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G = D5×C26order 260 = 22·5·13

Direct product of C26 and D5

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

Aliases: D5×C26, C10⋊C26, C1303C2, C654C22, C5⋊(C2×C26), SmallGroup(260,13)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C26
C1C5C65D5×C13 — D5×C26
C5 — D5×C26
C1C26

Generators and relations for D5×C26
 G = < a,b,c | a26=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C26
5C26
5C2×C26

Smallest permutation representation of D5×C26
On 130 points
Generators in S130
(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 83 36 77 114)(2 84 37 78 115)(3 85 38 53 116)(4 86 39 54 117)(5 87 40 55 118)(6 88 41 56 119)(7 89 42 57 120)(8 90 43 58 121)(9 91 44 59 122)(10 92 45 60 123)(11 93 46 61 124)(12 94 47 62 125)(13 95 48 63 126)(14 96 49 64 127)(15 97 50 65 128)(16 98 51 66 129)(17 99 52 67 130)(18 100 27 68 105)(19 101 28 69 106)(20 102 29 70 107)(21 103 30 71 108)(22 104 31 72 109)(23 79 32 73 110)(24 80 33 74 111)(25 81 34 75 112)(26 82 35 76 113)
(1 127)(2 128)(3 129)(4 130)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(53 98)(54 99)(55 100)(56 101)(57 102)(58 103)(59 104)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(77 96)(78 97)

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

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

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

104 conjugacy classes

class 1 2A2B2C5A5B10A10B13A···13L26A···26L26M···26AJ65A···65X130A···130X
order122255101013···1326···2626···2665···65130···130
size115522221···11···15···52···22···2

104 irreducible representations

dim1111112222
type+++++
imageC1C2C2C13C26C26D5D10D5×C13D5×C26
kernelD5×C26D5×C13C130D10D5C10C26C13C2C1
# reps121122412222424

Matrix representation of D5×C26 in GL3(𝔽131) generated by

13000
0390
0039
,
100
01301
011812
,
100
01300
01181
G:=sub<GL(3,GF(131))| [130,0,0,0,39,0,0,0,39],[1,0,0,0,130,118,0,1,12],[1,0,0,0,130,118,0,0,1] >;

D5×C26 in GAP, Magma, Sage, TeX

D_5\times C_{26}
% in TeX

G:=Group("D5xC26");
// GroupNames label

G:=SmallGroup(260,13);
// by ID

G=gap.SmallGroup(260,13);
# by ID

G:=PCGroup([4,-2,-2,-13,-5,3331]);
// Polycyclic

G:=Group<a,b,c|a^26=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D5×C26 in TeX

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