direct product, abelian, monomial, 3-elementary
Aliases: C32×C15, SmallGroup(135,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C15 |
| C1 — C32×C15 |
| C1 — C32×C15 |
Generators and relations for C32×C15
G = < a,b,c | a3=b3=c15=1, ab=ba, ac=ca, bc=cb >
Subgroups: 56, all normal (4 characteristic)
C1, C3, C5, C32, C15, C33, C3×C15, C32×C15
Quotients: C1, C3, C5, C32, C15, C33, C3×C15, C32×C15
►Regular action on 135 points
Generators in S135
(1 91 59)(2 92 60)(3 93 46)(4 94 47)(5 95 48)(6 96 49)(7 97 50)(8 98 51)(9 99 52)(10 100 53)(11 101 54)(12 102 55)(13 103 56)(14 104 57)(15 105 58)(16 119 61)(17 120 62)(18 106 63)(19 107 64)(20 108 65)(21 109 66)(22 110 67)(23 111 68)(24 112 69)(25 113 70)(26 114 71)(27 115 72)(28 116 73)(29 117 74)(30 118 75)(31 132 89)(32 133 90)(33 134 76)(34 135 77)(35 121 78)(36 122 79)(37 123 80)(38 124 81)(39 125 82)(40 126 83)(41 127 84)(42 128 85)(43 129 86)(44 130 87)(45 131 88)
(1 31 25)(2 32 26)(3 33 27)(4 34 28)(5 35 29)(6 36 30)(7 37 16)(8 38 17)(9 39 18)(10 40 19)(11 41 20)(12 42 21)(13 43 22)(14 44 23)(15 45 24)(46 76 72)(47 77 73)(48 78 74)(49 79 75)(50 80 61)(51 81 62)(52 82 63)(53 83 64)(54 84 65)(55 85 66)(56 86 67)(57 87 68)(58 88 69)(59 89 70)(60 90 71)(91 132 113)(92 133 114)(93 134 115)(94 135 116)(95 121 117)(96 122 118)(97 123 119)(98 124 120)(99 125 106)(100 126 107)(101 127 108)(102 128 109)(103 129 110)(104 130 111)(105 131 112)
(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)
G:=sub<Sym(135)| (1,91,59)(2,92,60)(3,93,46)(4,94,47)(5,95,48)(6,96,49)(7,97,50)(8,98,51)(9,99,52)(10,100,53)(11,101,54)(12,102,55)(13,103,56)(14,104,57)(15,105,58)(16,119,61)(17,120,62)(18,106,63)(19,107,64)(20,108,65)(21,109,66)(22,110,67)(23,111,68)(24,112,69)(25,113,70)(26,114,71)(27,115,72)(28,116,73)(29,117,74)(30,118,75)(31,132,89)(32,133,90)(33,134,76)(34,135,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86)(44,130,87)(45,131,88), (1,31,25)(2,32,26)(3,33,27)(4,34,28)(5,35,29)(6,36,30)(7,37,16)(8,38,17)(9,39,18)(10,40,19)(11,41,20)(12,42,21)(13,43,22)(14,44,23)(15,45,24)(46,76,72)(47,77,73)(48,78,74)(49,79,75)(50,80,61)(51,81,62)(52,82,63)(53,83,64)(54,84,65)(55,85,66)(56,86,67)(57,87,68)(58,88,69)(59,89,70)(60,90,71)(91,132,113)(92,133,114)(93,134,115)(94,135,116)(95,121,117)(96,122,118)(97,123,119)(98,124,120)(99,125,106)(100,126,107)(101,127,108)(102,128,109)(103,129,110)(104,130,111)(105,131,112), (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)>;
