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G = C10×D13order 260 = 22·5·13

Direct product of C10 and D13

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C10×D13, C26⋊C10, C1302C2, C653C22, C13⋊(C2×C10), SmallGroup(260,12)

Series: Derived Chief Lower central Upper central

C1C13 — C10×D13
C1C13C65C5×D13 — C10×D13
C13 — C10×D13
C1C10

Generators and relations for C10×D13
 G = < a,b,c | a10=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

13C2
13C2
13C22
13C10
13C10
13C2×C10

Smallest permutation representation of C10×D13
On 130 points
Generators in S130
(1 96 61 86 45 69 27 128 26 105)(2 97 62 87 46 70 28 129 14 106)(3 98 63 88 47 71 29 130 15 107)(4 99 64 89 48 72 30 118 16 108)(5 100 65 90 49 73 31 119 17 109)(6 101 53 91 50 74 32 120 18 110)(7 102 54 79 51 75 33 121 19 111)(8 103 55 80 52 76 34 122 20 112)(9 104 56 81 40 77 35 123 21 113)(10 92 57 82 41 78 36 124 22 114)(11 93 58 83 42 66 37 125 23 115)(12 94 59 84 43 67 38 126 24 116)(13 95 60 85 44 68 39 127 25 117)
(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)
(1 68)(2 67)(3 66)(4 78)(5 77)(6 76)(7 75)(8 74)(9 73)(10 72)(11 71)(12 70)(13 69)(14 84)(15 83)(16 82)(17 81)(18 80)(19 79)(20 91)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 95)(28 94)(29 93)(30 92)(31 104)(32 103)(33 102)(34 101)(35 100)(36 99)(37 98)(38 97)(39 96)(40 109)(41 108)(42 107)(43 106)(44 105)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 122)(54 121)(55 120)(56 119)(57 118)(58 130)(59 129)(60 128)(61 127)(62 126)(63 125)(64 124)(65 123)

G:=sub<Sym(130)| (1,96,61,86,45,69,27,128,26,105)(2,97,62,87,46,70,28,129,14,106)(3,98,63,88,47,71,29,130,15,107)(4,99,64,89,48,72,30,118,16,108)(5,100,65,90,49,73,31,119,17,109)(6,101,53,91,50,74,32,120,18,110)(7,102,54,79,51,75,33,121,19,111)(8,103,55,80,52,76,34,122,20,112)(9,104,56,81,40,77,35,123,21,113)(10,92,57,82,41,78,36,124,22,114)(11,93,58,83,42,66,37,125,23,115)(12,94,59,84,43,67,38,126,24,116)(13,95,60,85,44,68,39,127,25,117), (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), (1,68)(2,67)(3,66)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,72)(11,71)(12,70)(13,69)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,95)(28,94)(29,93)(30,92)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,109)(41,108)(42,107)(43,106)(44,105)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,122)(54,121)(55,120)(56,119)(57,118)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123)>;

G:=Group( (1,96,61,86,45,69,27,128,26,105)(2,97,62,87,46,70,28,129,14,106)(3,98,63,88,47,71,29,130,15,107)(4,99,64,89,48,72,30,118,16,108)(5,100,65,90,49,73,31,119,17,109)(6,101,53,91,50,74,32,120,18,110)(7,102,54,79,51,75,33,121,19,111)(8,103,55,80,52,76,34,122,20,112)(9,104,56,81,40,77,35,123,21,113)(10,92,57,82,41,78,36,124,22,114)(11,93,58,83,42,66,37,125,23,115)(12,94,59,84,43,67,38,126,24,116)(13,95,60,85,44,68,39,127,25,117), (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), (1,68)(2,67)(3,66)(4,78)(5,77)(6,76)(7,75)(8,74)(9,73)(10,72)(11,71)(12,70)(13,69)(14,84)(15,83)(16,82)(17,81)(18,80)(19,79)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,95)(28,94)(29,93)(30,92)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,109)(41,108)(42,107)(43,106)(44,105)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,122)(54,121)(55,120)(56,119)(57,118)(58,130)(59,129)(60,128)(61,127)(62,126)(63,125)(64,124)(65,123) );

G=PermutationGroup([(1,96,61,86,45,69,27,128,26,105),(2,97,62,87,46,70,28,129,14,106),(3,98,63,88,47,71,29,130,15,107),(4,99,64,89,48,72,30,118,16,108),(5,100,65,90,49,73,31,119,17,109),(6,101,53,91,50,74,32,120,18,110),(7,102,54,79,51,75,33,121,19,111),(8,103,55,80,52,76,34,122,20,112),(9,104,56,81,40,77,35,123,21,113),(10,92,57,82,41,78,36,124,22,114),(11,93,58,83,42,66,37,125,23,115),(12,94,59,84,43,67,38,126,24,116),(13,95,60,85,44,68,39,127,25,117)], [(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)], [(1,68),(2,67),(3,66),(4,78),(5,77),(6,76),(7,75),(8,74),(9,73),(10,72),(11,71),(12,70),(13,69),(14,84),(15,83),(16,82),(17,81),(18,80),(19,79),(20,91),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,95),(28,94),(29,93),(30,92),(31,104),(32,103),(33,102),(34,101),(35,100),(36,99),(37,98),(38,97),(39,96),(40,109),(41,108),(42,107),(43,106),(44,105),(45,117),(46,116),(47,115),(48,114),(49,113),(50,112),(51,111),(52,110),(53,122),(54,121),(55,120),(56,119),(57,118),(58,130),(59,129),(60,128),(61,127),(62,126),(63,125),(64,124),(65,123)])

80 conjugacy classes

class 1 2A2B2C5A5B5C5D10A10B10C10D10E···10L13A···13F26A···26F65A···65X130A···130X
order122255551010101010···1013···1326···2665···65130···130
size1113131111111113···132···22···22···22···2

80 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D13D26C5×D13C10×D13
kernelC10×D13C5×D13C130D26D13C26C10C5C2C1
# reps121484662424

Matrix representation of C10×D13 in GL3(𝔽131) generated by

13000
0890
0089
,
100
01191
09225
,
100
057110
011174
G:=sub<GL(3,GF(131))| [130,0,0,0,89,0,0,0,89],[1,0,0,0,119,92,0,1,25],[1,0,0,0,57,111,0,110,74] >;

C10×D13 in GAP, Magma, Sage, TeX

C_{10}\times D_{13}
% in TeX

G:=Group("C10xD13");
// GroupNames label

G:=SmallGroup(260,12);
// by ID

G=gap.SmallGroup(260,12);
# by ID

G:=PCGroup([4,-2,-2,-5,-13,3843]);
// Polycyclic

G:=Group<a,b,c|a^10=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×D13 in TeX

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