direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C27, C5⋊C54, C45.C6, C135⋊3C2, C15.C18, C9.(C3×D5), C3.(C9×D5), (C9×D5).C3, (C3×D5).C9, SmallGroup(270,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C27 |
Generators and relations for D5×C27
G = < a,b,c | a27=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C27
►On 135 points
Generators in S135
(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)
(1 28 130 62 91)(2 29 131 63 92)(3 30 132 64 93)(4 31 133 65 94)(5 32 134 66 95)(6 33 135 67 96)(7 34 109 68 97)(8 35 110 69 98)(9 36 111 70 99)(10 37 112 71 100)(11 38 113 72 101)(12 39 114 73 102)(13 40 115 74 103)(14 41 116 75 104)(15 42 117 76 105)(16 43 118 77 106)(17 44 119 78 107)(18 45 120 79 108)(19 46 121 80 82)(20 47 122 81 83)(21 48 123 55 84)(22 49 124 56 85)(23 50 125 57 86)(24 51 126 58 87)(25 52 127 59 88)(26 53 128 60 89)(27 54 129 61 90)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 82)(20 83)(21 84)(22 85)(23 86)(24 87)(25 88)(26 89)(27 90)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)
G:=sub<Sym(135)| (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), (1,28,130,62,91)(2,29,131,63,92)(3,30,132,64,93)(4,31,133,65,94)(5,32,134,66,95)(6,33,135,67,96)(7,34,109,68,97)(8,35,110,69,98)(9,36,111,70,99)(10,37,112,71,100)(11,38,113,72,101)(12,39,114,73,102)(13,40,115,74,103)(14,41,116,75,104)(15,42,117,76,105)(16,43,118,77,106)(17,44,119,78,107)(18,45,120,79,108)(19,46,121,80,82)(20,47,122,81,83)(21,48,123,55,84)(22,49,124,56,85)(23,50,125,57,86)(24,51,126,58,87)(25,52,127,59,88)(26,53,128,60,89)(27,54,129,61,90), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,55)(49,56)(50,57)(51,58)(52,59)(53,60)(54,61)>;
G:=Group( (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), (1,28,130,62,91)(2,29,131,63,92)(3,30,132,64,93)(4,31,133,65,94)(5,32,134,66,95)(6,33,135,67,96)(7,34,109,68,97)(8,35,110,69,98)(9,36,111,70,99)(10,37,112,71,100)(11,38,113,72,101)(12,39,114,73,102)(13,40,115,74,103)(14,41,116,75,104)(15,42,117,76,105)(16,43,118,77,106)(17,44,119,78,107)(18,45,120,79,108)(19,46,121,80,82)(20,47,122,81,83)(21,48,123,55,84)(22,49,124,56,85)(23,50,125,57,86)(24,51,126,58,87)(25,52,127,59,88)(26,53,128,60,89)(27,54,129,61,90), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,82)(20,83)(21,84)(22,85)(23,86)(24,87)(25,88)(26,89)(27,90)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,81)(48,55)(49,56)(50,57)(51,58)(52,59)(53,60)(54,61) );
G=PermutationGroup([[(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)], [(1,28,130,62,91),(2,29,131,63,92),(3,30,132,64,93),(4,31,133,65,94),(5,32,134,66,95),(6,33,135,67,96),(7,34,109,68,97),(8,35,110,69,98),(9,36,111,70,99),(10,37,112,71,100),(11,38,113,72,101),(12,39,114,73,102),(13,40,115,74,103),(14,41,116,75,104),(15,42,117,76,105),(16,43,118,77,106),(17,44,119,78,107),(18,45,120,79,108),(19,46,121,80,82),(20,47,122,81,83),(21,48,123,55,84),(22,49,124,56,85),(23,50,125,57,86),(24,51,126,58,87),(25,52,127,59,88),(26,53,128,60,89),(27,54,129,61,90)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,82),(20,83),(21,84),(22,85),(23,86),(24,87),(25,88),(26,89),(27,90),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,81),(48,55),(49,56),(50,57),(51,58),(52,59),(53,60),(54,61)]])
108 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 9A | ··· | 9F | 15A | 15B | 15C | 15D | 18A | ··· | 18F | 27A | ··· | 27R | 45A | ··· | 45L | 54A | ··· | 54R | 135A | ··· | 135AJ |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 9 | ··· | 9 | 15 | 15 | 15 | 15 | 18 | ··· | 18 | 27 | ··· | 27 | 45 | ··· | 45 | 54 | ··· | 54 | 135 | ··· | 135 |
| size | 1 | 5 | 1 | 1 | 2 | 2 | 5 | 5 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | D5 | C3×D5 | C9×D5 | D5×C27 |
| kernel | D5×C27 | C135 | C9×D5 | C45 | C3×D5 | C15 | D5 | C5 | C27 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 4 | 12 | 36 |
Matrix representation of D5×C27 ►in GL2(𝔽271) generated by
| 158 | 0 |
| 0 | 158 |
| 0 | 1 |
| 270 | 254 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(271))| [158,0,0,158],[0,270,1,254],[0,1,1,0] >;
D5×C27 in GAP, Magma, Sage, TeX
D_5\times C_{27} % in TeX
G:=Group("D5xC27"); // GroupNames label
G:=SmallGroup(270,2);
// by ID
G=gap.SmallGroup(270,2);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-5,36,57,5404]);
// Polycyclic
G:=Group<a,b,c|a^27=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export