G:=Group( (1,91,59)(2,92,60)(3,93,46)(4,94,47)(5,95,48)(6,96,49)(7,97,50)(8,98,51)(9,99,52)(10,100,53)(11,101,54)(12,102,55)(13,103,56)(14,104,57)(15,105,58)(16,119,61)(17,120,62)(18,106,63)(19,107,64)(20,108,65)(21,109,66)(22,110,67)(23,111,68)(24,112,69)(25,113,70)(26,114,71)(27,115,72)(28,116,73)(29,117,74)(30,118,75)(31,132,89)(32,133,90)(33,134,76)(34,135,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86)(44,130,87)(45,131,88), (1,31,25)(2,32,26)(3,33,27)(4,34,28)(5,35,29)(6,36,30)(7,37,16)(8,38,17)(9,39,18)(10,40,19)(11,41,20)(12,42,21)(13,43,22)(14,44,23)(15,45,24)(46,76,72)(47,77,73)(48,78,74)(49,79,75)(50,80,61)(51,81,62)(52,82,63)(53,83,64)(54,84,65)(55,85,66)(56,86,67)(57,87,68)(58,88,69)(59,89,70)(60,90,71)(91,132,113)(92,133,114)(93,134,115)(94,135,116)(95,121,117)(96,122,118)(97,123,119)(98,124,120)(99,125,106)(100,126,107)(101,127,108)(102,128,109)(103,129,110)(104,130,111)(105,131,112), (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) );
G=PermutationGroup([[(1,91,59),(2,92,60),(3,93,46),(4,94,47),(5,95,48),(6,96,49),(7,97,50),(8,98,51),(9,99,52),(10,100,53),(11,101,54),(12,102,55),(13,103,56),(14,104,57),(15,105,58),(16,119,61),(17,120,62),(18,106,63),(19,107,64),(20,108,65),(21,109,66),(22,110,67),(23,111,68),(24,112,69),(25,113,70),(26,114,71),(27,115,72),(28,116,73),(29,117,74),(30,118,75),(31,132,89),(32,133,90),(33,134,76),(34,135,77),(35,121,78),(36,122,79),(37,123,80),(38,124,81),(39,125,82),(40,126,83),(41,127,84),(42,128,85),(43,129,86),(44,130,87),(45,131,88)], [(1,31,25),(2,32,26),(3,33,27),(4,34,28),(5,35,29),(6,36,30),(7,37,16),(8,38,17),(9,39,18),(10,40,19),(11,41,20),(12,42,21),(13,43,22),(14,44,23),(15,45,24),(46,76,72),(47,77,73),(48,78,74),(49,79,75),(50,80,61),(51,81,62),(52,82,63),(53,83,64),(54,84,65),(55,85,66),(56,86,67),(57,87,68),(58,88,69),(59,89,70),(60,90,71),(91,132,113),(92,133,114),(93,134,115),(94,135,116),(95,121,117),(96,122,118),(97,123,119),(98,124,120),(99,125,106),(100,126,107),(101,127,108),(102,128,109),(103,129,110),(104,130,111),(105,131,112)], [(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)]])
C32×C15 is a maximal subgroup of
C33⋊D5
135 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 5A | 5B | 5C | 5D | 15A | ··· | 15CZ |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 15 | ··· | 15 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C5 | C15 |
| kernel | C32×C15 | C3×C15 | C33 | C32 |
| # reps | 1 | 26 | 4 | 104 |
Matrix representation of C32×C15 ►in GL3(𝔽31) generated by
| 5 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 5 |
| 5 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 25 |
| 25 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 28 |
G:=sub<GL(3,GF(31))| [5,0,0,0,25,0,0,0,5],[5,0,0,0,25,0,0,0,25],[25,0,0,0,7,0,0,0,28] >;
C32×C15 in GAP, Magma, Sage, TeX
C_3^2\times C_{15} % in TeX
G:=Group("C3^2xC15"); // GroupNames label
G:=SmallGroup(135,5);
// by ID
G=gap.SmallGroup(135,5);
# by ID
G:=PCGroup([4,-3,-3,-3,-5]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^15=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations