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G = C3×C45order 135 = 33·5

Abelian group of type [3,45]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C45, SmallGroup(135,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C45
C1C3C15C45 — 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
(1 132 57)(2 133 58)(3 134 59)(4 135 60)(5 91 61)(6 92 62)(7 93 63)(8 94 64)(9 95 65)(10 96 66)(11 97 67)(12 98 68)(13 99 69)(14 100 70)(15 101 71)(16 102 72)(17 103 73)(18 104 74)(19 105 75)(20 106 76)(21 107 77)(22 108 78)(23 109 79)(24 110 80)(25 111 81)(26 112 82)(27 113 83)(28 114 84)(29 115 85)(30 116 86)(31 117 87)(32 118 88)(33 119 89)(34 120 90)(35 121 46)(36 122 47)(37 123 48)(38 124 49)(39 125 50)(40 126 51)(41 127 52)(42 128 53)(43 129 54)(44 130 55)(45 131 56)
(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)| (1,132,57)(2,133,58)(3,134,59)(4,135,60)(5,91,61)(6,92,62)(7,93,63)(8,94,64)(9,95,65)(10,96,66)(11,97,67)(12,98,68)(13,99,69)(14,100,70)(15,101,71)(16,102,72)(17,103,73)(18,104,74)(19,105,75)(20,106,76)(21,107,77)(22,108,78)(23,109,79)(24,110,80)(25,111,81)(26,112,82)(27,113,83)(28,114,84)(29,115,85)(30,116,86)(31,117,87)(32,118,88)(33,119,89)(34,120,90)(35,121,46)(36,122,47)(37,123,48)(38,124,49)(39,125,50)(40,126,51)(41,127,52)(42,128,53)(43,129,54)(44,130,55)(45,131,56), (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( (1,132,57)(2,133,58)(3,134,59)(4,135,60)(5,91,61)(6,92,62)(7,93,63)(8,94,64)(9,95,65)(10,96,66)(11,97,67)(12,98,68)(13,99,69)(14,100,70)(15,101,71)(16,102,72)(17,103,73)(18,104,74)(19,105,75)(20,106,76)(21,107,77)(22,108,78)(23,109,79)(24,110,80)(25,111,81)(26,112,82)(27,113,83)(28,114,84)(29,115,85)(30,116,86)(31,117,87)(32,118,88)(33,119,89)(34,120,90)(35,121,46)(36,122,47)(37,123,48)(38,124,49)(39,125,50)(40,126,51)(41,127,52)(42,128,53)(43,129,54)(44,130,55)(45,131,56), (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([(1,132,57),(2,133,58),(3,134,59),(4,135,60),(5,91,61),(6,92,62),(7,93,63),(8,94,64),(9,95,65),(10,96,66),(11,97,67),(12,98,68),(13,99,69),(14,100,70),(15,101,71),(16,102,72),(17,103,73),(18,104,74),(19,105,75),(20,106,76),(21,107,77),(22,108,78),(23,109,79),(24,110,80),(25,111,81),(26,112,82),(27,113,83),(28,114,84),(29,115,85),(30,116,86),(31,117,87),(32,118,88),(33,119,89),(34,120,90),(35,121,46),(36,122,47),(37,123,48),(38,124,49),(39,125,50),(40,126,51),(41,127,52),(42,128,53),(43,129,54),(44,130,55),(45,131,56)], [(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···3H5A5B5C5D9A···9R15A···15AF45A···45BT
order13···355559···915···1545···45
size11···111111···11···11···1

135 irreducible representations

dim11111111
type+
imageC1C3C3C5C9C15C15C45
kernelC3×C45C45C3×C15C3×C9C15C9C32C3
# reps16241824872

Matrix representation of C3×C45 in GL2(𝔽181) generated by

480
01
,
10
087
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

Export

Subgroup lattice of C3×C45 in TeX

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