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G = D52order 104 = 23·13

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D52, C4⋊D13, C131D4, C521C2, D261C2, C2.4D26, C26.3C22, sometimes denoted D104 or Dih52 or Dih104, SmallGroup(104,6)

Series: Derived Chief Lower central Upper central

C1C26 — D52
C1C13C26D26 — D52
C13C26 — D52
C1C2C4

Generators and relations for D52
 G = < a,b | a52=b2=1, bab=a-1 >

26C2
26C2
13C22
13C22
2D13
2D13
13D4

Character table of D52

 class 12A2B2C413A13B13C13D13E13F26A26B26C26D26E26F52A52B52C52D52E52F52G52H52I52J52K52L
 size 1126262222222222222222222222222
ρ111111111111111111111111111111    trivial
ρ2111-1-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ311-1-11111111111111111111111111    linear of order 2
ρ411-11-1111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ52-2000222222-2-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ622002ζ1311132ζ1310133ζ137136ζ139134ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ138135ζ138135ζ1311132ζ139134ζ1310133ζ1310133ζ139134ζ1311132ζ138135ζ131213ζ137136ζ137136ζ131213    orthogonal lifted from D13
ρ72200-2ζ131213ζ138135ζ1310133ζ1311132ζ137136ζ139134ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ1391341391341312131311132138135138135131113213121313913413713613101331310133137136    orthogonal lifted from D26
ρ822002ζ138135ζ131213ζ1311132ζ1310133ζ139134ζ137136ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ137136ζ137136ζ138135ζ1310133ζ131213ζ131213ζ1310133ζ138135ζ137136ζ139134ζ1311132ζ1311132ζ139134    orthogonal lifted from D13
ρ92200-2ζ1311132ζ1310133ζ137136ζ139134ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ137136ζ131213ζ1381351381351311132139134131013313101331391341311132138135131213137136137136131213    orthogonal lifted from D26
ρ102200-2ζ139134ζ137136ζ131213ζ138135ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ131213ζ1311132ζ13101331310133139134138135137136137136138135139134131013313111321312131312131311132    orthogonal lifted from D26
ρ1122002ζ131213ζ138135ζ1310133ζ1311132ζ137136ζ139134ζ131213ζ1311132ζ138135ζ1310133ζ137136ζ139134ζ139134ζ131213ζ1311132ζ138135ζ138135ζ1311132ζ131213ζ139134ζ137136ζ1310133ζ1310133ζ137136    orthogonal lifted from D13
ρ1222002ζ1310133ζ1311132ζ139134ζ137136ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ131213ζ131213ζ1310133ζ137136ζ1311132ζ1311132ζ137136ζ1310133ζ131213ζ138135ζ139134ζ139134ζ138135    orthogonal lifted from D13
ρ132200-2ζ138135ζ131213ζ1311132ζ1310133ζ139134ζ137136ζ138135ζ1310133ζ131213ζ1311132ζ139134ζ1371361371361381351310133131213131213131013313813513713613913413111321311132139134    orthogonal lifted from D26
ρ1422002ζ137136ζ139134ζ138135ζ131213ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ138135ζ1310133ζ1311132ζ1311132ζ137136ζ131213ζ139134ζ139134ζ131213ζ137136ζ1311132ζ1310133ζ138135ζ138135ζ1310133    orthogonal lifted from D13
ρ152200-2ζ1310133ζ1311132ζ139134ζ137136ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ139134ζ138135ζ1312131312131310133137136131113213111321371361310133131213138135139134139134138135    orthogonal lifted from D26
ρ162200-2ζ137136ζ139134ζ138135ζ131213ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ138135ζ1310133ζ13111321311132137136131213139134139134131213137136131113213101331381351381351310133    orthogonal lifted from D26
ρ1722002ζ139134ζ137136ζ131213ζ138135ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ131213ζ1311132ζ1310133ζ1310133ζ139134ζ138135ζ137136ζ137136ζ138135ζ139134ζ1310133ζ1311132ζ131213ζ131213ζ1311132    orthogonal lifted from D13
ρ182-2000ζ138135ζ131213ζ1311132ζ1310133ζ139134ζ137136138135131013313121313111321391341371364ζ1374ζ13643ζ13843ζ13543ζ131043ζ13343ζ131243ζ13ζ43ζ131243ζ13ζ43ζ131043ζ133ζ43ζ13843ζ135ζ4ζ1374ζ136ζ4ζ1394ζ134ζ4ζ13114ζ1324ζ13114ζ1324ζ1394ζ134    orthogonal faithful
ρ192-2000ζ138135ζ131213ζ1311132ζ1310133ζ139134ζ13713613813513101331312131311132139134137136ζ4ζ1374ζ136ζ43ζ13843ζ135ζ43ζ131043ζ133ζ43ζ131243ζ1343ζ131243ζ1343ζ131043ζ13343ζ13843ζ1354ζ1374ζ1364ζ1394ζ1344ζ13114ζ132ζ4ζ13114ζ132ζ4ζ1394ζ134    orthogonal faithful
