direct product, abelian, monomial, 3-elementary
Aliases: C3×C45, SmallGroup(135,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C45 |
| C1 — C3×C45 |
| C1 — C3×C45 |
Generators and relations for C3×C45
G = < a,b | a3=b45=1, ab=ba >
Smallest permutation representation of C3×C45
►Regular action on 135 points
Generators in S135
(1 92 74)(2 93 75)(3 94 76)(4 95 77)(5 96 78)(6 97 79)(7 98 80)(8 99 81)(9 100 82)(10 101 83)(11 102 84)(12 103 85)(13 104 86)(14 105 87)(15 106 88)(16 107 89)(17 108 90)(18 109 46)(19 110 47)(20 111 48)(21 112 49)(22 113 50)(23 114 51)(24 115 52)(25 116 53)(26 117 54)(27 118 55)(28 119 56)(29 120 57)(30 121 58)(31 122 59)(32 123 60)(33 124 61)(34 125 62)(35 126 63)(36 127 64)(37 128 65)(38 129 66)(39 130 67)(40 131 68)(41 132 69)(42 133 70)(43 134 71)(44 135 72)(45 91 73)
(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)| (1,92,74)(2,93,75)(3,94,76)(4,95,77)(5,96,78)(6,97,79)(7,98,80)(8,99,81)(9,100,82)(10,101,83)(11,102,84)(12,103,85)(13,104,86)(14,105,87)(15,106,88)(16,107,89)(17,108,90)(18,109,46)(19,110,47)(20,111,48)(21,112,49)(22,113,50)(23,114,51)(24,115,52)(25,116,53)(26,117,54)(27,118,55)(28,119,56)(29,120,57)(30,121,58)(31,122,59)(32,123,60)(33,124,61)(34,125,62)(35,126,63)(36,127,64)(37,128,65)(38,129,66)(39,130,67)(40,131,68)(41,132,69)(42,133,70)(43,134,71)(44,135,72)(45,91,73), (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( (1,92,74)(2,93,75)(3,94,76)(4,95,77)(5,96,78)(6,97,79)(7,98,80)(8,99,81)(9,100,82)(10,101,83)(11,102,84)(12,103,85)(13,104,86)(14,105,87)(15,106,88)(16,107,89)(17,108,90)(18,109,46)(19,110,47)(20,111,48)(21,112,49)(22,113,50)(23,114,51)(24,115,52)(25,116,53)(26,117,54)(27,118,55)(28,119,56)(29,120,57)(30,121,58)(31,122,59)(32,123,60)(33,124,61)(34,125,62)(35,126,63)(36,127,64)(37,128,65)(38,129,66)(39,130,67)(40,131,68)(41,132,69)(42,133,70)(43,134,71)(44,135,72)(45,91,73), (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([[(1,92,74),(2,93,75),(3,94,76),(4,95,77),(5,96,78),(6,97,79),(7,98,80),(8,99,81),(9,100,82),(10,101,83),(11,102,84),(12,103,85),(13,104,86),(14,105,87),(15,106,88),(16,107,89),(17,108,90),(18,109,46),(19,110,47),(20,111,48),(21,112,49),(22,113,50),(23,114,51),(24,115,52),(25,116,53),(26,117,54),(27,118,55),(28,119,56),(29,120,57),(30,121,58),(31,122,59),(32,123,60),(33,124,61),(34,125,62),(35,126,63),(36,127,64),(37,128,65),(38,129,66),(39,130,67),(40,131,68),(41,132,69),(42,133,70),(43,134,71),(44,135,72),(45,91,73)], [(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)]])
C3×C45 is a maximal subgroup of
C3⋊D45
135 conjugacy classes
| class | 1 | 3A | ··· | 3H | 5A | 5B | 5C | 5D | 9A | ··· | 9R | 15A | ··· | 15AF | 45A | ··· | 45BT |
| order | 1 | 3 | ··· | 3 | 5 | 5 | 5 | 5 | 9 | ··· | 9 | 15 | ··· | 15 | 45 | ··· | 45 |
| size | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | |||||||
| image | C1 | C3 | C3 | C5 | C9 | C15 | C15 | C45 |
| kernel | C3×C45 | C45 | C3×C15 | C3×C9 | C15 | C9 | C32 | C3 |
| # reps | 1 | 6 | 2 | 4 | 18 | 24 | 8 | 72 |
Matrix representation of C3×C45 ►in GL2(𝔽181) generated by
| 48 | 0 |
| 0 | 1 |
| 1 | 0 |
| 0 | 87 |
G:=sub<GL(2,GF(181))| [48,0,0,1],[1,0,0,87] >;
C3×C45 in GAP, Magma, Sage, TeX
C_3\times C_{45} % in TeX
G:=Group("C3xC45"); // GroupNames label
G:=SmallGroup(135,2);
// by ID
G=gap.SmallGroup(135,2);
# by ID
G:=PCGroup([4,-3,-3,-5,-3,180]);
// Polycyclic
G:=Group<a,b|a^3=b^45=1,a*b=b*a>;
// generators/relations
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