direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D53, C53⋊C6, C159⋊2C2, SmallGroup(318,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C53 — C3×D53 |
Generators and relations for C3×D53
G = < a,b,c | a3=b53=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D53
►On 159 points
Generators in S159
(1 134 106)(2 135 54)(3 136 55)(4 137 56)(5 138 57)(6 139 58)(7 140 59)(8 141 60)(9 142 61)(10 143 62)(11 144 63)(12 145 64)(13 146 65)(14 147 66)(15 148 67)(16 149 68)(17 150 69)(18 151 70)(19 152 71)(20 153 72)(21 154 73)(22 155 74)(23 156 75)(24 157 76)(25 158 77)(26 159 78)(27 107 79)(28 108 80)(29 109 81)(30 110 82)(31 111 83)(32 112 84)(33 113 85)(34 114 86)(35 115 87)(36 116 88)(37 117 89)(38 118 90)(39 119 91)(40 120 92)(41 121 93)(42 122 94)(43 123 95)(44 124 96)(45 125 97)(46 126 98)(47 127 99)(48 128 100)(49 129 101)(50 130 102)(51 131 103)(52 132 104)(53 133 105)
(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)
(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(54 104)(55 103)(56 102)(57 101)(58 100)(59 99)(60 98)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(105 106)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)(121 146)(122 145)(123 144)(124 143)(125 142)(126 141)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)
G:=sub<Sym(159)| (1,134,106)(2,135,54)(3,136,55)(4,137,56)(5,138,57)(6,139,58)(7,140,59)(8,141,60)(9,142,61)(10,143,62)(11,144,63)(12,145,64)(13,146,65)(14,147,66)(15,148,67)(16,149,68)(17,150,69)(18,151,70)(19,152,71)(20,153,72)(21,154,73)(22,155,74)(23,156,75)(24,157,76)(25,158,77)(26,159,78)(27,107,79)(28,108,80)(29,109,81)(30,110,82)(31,111,83)(32,112,84)(33,113,85)(34,114,86)(35,115,87)(36,116,88)(37,117,89)(38,118,90)(39,119,91)(40,120,92)(41,121,93)(42,122,94)(43,123,95)(44,124,96)(45,125,97)(46,126,98)(47,127,99)(48,128,100)(49,129,101)(50,130,102)(51,131,103)(52,132,104)(53,133,105), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(105,106)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)>;
G:=Group( (1,134,106)(2,135,54)(3,136,55)(4,137,56)(5,138,57)(6,139,58)(7,140,59)(8,141,60)(9,142,61)(10,143,62)(11,144,63)(12,145,64)(13,146,65)(14,147,66)(15,148,67)(16,149,68)(17,150,69)(18,151,70)(19,152,71)(20,153,72)(21,154,73)(22,155,74)(23,156,75)(24,157,76)(25,158,77)(26,159,78)(27,107,79)(28,108,80)(29,109,81)(30,110,82)(31,111,83)(32,112,84)(33,113,85)(34,114,86)(35,115,87)(36,116,88)(37,117,89)(38,118,90)(39,119,91)(40,120,92)(41,121,93)(42,122,94)(43,123,95)(44,124,96)(45,125,97)(46,126,98)(47,127,99)(48,128,100)(49,129,101)(50,130,102)(51,131,103)(52,132,104)(53,133,105), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)(54,104)(55,103)(56,102)(57,101)(58,100)(59,99)(60,98)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,90)(69,89)(70,88)(71,87)(72,86)(73,85)(74,84)(75,83)(76,82)(77,81)(78,80)(105,106)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)(121,146)(122,145)(123,144)(124,143)(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134) );
G=PermutationGroup([[(1,134,106),(2,135,54),(3,136,55),(4,137,56),(5,138,57),(6,139,58),(7,140,59),(8,141,60),(9,142,61),(10,143,62),(11,144,63),(12,145,64),(13,146,65),(14,147,66),(15,148,67),(16,149,68),(17,150,69),(18,151,70),(19,152,71),(20,153,72),(21,154,73),(22,155,74),(23,156,75),(24,157,76),(25,158,77),(26,159,78),(27,107,79),(28,108,80),(29,109,81),(30,110,82),(31,111,83),(32,112,84),(33,113,85),(34,114,86),(35,115,87),(36,116,88),(37,117,89),(38,118,90),(39,119,91),(40,120,92),(41,121,93),(42,122,94),(43,123,95),(44,124,96),(45,125,97),(46,126,98),(47,127,99),(48,128,100),(49,129,101),(50,130,102),(51,131,103),(52,132,104),(53,133,105)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,28),(54,104),(55,103),(56,102),(57,101),(58,100),(59,99),(60,98),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,90),(69,89),(70,88),(71,87),(72,86),(73,85),(74,84),(75,83),(76,82),(77,81),(78,80),(105,106),(108,159),(109,158),(110,157),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147),(121,146),(122,145),(123,144),(124,143),(125,142),(126,141),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134)]])
84 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 53A | ··· | 53Z | 159A | ··· | 159AZ |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 53 | ··· | 53 | 159 | ··· | 159 |
| size | 1 | 53 | 1 | 1 | 53 | 53 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D53 | C3×D53 |
| kernel | C3×D53 | C159 | D53 | C53 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 26 | 52 |
Matrix representation of C3×D53 ►in GL2(𝔽3181) generated by
| 440 | 0 |
| 0 | 440 |
| 2416 | 1 |
| 574 | 1018 |
| 287 | 2793 |
| 2803 | 2894 |
G:=sub<GL(2,GF(3181))| [440,0,0,440],[2416,574,1,1018],[287,2803,2793,2894] >;
C3×D53 in GAP, Magma, Sage, TeX
C_3\times D_{53} % in TeX
G:=Group("C3xD53"); // GroupNames label
G:=SmallGroup(318,2);
// by ID
G=gap.SmallGroup(318,2);
# by ID
G:=PCGroup([3,-2,-3,-53,2810]);
// Polycyclic
G:=Group<a,b,c|a^3=b^53=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export