),(56,71),(57,70),(58,69),(59,68),(60,67),(61,66),(62,65),(63,64),(76,88),(77,87),(78,86),(79,85),(80,84),(81,83),(89,100),(90,99),(91,98),(92,97),(93,96),(94,95),(101,125),(102,124),(103,123),(104,122),(105,121),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(126,133),(127,132),(128,131),(129,130),(134,150),(135,149),(136,148),(137,147),(138,146),(139,145),(140,144),(141,143)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 10A | 10B | 15A | 15B | 15C | 15D | 25A | ··· | 25J | 30A | 30B | 30C | 30D | 50A | ··· | 50J | 75A | ··· | 75T | 150A | ··· | 150T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 15 | 15 | 15 | 15 | 25 | ··· | 25 | 30 | 30 | 30 | 30 | 50 | ··· | 50 | 75 | ··· | 75 | 150 | ··· | 150 |
| size | 1 | 1 | 25 | 25 | 1 | 1 | 2 | 2 | 1 | 1 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | D25 | C6×D5 | D50 | C3×D25 | C6×D25 |
| kernel | C6×D25 | C3×D25 | C150 | D50 | D25 | C50 | C30 | C15 | C10 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 10 | 4 | 10 | 20 | 20 |
Matrix representation of C6×D25 ►in GL3(𝔽151) generated by
| 150 | 0 | 0 |
| 0 | 32 | 0 |
| 0 | 0 | 32 |
| 1 | 0 | 0 |
| 0 | 36 | 80 |
| 0 | 71 | 11 |
| 150 | 0 | 0 |
| 0 | 11 | 74 |
| 0 | 80 | 140 |
G:=sub<GL(3,GF(151))| [150,0,0,0,32,0,0,0,32],[1,0,0,0,36,71,0,80,11],[150,0,0,0,11,80,0,74,140] >;
C6×D25 in GAP, Magma, Sage, TeX
C_6\times D_{25} % in TeX
G:=Group("C6xD25"); // GroupNames label
G:=SmallGroup(300,9);
// by ID
G=gap.SmallGroup(300,9);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,2163,418,6004]);
// Polycyclic
G:=Group<a,b,c|a^6=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export