Copied to
clipboard

G = Dic26order 104 = 23·13

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic26, C13⋊Q8, C4.D13, C52.1C2, C2.3D26, C26.1C22, Dic13.1C2, SmallGroup(104,4)

Series: Derived Chief Lower central Upper central

C1C26 — Dic26
C1C13C26Dic13 — Dic26
C13C26 — Dic26
C1C2C4

Generators and relations for Dic26
 G = < a,b | a52=1, b2=a26, bab-1=a-1 >

13C4
13C4
13Q8

Character table of Dic26

 class 124A4B4C13A13B13C13D13E13F26A26B26C26D26E26F52A52B52C52D52E52F52G52H52I52J52K52L
 size 1122626222222222222222222222222
ρ111111111111111111111111111111    trivial
ρ211-1-11111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ3111-1-1111111111111111111111111    linear of order 2
ρ411-11-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ522-200ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213ζ137136ζ131213ζ139134ζ138135ζ1310133ζ13111321311132137136131213139134139134131213137136131113213101331381351381351310133    orthogonal lifted from D26
ρ622200ζ139134ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ139134ζ139134ζ131213ζ1311132ζ138135ζ138135ζ1311132ζ131213ζ139134ζ137136ζ1310133ζ1310133ζ137136    orthogonal lifted from D13
ρ722-200ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135ζ139134ζ138135ζ137136ζ131213ζ1311132ζ13101331310133139134138135137136137136138135139134131013313111321312131312131311132    orthogonal lifted from D26
ρ822200ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213ζ137136ζ131213ζ139134ζ138135ζ1310133ζ1311132ζ1311132ζ137136ζ131213ζ139134ζ139134ζ131213ζ137136ζ1311132ζ1310133ζ138135ζ138135ζ1310133    orthogonal lifted from D13
ρ922-200ζ139134ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ1391341391341312131311132138135138135131113213121313913413713613101331310133137136    orthogonal lifted from D26
ρ1022-200ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ1312131312131310133137136131113213111321371361310133131213138135139134139134138135    orthogonal lifted from D26
ρ1122-200ζ137136ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ1371361371361381351310133131213131213131013313813513713613913413111321311132139134    orthogonal lifted from D26
ρ1222200ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ131213ζ131213ζ1310133ζ137136ζ1311132ζ1311132ζ137136ζ1310133ζ131213ζ138135ζ139134ζ139134ζ138135    orthogonal lifted from D13
ρ1322200ζ137136ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ137136ζ137136ζ138135ζ1310133ζ131213ζ131213ζ1310133ζ138135ζ137136ζ139134ζ1311132ζ1311132ζ139134    orthogonal lifted from D13
ρ1422200ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ138135ζ138135ζ1311132ζ139134ζ1310133ζ1310133ζ139134ζ1311132ζ138135ζ131213ζ137136ζ137136ζ131213    orthogonal lifted from D13
ρ1522200ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135ζ139134ζ138135ζ137136ζ131213ζ1311132ζ1310133ζ1310133ζ139134ζ138135ζ137136ζ137136ζ138135ζ139134ζ1310133ζ1311132ζ131213ζ131213ζ1311132    orthogonal lifted from D13
ρ1622-200ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ1381351381351311132139134131013313101331391341311132138135131213137136137136131213    orthogonal lifted from D26
ρ172-2000222222-2-2-2-2-2-2000000000000    symplectic lifted from Q8, Schur index 2
