direct product, abelian, monomial, 3-elementary
Aliases: C3×C60, SmallGroup(180,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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
(1 145 76)(2 146 77)(3 147 78)(4 148 79)(5 149 80)(6 150 81)(7 151 82)(8 152 83)(9 153 84)(10 154 85)(11 155 86)(12 156 87)(13 157 88)(14 158 89)(15 159 90)(16 160 91)(17 161 92)(18 162 93)(19 163 94)(20 164 95)(21 165 96)(22 166 97)(23 167 98)(24 168 99)(25 169 100)(26 170 101)(27 171 102)(28 172 103)(29 173 104)(30 174 105)(31 175 106)(32 176 107)(33 177 108)(34 178 109)(35 179 110)(36 180 111)(37 121 112)(38 122 113)(39 123 114)(40 124 115)(41 125 116)(42 126 117)(43 127 118)(44 128 119)(45 129 120)(46 130 61)(47 131 62)(48 132 63)(49 133 64)(50 134 65)(51 135 66)(52 136 67)(53 137 68)(54 138 69)(55 139 70)(56 140 71)(57 141 72)(58 142 73)(59 143 74)(60 144 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)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)
G:=sub<Sym(180)| (1,145,76)(2,146,77)(3,147,78)(4,148,79)(5,149,80)(6,150,81)(7,151,82)(8,152,83)(9,153,84)(10,154,85)(11,155,86)(12,156,87)(13,157,88)(14,158,89)(15,159,90)(16,160,91)(17,161,92)(18,162,93)(19,163,94)(20,164,95)(21,165,96)(22,166,97)(23,167,98)(24,168,99)(25,169,100)(26,170,101)(27,171,102)(28,172,103)(29,173,104)(30,174,105)(31,175,106)(32,176,107)(33,177,108)(34,178,109)(35,179,110)(36,180,111)(37,121,112)(38,122,113)(39,123,114)(40,124,115)(41,125,116)(42,126,117)(43,127,118)(44,128,119)(45,129,120)(46,130,61)(47,131,62)(48,132,63)(49,133,64)(50,134,65)(51,135,66)(52,136,67)(53,137,68)(54,138,69)(55,139,70)(56,140,71)(57,141,72)(58,142,73)(59,143,74)(60,144,75), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)>;
G:=Group( (1,145,76)(2,146,77)(3,147,78)(4,148,79)(5,149,80)(6,150,81)(7,151,82)(8,152,83)(9,153,84)(10,154,85)(11,155,86)(12,156,87)(13,157,88)(14,158,89)(15,159,90)(16,160,91)(17,161,92)(18,162,93)(19,163,94)(20,164,95)(21,165,96)(22,166,97)(23,167,98)(24,168,99)(25,169,100)(26,170,101)(27,171,102)(28,172,103)(29,173,104)(30,174,105)(31,175,106)(32,176,107)(33,177,108)(34,178,109)(35,179,110)(36,180,111)(37,121,112)(38,122,113)(39,123,114)(40,124,115)(41,125,116)(42,126,117)(43,127,118)(44,128,119)(45,129,120)(46,130,61)(47,131,62)(48,132,63)(49,133,64)(50,134,65)(51,135,66)(52,136,67)(53,137,68)(54,138,69)(55,139,70)(56,140,71)(57,141,72)(58,142,73)(59,143,74)(60,144,75), (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)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180) );
G=PermutationGroup([[(1,145,76),(2,146,77),(3,147,78),(4,148,79),(5,149,80),(6,150,81),(7,151,82),(8,152,83),(9,153,84),(10,154,85),(11,155,86),(12,156,87),(13,157,88),(14,158,89),(15,159,90),(16,160,91),(17,161,92),(18,162,93),(19,163,94),(20,164,95),(21,165,96),(22,166,97),(23,167,98),(24,168,99),(25,169,100),(26,170,101),(27,171,102),(28,172,103),(29,173,104),(30,174,105),(31,175,106),(32,176,107),(33,177,108),(34,178,109),(35,179,110),(36,180,111),(37,121,112),(38,122,113),(39,123,114),(40,124,115),(41,125,116),(42,126,117),(43,127,118),(44,128,119),(45,129,120),(46,130,61),(47,131,62),(48,132,63),(49,133,64),(50,134,65),(51,135,66),(52,136,67),(53,137,68),(54,138,69),(55,139,70),(56,140,71),(57,141,72),(58,142,73),(59,143,74),(60,144,75)], [(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),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)]])
C3×C60 is a maximal subgroup of
C60.S3 C12.D15 C60⋊S3
180 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 4A | 4B | 5A | 5B | 5C | 5D | 6A | ··· | 6H | 10A | 10B | 10C | 10D | 12A | ··· | 12P | 15A | ··· | 15AF | 20A | ··· | 20H | 30A | ··· | 30AF | 60A | ··· | 60BL |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 |
| kernel | C3×C60 | C3×C30 | C60 | C3×C15 | C3×C12 | C30 | C3×C6 | C15 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 8 | 2 | 4 | 8 | 4 | 16 | 32 | 8 | 32 | 64 |
Matrix representation of C3×C60 ►in GL2(𝔽61) generated by
| 13 | 0 |
| 0 | 1 |
| 21 | 0 |
| 0 | 22 |
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