direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D59, C59⋊C6, C177⋊2C2, SmallGroup(354,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C59 — C3×D59 |
Generators and relations for C3×D59
G = < a,b,c | a3=b59=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D59
►On 177 points
Generators in S177
(1 163 117)(2 164 118)(3 165 60)(4 166 61)(5 167 62)(6 168 63)(7 169 64)(8 170 65)(9 171 66)(10 172 67)(11 173 68)(12 174 69)(13 175 70)(14 176 71)(15 177 72)(16 119 73)(17 120 74)(18 121 75)(19 122 76)(20 123 77)(21 124 78)(22 125 79)(23 126 80)(24 127 81)(25 128 82)(26 129 83)(27 130 84)(28 131 85)(29 132 86)(30 133 87)(31 134 88)(32 135 89)(33 136 90)(34 137 91)(35 138 92)(36 139 93)(37 140 94)(38 141 95)(39 142 96)(40 143 97)(41 144 98)(42 145 99)(43 146 100)(44 147 101)(45 148 102)(46 149 103)(47 150 104)(48 151 105)(49 152 106)(50 153 107)(51 154 108)(52 155 109)(53 156 110)(54 157 111)(55 158 112)(56 159 113)(57 160 114)(58 161 115)(59 162 116)
(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)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(60 114)(61 113)(62 112)(63 111)(64 110)(65 109)(66 108)(67 107)(68 106)(69 105)(70 104)(71 103)(72 102)(73 101)(74 100)(75 99)(76 98)(77 97)(78 96)(79 95)(80 94)(81 93)(82 92)(83 91)(84 90)(85 89)(86 88)(115 118)(116 117)(119 147)(120 146)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(148 177)(149 176)(150 175)(151 174)(152 173)(153 172)(154 171)(155 170)(156 169)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)
G:=sub<Sym(177)| (1,163,117)(2,164,118)(3,165,60)(4,166,61)(5,167,62)(6,168,63)(7,169,64)(8,170,65)(9,171,66)(10,172,67)(11,173,68)(12,174,69)(13,175,70)(14,176,71)(15,177,72)(16,119,73)(17,120,74)(18,121,75)(19,122,76)(20,123,77)(21,124,78)(22,125,79)(23,126,80)(24,127,81)(25,128,82)(26,129,83)(27,130,84)(28,131,85)(29,132,86)(30,133,87)(31,134,88)(32,135,89)(33,136,90)(34,137,91)(35,138,92)(36,139,93)(37,140,94)(38,141,95)(39,142,96)(40,143,97)(41,144,98)(42,145,99)(43,146,100)(44,147,101)(45,148,102)(46,149,103)(47,150,104)(48,151,105)(49,152,106)(50,153,107)(51,154,108)(52,155,109)(53,156,110)(54,157,111)(55,158,112)(56,159,113)(57,160,114)(58,161,115)(59,162,116), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,114)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(115,118)(116,117)(119,147)(120,146)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,171)(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)>;
G:=Group( (1,163,117)(2,164,118)(3,165,60)(4,166,61)(5,167,62)(6,168,63)(7,169,64)(8,170,65)(9,171,66)(10,172,67)(11,173,68)(12,174,69)(13,175,70)(14,176,71)(15,177,72)(16,119,73)(17,120,74)(18,121,75)(19,122,76)(20,123,77)(21,124,78)(22,125,79)(23,126,80)(24,127,81)(25,128,82)(26,129,83)(27,130,84)(28,131,85)(29,132,86)(30,133,87)(31,134,88)(32,135,89)(33,136,90)(34,137,91)(35,138,92)(36,139,93)(37,140,94)(38,141,95)(39,142,96)(40,143,97)(41,144,98)(42,145,99)(43,146,100)(44,147,101)(45,148,102)(46,149,103)(47,150,104)(48,151,105)(49,152,106)(50,153,107)(51,154,108)(52,155,109)(53,156,110)(54,157,111)(55,158,112)(56,159,113)(57,160,114)(58,161,115)(59,162,116), (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), (1,59)(2,58)(3,57)(4,56)(5,55)(6,54)(7,53)(8,52)(9,51)(10,50)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,36)(25,35)(26,34)(27,33)(28,32)(29,31)(60,114)(61,113)(62,112)(63,111)(64,110)(65,109)(66,108)(67,107)(68,106)(69,105)(70,104)(71,103)(72,102)(73,101)(74,100)(75,99)(76,98)(77,97)(78,96)(79,95)(80,94)(81,93)(82,92)(83,91)(84,90)(85,89)(86,88)(115,118)(116,117)(119,147)(120,146)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,171)(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163) );
G=PermutationGroup([[(1,163,117),(2,164,118),(3,165,60),(4,166,61),(5,167,62),(6,168,63),(7,169,64),(8,170,65),(9,171,66),(10,172,67),(11,173,68),(12,174,69),(13,175,70),(14,176,71),(15,177,72),(16,119,73),(17,120,74),(18,121,75),(19,122,76),(20,123,77),(21,124,78),(22,125,79),(23,126,80),(24,127,81),(25,128,82),(26,129,83),(27,130,84),(28,131,85),(29,132,86),(30,133,87),(31,134,88),(32,135,89),(33,136,90),(34,137,91),(35,138,92),(36,139,93),(37,140,94),(38,141,95),(39,142,96),(40,143,97),(41,144,98),(42,145,99),(43,146,100),(44,147,101),(45,148,102),(46,149,103),(47,150,104),(48,151,105),(49,152,106),(50,153,107),(51,154,108),(52,155,109),(53,156,110),(54,157,111),(55,158,112),(56,159,113),(57,160,114),(58,161,115),(59,162,116)], [(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)], [(1,59),(2,58),(3,57),(4,56),(5,55),(6,54),(7,53),(8,52),(9,51),(10,50),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,36),(25,35),(26,34),(27,33),(28,32),(29,31),(60,114),(61,113),(62,112),(63,111),(64,110),(65,109),(66,108),(67,107),(68,106),(69,105),(70,104),(71,103),(72,102),(73,101),(74,100),(75,99),(76,98),(77,97),(78,96),(79,95),(80,94),(81,93),(82,92),(83,91),(84,90),(85,89),(86,88),(115,118),(116,117),(119,147),(120,146),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(148,177),(149,176),(150,175),(151,174),(152,173),(153,172),(154,171),(155,170),(156,169),(157,168),(158,167),(159,166),(160,165),(161,164),(162,163)]])
93 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 59A | ··· | 59AC | 177A | ··· | 177BF |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 59 | ··· | 59 | 177 | ··· | 177 |
| size | 1 | 59 | 1 | 1 | 59 | 59 | 2 | ··· | 2 | 2 | ··· | 2 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D59 | C3×D59 |
| kernel | C3×D59 | C177 | D59 | C59 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 29 | 58 |
Matrix representation of C3×D59 ►in GL2(𝔽709) generated by
| 227 | 0 |
| 0 | 227 |
| 0 | 1 |
| 708 | 65 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(709))| [227,0,0,227],[0,708,1,65],[0,1,1,0] >;
C3×D59 in GAP, Magma, Sage, TeX
C_3\times D_{59} % in TeX
G:=Group("C3xD59"); // GroupNames label
G:=SmallGroup(354,2);
// by ID
G=gap.SmallGroup(354,2);
# by ID
G:=PCGroup([3,-2,-3,-59,3134]);
// Polycyclic
G:=Group<a,b,c|a^3=b^59=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export