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G = C3×C51order 153 = 32·17

Abelian group of type [3,51]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C51, SmallGroup(153,2)

Series: Derived Chief Lower central Upper central

C1 — C3×C51
C1C17C51 — 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
(1 152 83)(2 153 84)(3 103 85)(4 104 86)(5 105 87)(6 106 88)(7 107 89)(8 108 90)(9 109 91)(10 110 92)(11 111 93)(12 112 94)(13 113 95)(14 114 96)(15 115 97)(16 116 98)(17 117 99)(18 118 100)(19 119 101)(20 120 102)(21 121 52)(22 122 53)(23 123 54)(24 124 55)(25 125 56)(26 126 57)(27 127 58)(28 128 59)(29 129 60)(30 130 61)(31 131 62)(32 132 63)(33 133 64)(34 134 65)(35 135 66)(36 136 67)(37 137 68)(38 138 69)(39 139 70)(40 140 71)(41 141 72)(42 142 73)(43 143 74)(44 144 75)(45 145 76)(46 146 77)(47 147 78)(48 148 79)(49 149 80)(50 150 81)(51 151 82)
(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)| (1,152,83)(2,153,84)(3,103,85)(4,104,86)(5,105,87)(6,106,88)(7,107,89)(8,108,90)(9,109,91)(10,110,92)(11,111,93)(12,112,94)(13,113,95)(14,114,96)(15,115,97)(16,116,98)(17,117,99)(18,118,100)(19,119,101)(20,120,102)(21,121,52)(22,122,53)(23,123,54)(24,124,55)(25,125,56)(26,126,57)(27,127,58)(28,128,59)(29,129,60)(30,130,61)(31,131,62)(32,132,63)(33,133,64)(34,134,65)(35,135,66)(36,136,67)(37,137,68)(38,138,69)(39,139,70)(40,140,71)(41,141,72)(42,142,73)(43,143,74)(44,144,75)(45,145,76)(46,146,77)(47,147,78)(48,148,79)(49,149,80)(50,150,81)(51,151,82), (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( (1,152,83)(2,153,84)(3,103,85)(4,104,86)(5,105,87)(6,106,88)(7,107,89)(8,108,90)(9,109,91)(10,110,92)(11,111,93)(12,112,94)(13,113,95)(14,114,96)(15,115,97)(16,116,98)(17,117,99)(18,118,100)(19,119,101)(20,120,102)(21,121,52)(22,122,53)(23,123,54)(24,124,55)(25,125,56)(26,126,57)(27,127,58)(28,128,59)(29,129,60)(30,130,61)(31,131,62)(32,132,63)(33,133,64)(34,134,65)(35,135,66)(36,136,67)(37,137,68)(38,138,69)(39,139,70)(40,140,71)(41,141,72)(42,142,73)(43,143,74)(44,144,75)(45,145,76)(46,146,77)(47,147,78)(48,148,79)(49,149,80)(50,150,81)(51,151,82), (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([(1,152,83),(2,153,84),(3,103,85),(4,104,86),(5,105,87),(6,106,88),(7,107,89),(8,108,90),(9,109,91),(10,110,92),(11,111,93),(12,112,94),(13,113,95),(14,114,96),(15,115,97),(16,116,98),(17,117,99),(18,118,100),(19,119,101),(20,120,102),(21,121,52),(22,122,53),(23,123,54),(24,124,55),(25,125,56),(26,126,57),(27,127,58),(28,128,59),(29,129,60),(30,130,61),(31,131,62),(32,132,63),(33,133,64),(34,134,65),(35,135,66),(36,136,67),(37,137,68),(38,138,69),(39,139,70),(40,140,71),(41,141,72),(42,142,73),(43,143,74),(44,144,75),(45,145,76),(46,146,77),(47,147,78),(48,148,79),(49,149,80),(50,150,81),(51,151,82)], [(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···3H17A···17P51A···51DX
order13···317···1751···51
size11···11···11···1

153 irreducible representations

dim1111
type+
imageC1C3C17C51
kernelC3×C51C51C32C3
# reps1816128

Matrix representation of C3×C51 in GL2(𝔽103) generated by

10
056
,
680
0100
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

Export

Subgroup lattice of C3×C51 in TeX

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