direct product, abelian, monomial, 3-elementary
Aliases: C3×C57, SmallGroup(171,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3×C57 |
| C1 — C3×C57 |
| C1 — C3×C57 |
Generators and relations for C3×C57
G = < a,b | a3=b57=1, ab=ba >
Smallest permutation representation of C3×C57
►Regular action on 171 points
Generators in S171
(1 127 97)(2 128 98)(3 129 99)(4 130 100)(5 131 101)(6 132 102)(7 133 103)(8 134 104)(9 135 105)(10 136 106)(11 137 107)(12 138 108)(13 139 109)(14 140 110)(15 141 111)(16 142 112)(17 143 113)(18 144 114)(19 145 58)(20 146 59)(21 147 60)(22 148 61)(23 149 62)(24 150 63)(25 151 64)(26 152 65)(27 153 66)(28 154 67)(29 155 68)(30 156 69)(31 157 70)(32 158 71)(33 159 72)(34 160 73)(35 161 74)(36 162 75)(37 163 76)(38 164 77)(39 165 78)(40 166 79)(41 167 80)(42 168 81)(43 169 82)(44 170 83)(45 171 84)(46 115 85)(47 116 86)(48 117 87)(49 118 88)(50 119 89)(51 120 90)(52 121 91)(53 122 92)(54 123 93)(55 124 94)(56 125 95)(57 126 96)
(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)
G:=sub<Sym(171)| (1,127,97)(2,128,98)(3,129,99)(4,130,100)(5,131,101)(6,132,102)(7,133,103)(8,134,104)(9,135,105)(10,136,106)(11,137,107)(12,138,108)(13,139,109)(14,140,110)(15,141,111)(16,142,112)(17,143,113)(18,144,114)(19,145,58)(20,146,59)(21,147,60)(22,148,61)(23,149,62)(24,150,63)(25,151,64)(26,152,65)(27,153,66)(28,154,67)(29,155,68)(30,156,69)(31,157,70)(32,158,71)(33,159,72)(34,160,73)(35,161,74)(36,162,75)(37,163,76)(38,164,77)(39,165,78)(40,166,79)(41,167,80)(42,168,81)(43,169,82)(44,170,83)(45,171,84)(46,115,85)(47,116,86)(48,117,87)(49,118,88)(50,119,89)(51,120,90)(52,121,91)(53,122,92)(54,123,93)(55,124,94)(56,125,95)(57,126,96), (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)>;
G:=Group( (1,127,97)(2,128,98)(3,129,99)(4,130,100)(5,131,101)(6,132,102)(7,133,103)(8,134,104)(9,135,105)(10,136,106)(11,137,107)(12,138,108)(13,139,109)(14,140,110)(15,141,111)(16,142,112)(17,143,113)(18,144,114)(19,145,58)(20,146,59)(21,147,60)(22,148,61)(23,149,62)(24,150,63)(25,151,64)(26,152,65)(27,153,66)(28,154,67)(29,155,68)(30,156,69)(31,157,70)(32,158,71)(33,159,72)(34,160,73)(35,161,74)(36,162,75)(37,163,76)(38,164,77)(39,165,78)(40,166,79)(41,167,80)(42,168,81)(43,169,82)(44,170,83)(45,171,84)(46,115,85)(47,116,86)(48,117,87)(49,118,88)(50,119,89)(51,120,90)(52,121,91)(53,122,92)(54,123,93)(55,124,94)(56,125,95)(57,126,96), (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) );
G=PermutationGroup([[(1,127,97),(2,128,98),(3,129,99),(4,130,100),(5,131,101),(6,132,102),(7,133,103),(8,134,104),(9,135,105),(10,136,106),(11,137,107),(12,138,108),(13,139,109),(14,140,110),(15,141,111),(16,142,112),(17,143,113),(18,144,114),(19,145,58),(20,146,59),(21,147,60),(22,148,61),(23,149,62),(24,150,63),(25,151,64),(26,152,65),(27,153,66),(28,154,67),(29,155,68),(30,156,69),(31,157,70),(32,158,71),(33,159,72),(34,160,73),(35,161,74),(36,162,75),(37,163,76),(38,164,77),(39,165,78),(40,166,79),(41,167,80),(42,168,81),(43,169,82),(44,170,83),(45,171,84),(46,115,85),(47,116,86),(48,117,87),(49,118,88),(50,119,89),(51,120,90),(52,121,91),(53,122,92),(54,123,93),(55,124,94),(56,125,95),(57,126,96)], [(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)]])
C3×C57 is a maximal subgroup of
C3⋊D57
171 conjugacy classes
| class | 1 | 3A | ··· | 3H | 19A | ··· | 19R | 57A | ··· | 57EN |
| order | 1 | 3 | ··· | 3 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
171 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C19 | C57 |
| kernel | C3×C57 | C57 | C32 | C3 |
| # reps | 1 | 8 | 18 | 144 |
Matrix representation of C3×C57 ►in GL2(𝔽229) generated by
| 94 | 0 |
| 0 | 94 |
| 75 | 0 |
| 0 | 3 |
G:=sub<GL(2,GF(229))| [94,0,0,94],[75,0,0,3] >;
C3×C57 in GAP, Magma, Sage, TeX
C_3\times C_{57} % in TeX
G:=Group("C3xC57"); // GroupNames label
G:=SmallGroup(171,5);
// by ID
G=gap.SmallGroup(171,5);
# by ID
G:=PCGroup([3,-3,-3,-19]);
// Polycyclic
G:=Group<a,b|a^3=b^57=1,a*b=b*a>;
// generators/relations
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