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G = C3×C60order 180 = 22·32·5

Abelian group of type [3,60]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C60, SmallGroup(180,18)

Series: Derived Chief Lower central Upper central

C1 — C3×C60
C1C2C10C30C3×C30 — C3×C60
C1 — C3×C60
C1 — C3×C60

Generators and relations for C3×C60
 G = < a,b | a3=b60=1, ab=ba >


Smallest permutation representation of C3×C60
Regular action on 180 points
Generators in S180
(1 77 147)(2 78 148)(3 79 149)(4 80 150)(5 81 151)(6 82 152)(7 83 153)(8 84 154)(9 85 155)(10 86 156)(11 87 157)(12 88 158)(13 89 159)(14 90 160)(15 91 161)(16 92 162)(17 93 163)(18 94 164)(19 95 165)(20 96 166)(21 97 167)(22 98 168)(23 99 169)(24 100 170)(25 101 171)(26 102 172)(27 103 173)(28 104 174)(29 105 175)(30 106 176)(31 107 177)(32 108 178)(33 109 179)(34 110 180)(35 111 121)(36 112 122)(37 113 123)(38 114 124)(39 115 125)(40 116 126)(41 117 127)(42 118 128)(43 119 129)(44 120 130)(45 61 131)(46 62 132)(47 63 133)(48 64 134)(49 65 135)(50 66 136)(51 67 137)(52 68 138)(53 69 139)(54 70 140)(55 71 141)(56 72 142)(57 73 143)(58 74 144)(59 75 145)(60 76 146)
(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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)

G:=sub<Sym(180)| (1,77,147)(2,78,148)(3,79,149)(4,80,150)(5,81,151)(6,82,152)(7,83,153)(8,84,154)(9,85,155)(10,86,156)(11,87,157)(12,88,158)(13,89,159)(14,90,160)(15,91,161)(16,92,162)(17,93,163)(18,94,164)(19,95,165)(20,96,166)(21,97,167)(22,98,168)(23,99,169)(24,100,170)(25,101,171)(26,102,172)(27,103,173)(28,104,174)(29,105,175)(30,106,176)(31,107,177)(32,108,178)(33,109,179)(34,110,180)(35,111,121)(36,112,122)(37,113,123)(38,114,124)(39,115,125)(40,116,126)(41,117,127)(42,118,128)(43,119,129)(44,120,130)(45,61,131)(46,62,132)(47,63,133)(48,64,134)(49,65,135)(50,66,136)(51,67,137)(52,68,138)(53,69,139)(54,70,140)(55,71,141)(56,72,142)(57,73,143)(58,74,144)(59,75,145)(60,76,146), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)>;

G:=Group( (1,77,147)(2,78,148)(3,79,149)(4,80,150)(5,81,151)(6,82,152)(7,83,153)(8,84,154)(9,85,155)(10,86,156)(11,87,157)(12,88,158)(13,89,159)(14,90,160)(15,91,161)(16,92,162)(17,93,163)(18,94,164)(19,95,165)(20,96,166)(21,97,167)(22,98,168)(23,99,169)(24,100,170)(25,101,171)(26,102,172)(27,103,173)(28,104,174)(29,105,175)(30,106,176)(31,107,177)(32,108,178)(33,109,179)(34,110,180)(35,111,121)(36,112,122)(37,113,123)(38,114,124)(39,115,125)(40,116,126)(41,117,127)(42,118,128)(43,119,129)(44,120,130)(45,61,131)(46,62,132)(47,63,133)(48,64,134)(49,65,135)(50,66,136)(51,67,137)(52,68,138)(53,69,139)(54,70,140)(55,71,141)(56,72,142)(57,73,143)(58,74,144)(59,75,145)(60,76,146), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180) );

G=PermutationGroup([(1,77,147),(2,78,148),(3,79,149),(4,80,150),(5,81,151),(6,82,152),(7,83,153),(8,84,154),(9,85,155),(10,86,156),(11,87,157),(12,88,158),(13,89,159),(14,90,160),(15,91,161),(16,92,162),(17,93,163),(18,94,164),(19,95,165),(20,96,166),(21,97,167),(22,98,168),(23,99,169),(24,100,170),(25,101,171),(26,102,172),(27,103,173),(28,104,174),(29,105,175),(30,106,176),(31,107,177),(32,108,178),(33,109,179),(34,110,180),(35,111,121),(36,112,122),(37,113,123),(38,114,124),(39,115,125),(40,116,126),(41,117,127),(42,118,128),(43,119,129),(44,120,130),(45,61,131),(46,62,132),(47,63,133),(48,64,134),(49,65,135),(50,66,136),(51,67,137),(52,68,138),(53,69,139),(54,70,140),(55,71,141),(56,72,142),(57,73,143),(58,74,144),(59,75,145),(60,76,146)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)])

C3×C60 is a maximal subgroup of   C60.S3  C12.D15  C60⋊S3

180 conjugacy classes

class 1  2 3A···3H4A4B5A5B5C5D6A···6H10A10B10C10D12A···12P15A···15AF20A···20H30A···30AF60A···60BL
order123···34455556···61010101012···1215···1520···2030···3060···60
size111···11111111···111111···11···11···11···11···1

180 irreducible representations

dim111111111111
type++
imageC1C2C3C4C5C6C10C12C15C20C30C60
kernelC3×C60C3×C30C60C3×C15C3×C12C30C3×C6C15C12C32C6C3
# reps1182484163283264

Matrix representation of C3×C60 in GL2(𝔽61) generated by

130
01
,
210
022
G:=sub<GL(2,GF(61))| [13,0,0,1],[21,0,0,22] >;

C3×C60 in GAP, Magma, Sage, TeX

C_3\times C_{60}
% in TeX

G:=Group("C3xC60");
// GroupNames label

G:=SmallGroup(180,18);
// by ID

G=gap.SmallGroup(180,18);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-2,450]);
// Polycyclic

G:=Group<a,b|a^3=b^60=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C60 in TeX

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