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G = C3×C57order 171 = 32·19

Abelian group of type [3,57]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C57, SmallGroup(171,5)

Series: Derived Chief Lower central Upper central

C1 — C3×C57
C1C19C57 — 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
(1 164 103)(2 165 104)(3 166 105)(4 167 106)(5 168 107)(6 169 108)(7 170 109)(8 171 110)(9 115 111)(10 116 112)(11 117 113)(12 118 114)(13 119 58)(14 120 59)(15 121 60)(16 122 61)(17 123 62)(18 124 63)(19 125 64)(20 126 65)(21 127 66)(22 128 67)(23 129 68)(24 130 69)(25 131 70)(26 132 71)(27 133 72)(28 134 73)(29 135 74)(30 136 75)(31 137 76)(32 138 77)(33 139 78)(34 140 79)(35 141 80)(36 142 81)(37 143 82)(38 144 83)(39 145 84)(40 146 85)(41 147 86)(42 148 87)(43 149 88)(44 150 89)(45 151 90)(46 152 91)(47 153 92)(48 154 93)(49 155 94)(50 156 95)(51 157 96)(52 158 97)(53 159 98)(54 160 99)(55 161 100)(56 162 101)(57 163 102)
(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)(115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171)

G:=sub<Sym(171)| (1,164,103)(2,165,104)(3,166,105)(4,167,106)(5,168,107)(6,169,108)(7,170,109)(8,171,110)(9,115,111)(10,116,112)(11,117,113)(12,118,114)(13,119,58)(14,120,59)(15,121,60)(16,122,61)(17,123,62)(18,124,63)(19,125,64)(20,126,65)(21,127,66)(22,128,67)(23,129,68)(24,130,69)(25,131,70)(26,132,71)(27,133,72)(28,134,73)(29,135,74)(30,136,75)(31,137,76)(32,138,77)(33,139,78)(34,140,79)(35,141,80)(36,142,81)(37,143,82)(38,144,83)(39,145,84)(40,146,85)(41,147,86)(42,148,87)(43,149,88)(44,150,89)(45,151,90)(46,152,91)(47,153,92)(48,154,93)(49,155,94)(50,156,95)(51,157,96)(52,158,97)(53,159,98)(54,160,99)(55,161,100)(56,162,101)(57,163,102), (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)>;

G:=Group( (1,164,103)(2,165,104)(3,166,105)(4,167,106)(5,168,107)(6,169,108)(7,170,109)(8,171,110)(9,115,111)(10,116,112)(11,117,113)(12,118,114)(13,119,58)(14,120,59)(15,121,60)(16,122,61)(17,123,62)(18,124,63)(19,125,64)(20,126,65)(21,127,66)(22,128,67)(23,129,68)(24,130,69)(25,131,70)(26,132,71)(27,133,72)(28,134,73)(29,135,74)(30,136,75)(31,137,76)(32,138,77)(33,139,78)(34,140,79)(35,141,80)(36,142,81)(37,143,82)(38,144,83)(39,145,84)(40,146,85)(41,147,86)(42,148,87)(43,149,88)(44,150,89)(45,151,90)(46,152,91)(47,153,92)(48,154,93)(49,155,94)(50,156,95)(51,157,96)(52,158,97)(53,159,98)(54,160,99)(55,161,100)(56,162,101)(57,163,102), (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)(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171) );

G=PermutationGroup([(1,164,103),(2,165,104),(3,166,105),(4,167,106),(5,168,107),(6,169,108),(7,170,109),(8,171,110),(9,115,111),(10,116,112),(11,117,113),(12,118,114),(13,119,58),(14,120,59),(15,121,60),(16,122,61),(17,123,62),(18,124,63),(19,125,64),(20,126,65),(21,127,66),(22,128,67),(23,129,68),(24,130,69),(25,131,70),(26,132,71),(27,133,72),(28,134,73),(29,135,74),(30,136,75),(31,137,76),(32,138,77),(33,139,78),(34,140,79),(35,141,80),(36,142,81),(37,143,82),(38,144,83),(39,145,84),(40,146,85),(41,147,86),(42,148,87),(43,149,88),(44,150,89),(45,151,90),(46,152,91),(47,153,92),(48,154,93),(49,155,94),(50,156,95),(51,157,96),(52,158,97),(53,159,98),(54,160,99),(55,161,100),(56,162,101),(57,163,102)], [(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),(115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171)])

C3×C57 is a maximal subgroup of   C3⋊D57

171 conjugacy classes

class 1 3A···3H19A···19R57A···57EN
order13···319···1957···57
size11···11···11···1

171 irreducible representations

dim1111
type+
imageC1C3C19C57
kernelC3×C57C57C32C3
# reps1818144

Matrix representation of C3×C57 in GL2(𝔽229) generated by

940
094
,
750
03
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

Export

Subgroup lattice of C3×C57 in TeX

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