direct product, abelian, monomial, 3-elementary
Aliases: C3×C51, SmallGroup(153,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C51 |
| C1 — C3×C51 |
| C1 — C3×C51 |
Generators and relations for C3×C51
G = < a,b | a3=b51=1, ab=ba >
Smallest permutation representation of C3×C51
►Regular action on 153 points
Generators in S153
(1 126 52)(2 127 53)(3 128 54)(4 129 55)(5 130 56)(6 131 57)(7 132 58)(8 133 59)(9 134 60)(10 135 61)(11 136 62)(12 137 63)(13 138 64)(14 139 65)(15 140 66)(16 141 67)(17 142 68)(18 143 69)(19 144 70)(20 145 71)(21 146 72)(22 147 73)(23 148 74)(24 149 75)(25 150 76)(26 151 77)(27 152 78)(28 153 79)(29 103 80)(30 104 81)(31 105 82)(32 106 83)(33 107 84)(34 108 85)(35 109 86)(36 110 87)(37 111 88)(38 112 89)(39 113 90)(40 114 91)(41 115 92)(42 116 93)(43 117 94)(44 118 95)(45 119 96)(46 120 97)(47 121 98)(48 122 99)(49 123 100)(50 124 101)(51 125 102)
(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)
G:=sub<Sym(153)| (1,126,52)(2,127,53)(3,128,54)(4,129,55)(5,130,56)(6,131,57)(7,132,58)(8,133,59)(9,134,60)(10,135,61)(11,136,62)(12,137,63)(13,138,64)(14,139,65)(15,140,66)(16,141,67)(17,142,68)(18,143,69)(19,144,70)(20,145,71)(21,146,72)(22,147,73)(23,148,74)(24,149,75)(25,150,76)(26,151,77)(27,152,78)(28,153,79)(29,103,80)(30,104,81)(31,105,82)(32,106,83)(33,107,84)(34,108,85)(35,109,86)(36,110,87)(37,111,88)(38,112,89)(39,113,90)(40,114,91)(41,115,92)(42,116,93)(43,117,94)(44,118,95)(45,119,96)(46,120,97)(47,121,98)(48,122,99)(49,123,100)(50,124,101)(51,125,102), (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)>;
G:=Group( (1,126,52)(2,127,53)(3,128,54)(4,129,55)(5,130,56)(6,131,57)(7,132,58)(8,133,59)(9,134,60)(10,135,61)(11,136,62)(12,137,63)(13,138,64)(14,139,65)(15,140,66)(16,141,67)(17,142,68)(18,143,69)(19,144,70)(20,145,71)(21,146,72)(22,147,73)(23,148,74)(24,149,75)(25,150,76)(26,151,77)(27,152,78)(28,153,79)(29,103,80)(30,104,81)(31,105,82)(32,106,83)(33,107,84)(34,108,85)(35,109,86)(36,110,87)(37,111,88)(38,112,89)(39,113,90)(40,114,91)(41,115,92)(42,116,93)(43,117,94)(44,118,95)(45,119,96)(46,120,97)(47,121,98)(48,122,99)(49,123,100)(50,124,101)(51,125,102), (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) );
G=PermutationGroup([[(1,126,52),(2,127,53),(3,128,54),(4,129,55),(5,130,56),(6,131,57),(7,132,58),(8,133,59),(9,134,60),(10,135,61),(11,136,62),(12,137,63),(13,138,64),(14,139,65),(15,140,66),(16,141,67),(17,142,68),(18,143,69),(19,144,70),(20,145,71),(21,146,72),(22,147,73),(23,148,74),(24,149,75),(25,150,76),(26,151,77),(27,152,78),(28,153,79),(29,103,80),(30,104,81),(31,105,82),(32,106,83),(33,107,84),(34,108,85),(35,109,86),(36,110,87),(37,111,88),(38,112,89),(39,113,90),(40,114,91),(41,115,92),(42,116,93),(43,117,94),(44,118,95),(45,119,96),(46,120,97),(47,121,98),(48,122,99),(49,123,100),(50,124,101),(51,125,102)], [(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)]])
C3×C51 is a maximal subgroup of
C3⋊D51
153 conjugacy classes
| class | 1 | 3A | ··· | 3H | 17A | ··· | 17P | 51A | ··· | 51DX |
| order | 1 | 3 | ··· | 3 | 17 | ··· | 17 | 51 | ··· | 51 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
153 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C17 | C51 |
| kernel | C3×C51 | C51 | C32 | C3 |
| # reps | 1 | 8 | 16 | 128 |
Matrix representation of C3×C51 ►in GL2(𝔽103) generated by
| 1 | 0 |
| 0 | 56 |
| 68 | 0 |
| 0 | 100 |
G:=sub<GL(2,GF(103))| [1,0,0,56],[68,0,0,100] >;
C3×C51 in GAP, Magma, Sage, TeX
C_3\times C_{51} % in TeX
G:=Group("C3xC51"); // GroupNames label
G:=SmallGroup(153,2);
// by ID
G=gap.SmallGroup(153,2);
# by ID
G:=PCGroup([3,-3,-3,-17]);
// Polycyclic
G:=Group<a,b|a^3=b^51=1,a*b=b*a>;
// generators/relations
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