ρ202-2000ζ1310133ζ1311132ζ139134ζ137136ζ138135ζ13121313101331371361311132139134138135131213ζ43ζ131243ζ1343ζ131043ζ1334ζ1374ζ1364ζ13114ζ132ζ4ζ13114ζ132ζ4ζ1374ζ136ζ43ζ131043ζ13343ζ131243ζ1343ζ13843ζ1354ζ1394ζ134ζ4ζ1394ζ134ζ43ζ13843ζ135    orthogonal faithful
ρ212-2000ζ1310133ζ1311132ζ139134ζ137136ζ138135ζ1312131310133137136131113213913413813513121343ζ131243ζ13ζ43ζ131043ζ133ζ4ζ1374ζ136ζ4ζ13114ζ1324ζ13114ζ1324ζ1374ζ13643ζ131043ζ133ζ43ζ131243ζ13ζ43ζ13843ζ135ζ4ζ1394ζ1344ζ1394ζ13443ζ13843ζ135    orthogonal faithful
ρ222-2000ζ139134ζ137136ζ131213ζ138135ζ1311132ζ131013313913413813513713613121313111321310133ζ43ζ131043ζ1334ζ1394ζ134ζ43ζ13843ζ1354ζ1374ζ136ζ4ζ1374ζ13643ζ13843ζ135ζ4ζ1394ζ13443ζ131043ζ133ζ4ζ13114ζ13243ζ131243ζ13ζ43ζ131243ζ134ζ13114ζ132    orthogonal faithful
ρ232-2000ζ131213ζ138135ζ1310133ζ1311132ζ137136ζ139134131213131113213813513101331371361391344ζ1394ζ134ζ43ζ131243ζ13ζ4ζ13114ζ13243ζ13843ζ135ζ43ζ13843ζ1354ζ13114ζ13243ζ131243ζ13ζ4ζ1394ζ1344ζ1374ζ136ζ43ζ131043ζ13343ζ131043ζ133ζ4ζ1374ζ136    orthogonal faithful
ρ242-2000ζ1311132ζ1310133ζ137136ζ139134ζ131213ζ13813513111321391341310133137136131213138135ζ43ζ13843ζ135ζ4ζ13114ζ1324ζ1394ζ13443ζ131043ζ133ζ43ζ131043ζ133ζ4ζ1394ζ1344ζ13114ζ13243ζ13843ζ135ζ43ζ131243ζ13ζ4ζ1374ζ1364ζ1374ζ13643ζ131243ζ13    orthogonal faithful
ρ252-2000ζ1311132ζ1310133ζ137136ζ139134ζ131213ζ1381351311132139134131013313713613121313813543ζ13843ζ1354ζ13114ζ132ζ4ζ1394ζ134ζ43ζ131043ζ13343ζ131043ζ1334ζ1394ζ134ζ4ζ13114ζ132ζ43ζ13843ζ13543ζ131243ζ134ζ1374ζ136ζ4ζ1374ζ136ζ43ζ131243ζ13    orthogonal faithful
ρ262-2000ζ131213ζ138135ζ1310133ζ1311132ζ137136ζ13913413121313111321381351310133137136139134ζ4ζ1394ζ13443ζ131243ζ134ζ13114ζ132ζ43ζ13843ζ13543ζ13843ζ135ζ4ζ13114ζ132ζ43ζ131243ζ134ζ1394ζ134ζ4ζ1374ζ13643ζ131043ζ133ζ43ζ131043ζ1334ζ1374ζ136    orthogonal faithful
ρ272-2000ζ137136ζ139134ζ138135ζ131213ζ1310133ζ1311132137136131213139134138135131013313111324ζ13114ζ132ζ4ζ1374ζ13643ζ131243ζ134ζ1394ζ134ζ4ζ1394ζ134ζ43ζ131243ζ134ζ1374ζ136ζ4ζ13114ζ132ζ43ζ131043ζ13343ζ13843ζ135ζ43ζ13843ζ13543ζ131043ζ133    orthogonal faithful
ρ282-2000ζ137136ζ139134ζ138135ζ131213ζ1310133ζ131113213713613121313913413813513101331311132ζ4ζ13114ζ1324ζ1374ζ136ζ43ζ131243ζ13ζ4ζ1394ζ1344ζ1394ζ13443ζ131243ζ13ζ4ζ1374ζ1364ζ13114ζ13243ζ131043ζ133ζ43ζ13843ζ13543ζ13843ζ135ζ43ζ131043ζ133    orthogonal faithful
ρ292-2000ζ139134ζ137136ζ131213ζ138135ζ1311132ζ13101331391341381351371361312131311132131013343ζ131043ζ133ζ4ζ1394ζ13443ζ13843ζ135ζ4ζ1374ζ1364ζ1374ζ136ζ43ζ13843ζ1354ζ1394ζ134ζ43ζ131043ζ1334ζ13114ζ132ζ43ζ131243ζ1343ζ131243ζ13ζ4ζ13114ζ132    orthogonal faithful

Smallest permutation representation of D52
On 52 points
Generators in S52
(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)

G:=sub<Sym(52)| (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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)>;

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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27) );

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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,40),(14,39),(15,38),(16,37),(17,36),(18,35),(19,34),(20,33),(21,32),(22,31),(23,30),(24,29),(25,28),(26,27)])

D52 is a maximal subgroup of
C104⋊C2  D104  D4⋊D13  Q8⋊D13  D525C2  D4×D13  D52⋊C2  D52⋊C3  C3⋊D52  D156
D52 is a maximal quotient of
C104⋊C2  D104  Dic52  C523C4  D26⋊C4  C3⋊D52  D156

Matrix representation of D52 in GL2(𝔽53) generated by

2138
157
,
2138
4732
G:=sub<GL(2,GF(53))| [21,15,38,7],[21,47,38,32] >;

D52 in GAP, Magma, Sage, TeX

D_{52}
% in TeX

G:=Group("D52");
// GroupNames label

G:=SmallGroup(104,6);
// by ID

G=gap.SmallGroup(104,6);
# by ID

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

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

Export

Subgroup lattice of D52 in TeX
Character table of D52 in TeX

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