ρ182-2000ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ138135139134138135137136131213131113213101334ζ13104ζ133ζ43ζ13943ζ134ζ43ζ13843ζ135ζ43ζ13743ζ13643ζ13743ζ13643ζ13843ζ13543ζ13943ζ134ζ4ζ13104ζ133ζ4ζ13114ζ132ζ4ζ13124ζ134ζ13124ζ134ζ13114ζ132    symplectic faithful, Schur index 2
ρ192-2000ζ139134ζ138135ζ1310133ζ137136ζ131213ζ131113213121313111321381351310133137136139134ζ43ζ13943ζ1344ζ13124ζ13ζ4ζ13114ζ13243ζ13843ζ135ζ43ζ13843ζ1354ζ13114ζ132ζ4ζ13124ζ1343ζ13943ζ134ζ43ζ13743ζ1364ζ13104ζ133ζ4ζ13104ζ13343ζ13743ζ136    symplectic faithful, Schur index 2
ρ202-2000ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ1391341311132139134131013313713613121313813543ζ13843ζ1354ζ13114ζ13243ζ13943ζ1344ζ13104ζ133ζ4ζ13104ζ133ζ43ζ13943ζ134ζ4ζ13114ζ132ζ43ζ13843ζ135ζ4ζ13124ζ13ζ43ζ13743ζ13643ζ13743ζ1364ζ13124ζ13    symplectic faithful, Schur index 2
ρ212-2000ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ13121313713613121313913413813513101331311132ζ4ζ13114ζ132ζ43ζ13743ζ1364ζ13124ζ1343ζ13943ζ134ζ43ζ13943ζ134ζ4ζ13124ζ1343ζ13743ζ1364ζ13114ζ132ζ4ζ13104ζ133ζ43ζ13843ζ13543ζ13843ζ1354ζ13104ζ133    symplectic faithful, Schur index 2
ρ222-2000ζ1311132ζ139134ζ138135ζ1310133ζ137136ζ131213137136131213139134138135131013313111324ζ13114ζ13243ζ13743ζ136ζ4ζ13124ζ13ζ43ζ13943ζ13443ζ13943ζ1344ζ13124ζ13ζ43ζ13743ζ136ζ4ζ13114ζ1324ζ13104ζ13343ζ13843ζ135ζ43ζ13843ζ135ζ4ζ13104ζ133    symplectic faithful, Schur index 2
ρ232-2000ζ1310133ζ137136ζ131213ζ1311132ζ139134ζ13813513913413813513713613121313111321310133ζ4ζ13104ζ13343ζ13943ζ13443ζ13843ζ13543ζ13743ζ136ζ43ζ13743ζ136ζ43ζ13843ζ135ζ43ζ13943ζ1344ζ13104ζ1334ζ13114ζ1324ζ13124ζ13ζ4ζ13124ζ13ζ4ζ13114ζ132    symplectic faithful, Schur index 2
ρ242-2000ζ137136ζ131213ζ1311132ζ139134ζ138135ζ131013313813513101331312131311132139134137136ζ43ζ13743ζ13643ζ13843ζ135ζ4ζ13104ζ133ζ4ζ13124ζ134ζ13124ζ134ζ13104ζ133ζ43ζ13843ζ13543ζ13743ζ13643ζ13943ζ134ζ4ζ13114ζ1324ζ13114ζ132ζ43ζ13943ζ134    symplectic faithful, Schur index 2
ρ252-2000ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ137136131013313713613111321391341381351312134ζ13124ζ13ζ4ζ13104ζ133ζ43ζ13743ζ1364ζ13114ζ132ζ4ζ13114ζ13243ζ13743ζ1364ζ13104ζ133ζ4ζ13124ζ1343ζ13843ζ135ζ43ζ13943ζ13443ζ13943ζ134ζ43ζ13843ζ135    symplectic faithful, Schur index 2
ρ262-2000ζ131213ζ1311132ζ139134ζ138135ζ1310133ζ13713613101331371361311132139134138135131213ζ4ζ13124ζ134ζ13104ζ13343ζ13743ζ136ζ4ζ13114ζ1324ζ13114ζ132ζ43ζ13743ζ136ζ4ζ13104ζ1334ζ13124ζ13ζ43ζ13843ζ13543ζ13943ζ134ζ43ζ13943ζ13443ζ13843ζ135    symplectic faithful, Schur index 2
ρ272-2000ζ137136ζ131213ζ1311132ζ139134ζ138135ζ13101331381351310133131213131113213913413713643ζ13743ζ136ζ43ζ13843ζ1354ζ13104ζ1334ζ13124ζ13ζ4ζ13124ζ13ζ4ζ13104ζ13343ζ13843ζ135ζ43ζ13743ζ136ζ43ζ13943ζ1344ζ13114ζ132ζ4ζ13114ζ13243ζ13943ζ134    symplectic faithful, Schur index 2
ρ282-2000ζ138135ζ1310133ζ137136ζ131213ζ1311132ζ13913413111321391341310133137136131213138135ζ43ζ13843ζ135ζ4ζ13114ζ132ζ43ζ13943ζ134ζ4ζ13104ζ1334ζ13104ζ13343ζ13943ζ1344ζ13114ζ13243ζ13843ζ1354ζ13124ζ1343ζ13743ζ136ζ43ζ13743ζ136ζ4ζ13124ζ13    symplectic faithful, Schur index 2
ρ292-2000ζ139134ζ138135ζ1310133ζ137136ζ131213ζ13111321312131311132138135131013313713613913443ζ13943ζ134ζ4ζ13124ζ134ζ13114ζ132ζ43ζ13843ζ13543ζ13843ζ135ζ4ζ13114ζ1324ζ13124ζ13ζ43ζ13943ζ13443ζ13743ζ136ζ4ζ13104ζ1334ζ13104ζ133ζ43ζ13743ζ136    symplectic faithful, Schur index 2

