direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C5×C27⋊C3, C135⋊C3, C27⋊C15, C9.C45, C45.C9, C32.C45, C45.2C32, (C3×C15).C9, C3.3(C3×C45), (C3×C9).3C15, C9.2(C3×C15), (C3×C45).3C3, C15.3(C3×C9), SmallGroup(405,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C27⋊C3
G = < a,b,c | a5=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C5×C27⋊C3
►On 135 points
Generators in S135
(1 28 101 131 76)(2 29 102 132 77)(3 30 103 133 78)(4 31 104 134 79)(5 32 105 135 80)(6 33 106 109 81)(7 34 107 110 55)(8 35 108 111 56)(9 36 82 112 57)(10 37 83 113 58)(11 38 84 114 59)(12 39 85 115 60)(13 40 86 116 61)(14 41 87 117 62)(15 42 88 118 63)(16 43 89 119 64)(17 44 90 120 65)(18 45 91 121 66)(19 46 92 122 67)(20 47 93 123 68)(21 48 94 124 69)(22 49 95 125 70)(23 50 96 126 71)(24 51 97 127 72)(25 52 98 128 73)(26 53 99 129 74)(27 54 100 130 75)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)(56 74 65)(57 66 75)(59 77 68)(60 69 78)(62 80 71)(63 72 81)(82 91 100)(84 102 93)(85 94 103)(87 105 96)(88 97 106)(90 108 99)(109 118 127)(111 129 120)(112 121 130)(114 132 123)(115 124 133)(117 135 126)
G:=sub<Sym(135)| (1,28,101,131,76)(2,29,102,132,77)(3,30,103,133,78)(4,31,104,134,79)(5,32,105,135,80)(6,33,106,109,81)(7,34,107,110,55)(8,35,108,111,56)(9,36,82,112,57)(10,37,83,113,58)(11,38,84,114,59)(12,39,85,115,60)(13,40,86,116,61)(14,41,87,117,62)(15,42,88,118,63)(16,43,89,119,64)(17,44,90,120,65)(18,45,91,121,66)(19,46,92,122,67)(20,47,93,123,68)(21,48,94,124,69)(22,49,95,125,70)(23,50,96,126,71)(24,51,97,127,72)(25,52,98,128,73)(26,53,99,129,74)(27,54,100,130,75), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(56,74,65)(57,66,75)(59,77,68)(60,69,78)(62,80,71)(63,72,81)(82,91,100)(84,102,93)(85,94,103)(87,105,96)(88,97,106)(90,108,99)(109,118,127)(111,129,120)(112,121,130)(114,132,123)(115,124,133)(117,135,126)>;
G:=Group( (1,28,101,131,76)(2,29,102,132,77)(3,30,103,133,78)(4,31,104,134,79)(5,32,105,135,80)(6,33,106,109,81)(7,34,107,110,55)(8,35,108,111,56)(9,36,82,112,57)(10,37,83,113,58)(11,38,84,114,59)(12,39,85,115,60)(13,40,86,116,61)(14,41,87,117,62)(15,42,88,118,63)(16,43,89,119,64)(17,44,90,120,65)(18,45,91,121,66)(19,46,92,122,67)(20,47,93,123,68)(21,48,94,124,69)(22,49,95,125,70)(23,50,96,126,71)(24,51,97,127,72)(25,52,98,128,73)(26,53,99,129,74)(27,54,100,130,75), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)(56,74,65)(57,66,75)(59,77,68)(60,69,78)(62,80,71)(63,72,81)(82,91,100)(84,102,93)(85,94,103)(87,105,96)(88,97,106)(90,108,99)(109,118,127)(111,129,120)(112,121,130)(114,132,123)(115,124,133)(117,135,126) );
G=PermutationGroup([[(1,28,101,131,76),(2,29,102,132,77),(3,30,103,133,78),(4,31,104,134,79),(5,32,105,135,80),(6,33,106,109,81),(7,34,107,110,55),(8,35,108,111,56),(9,36,82,112,57),(10,37,83,113,58),(11,38,84,114,59),(12,39,85,115,60),(13,40,86,116,61),(14,41,87,117,62),(15,42,88,118,63),(16,43,89,119,64),(17,44,90,120,65),(18,45,91,121,66),(19,46,92,122,67),(20,47,93,123,68),(21,48,94,124,69),(22,49,95,125,70),(23,50,96,126,71),(24,51,97,127,72),(25,52,98,128,73),(26,53,99,129,74),(27,54,100,130,75)], [(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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54),(56,74,65),(57,66,75),(59,77,68),(60,69,78),(62,80,71),(63,72,81),(82,91,100),(84,102,93),(85,94,103),(87,105,96),(88,97,106),(90,108,99),(109,118,127),(111,129,120),(112,121,130),(114,132,123),(115,124,133),(117,135,126)]])
165 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 5A | 5B | 5C | 5D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 15A | ··· | 15H | 15I | ··· | 15P | 27A | ··· | 27R | 45A | ··· | 45X | 45Y | ··· | 45AN | 135A | ··· | 135BT |
| order | 1 | 3 | 3 | 3 | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 15 | ··· | 15 | 15 | ··· | 15 | 27 | ··· | 27 | 45 | ··· | 45 | 45 | ··· | 45 | 135 | ··· | 135 |
| size | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
165 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||||||||
| image | C1 | C3 | C3 | C5 | C9 | C9 | C15 | C15 | C45 | C45 | C27⋊C3 | C5×C27⋊C3 |
| kernel | C5×C27⋊C3 | C135 | C3×C45 | C27⋊C3 | C45 | C3×C15 | C27 | C3×C9 | C9 | C32 | C5 | C1 |
| # reps | 1 | 6 | 2 | 4 | 12 | 6 | 24 | 8 | 48 | 24 | 6 | 24 |
Matrix representation of C5×C27⋊C3 ►in GL3(𝔽271) generated by
| 10 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 10 |
| 70 | 168 | 195 |
| 0 | 0 | 242 |
| 191 | 145 | 201 |
| 1 | 115 | 78 |
| 0 | 242 | 0 |
| 0 | 0 | 28 |
G:=sub<GL(3,GF(271))| [10,0,0,0,10,0,0,0,10],[70,0,191,168,0,145,195,242,201],[1,0,0,115,242,0,78,0,28] >;
C5×C27⋊C3 in GAP, Magma, Sage, TeX
C_5\times C_{27}\rtimes C_3 % in TeX
G:=Group("C5xC27:C3"); // GroupNames label
G:=SmallGroup(405,6);
// by ID
G=gap.SmallGroup(405,6);
# by ID
G:=PCGroup([5,-3,-3,-5,-3,-3,225,1381,78]);
// Polycyclic
G:=Group<a,b,c|a^5=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
Export