Smallest permutation representation of Dic26
Regular action on 104 points
Generators in S104
(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)
(1 103 27 77)(2 102 28 76)(3 101 29 75)(4 100 30 74)(5 99 31 73)(6 98 32 72)(7 97 33 71)(8 96 34 70)(9 95 35 69)(10 94 36 68)(11 93 37 67)(12 92 38 66)(13 91 39 65)(14 90 40 64)(15 89 41 63)(16 88 42 62)(17 87 43 61)(18 86 44 60)(19 85 45 59)(20 84 46 58)(21 83 47 57)(22 82 48 56)(23 81 49 55)(24 80 50 54)(25 79 51 53)(26 78 52 104)

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

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), (1,103,27,77)(2,102,28,76)(3,101,29,75)(4,100,30,74)(5,99,31,73)(6,98,32,72)(7,97,33,71)(8,96,34,70)(9,95,35,69)(10,94,36,68)(11,93,37,67)(12,92,38,66)(13,91,39,65)(14,90,40,64)(15,89,41,63)(16,88,42,62)(17,87,43,61)(18,86,44,60)(19,85,45,59)(20,84,46,58)(21,83,47,57)(22,82,48,56)(23,81,49,55)(24,80,50,54)(25,79,51,53)(26,78,52,104) );

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)], [(1,103,27,77),(2,102,28,76),(3,101,29,75),(4,100,30,74),(5,99,31,73),(6,98,32,72),(7,97,33,71),(8,96,34,70),(9,95,35,69),(10,94,36,68),(11,93,37,67),(12,92,38,66),(13,91,39,65),(14,90,40,64),(15,89,41,63),(16,88,42,62),(17,87,43,61),(18,86,44,60),(19,85,45,59),(20,84,46,58),(21,83,47,57),(22,82,48,56),(23,81,49,55),(24,80,50,54),(25,79,51,53),(26,78,52,104)])

Dic26 is a maximal subgroup of
C104⋊C2  Dic52  D4.D13  C13⋊Q16  D525C2  D42D13  Q8×D13  Dic26⋊C3  C39⋊Q8  Dic78
Dic26 is a maximal quotient of
C26.D4  C523C4  C39⋊Q8  Dic78

Matrix representation of Dic26 in GL2(𝔽53) generated by

219
048
,
4344
2310
G:=sub<GL(2,GF(53))| [21,0,9,48],[43,23,44,10] >;

Dic26 in GAP, Magma, Sage, TeX

{\rm Dic}_{26}
% in TeX

G:=Group("Dic26");
// GroupNames label

G:=SmallGroup(104,4);
// by ID

G=gap.SmallGroup(104,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-13,16,49,21,1539]);
// Polycyclic

G:=Group<a,b|a^52=1,b^2=a^26,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic26 in TeX
Character table of Dic26 in TeX

׿
×
𝔽