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G = D39order 78 = 2·3·13

Dihedral group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D39, C13⋊S3, C3⋊D13, C391C2, sometimes denoted D78 or Dih39 or Dih78, SmallGroup(78,5)

Series: Derived Chief Lower central Upper central

C1C39 — D39
C1C13C39 — D39
C39 — D39
C1

Generators and relations for D39
 G = < a,b | a39=b2=1, bab=a-1 >

39C2
13S3
3D13

Character table of D39

 class 12313A13B13C13D13E13F39A39B39C39D39E39F39G39H39I39J39K39L
 size 1392222222222222222222
ρ1111111111111111111111    trivial
ρ21-11111111111111111111    linear of order 2
ρ320-1222222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ4202ζ1311132ζ139134ζ138135ζ137136ζ1310133ζ131213ζ1311132ζ1310133ζ139134ζ138135ζ137136ζ137136ζ138135ζ139134ζ1310133ζ1311132ζ131213ζ131213    orthogonal lifted from D13
ρ5202ζ131213ζ1311132ζ139134ζ1310133ζ138135ζ137136ζ131213ζ138135ζ1311132ζ139134ζ1310133ζ1310133ζ139134ζ1311132ζ138135ζ131213ζ137136ζ137136    orthogonal lifted from D13
ρ6202ζ139134ζ138135ζ1310133ζ131213ζ137136ζ1311132ζ139134ζ137136ζ138135ζ1310133ζ131213ζ131213ζ1310133ζ138135ζ137136ζ139134ζ1311132ζ1311132    orthogonal lifted from D13
ρ7202ζ138135ζ1310133ζ137136ζ1311132ζ131213ζ139134ζ138135ζ131213ζ1310133ζ137136ζ1311132ζ1311132ζ137136ζ1310133ζ131213ζ138135ζ139134ζ139134    orthogonal lifted from D13
ρ8202ζ1310133ζ137136ζ131213ζ139134ζ1311132ζ138135ζ1310133ζ1311132ζ137136ζ131213ζ139134ζ139134ζ131213ζ137136ζ1311132ζ1310133ζ138135ζ138135    orthogonal lifted from D13
ρ9202ζ137136ζ131213ζ1311132ζ138135ζ139134ζ1310133ζ137136ζ139134ζ131213ζ1311132ζ138135ζ138135ζ1311132ζ131213ζ139134ζ137136ζ1310133ζ1310133    orthogonal lifted from D13
ρ1020-1ζ139134ζ138135ζ1310133ζ131213ζ137136ζ1311132ζ3ζ1393ζ13413432ζ13732ζ136137ζ32ζ13832ζ135135ζ32ζ131032ζ1331333ζ13123ζ131312ζ3ζ13123ζ1313ζ3ζ13103ζ13313332ζ13832ζ135138ζ32ζ13732ζ136136ζ32ζ13932ζ1341343ζ13113ζ1321311ζ3ζ13113ζ132132    orthogonal faithful
ρ1120-1ζ1311132ζ139134ζ138135ζ137136ζ1310133ζ1312133ζ13113ζ1321311ζ32ζ131032ζ133133ζ32ζ13932ζ134134ζ32ζ13832ζ135135ζ32ζ13732ζ13613632ζ13732ζ13613732ζ13832ζ135138ζ3ζ1393ζ134134ζ3ζ13103ζ133133ζ3ζ13113ζ132132ζ3ζ13123ζ13133ζ13123ζ131312    orthogonal faithful
ρ1220-1ζ1310133ζ137136ζ131213ζ139134ζ1311132ζ138135ζ32ζ131032ζ133133ζ3ζ13113ζ132132ζ32ζ13732ζ1361363ζ13123ζ131312ζ3ζ1393ζ134134ζ32ζ13932ζ134134ζ3ζ13123ζ131332ζ13732ζ1361373ζ13113ζ1321311ζ3ζ13103ζ133133ζ32ζ13832ζ13513532ζ13832ζ135138    orthogonal faithful
ρ1320-1ζ1310133ζ137136ζ131213ζ139134ζ1311132ζ138135ζ3ζ13103ζ1331333ζ13113ζ132131132ζ13732ζ136137ζ3ζ13123ζ1313ζ32ζ13932ζ134134ζ3ζ1393ζ1341343ζ13123ζ131312ζ32ζ13732ζ136136ζ3ζ13113ζ132132ζ32ζ131032ζ13313332ζ13832ζ135138ζ32ζ13832ζ135135    orthogonal faithful
ρ1420-1ζ138135ζ1310133ζ137136ζ1311132ζ131213ζ139134ζ32ζ13832ζ1351353ζ13123ζ131312ζ3ζ13103ζ133133ζ32ζ13732ζ1361363ζ13113ζ1321311ζ3ζ13113ζ13213232ζ13732ζ136137ζ32ζ131032ζ133133ζ3ζ13123ζ131332ζ13832ζ135138ζ32ζ13932ζ134134ζ3ζ1393ζ134134    orthogonal faithful
ρ1520-1ζ138135ζ1310133ζ137136ζ1311132ζ131213ζ13913432ζ13832ζ135138ζ3ζ13123ζ1313ζ32ζ131032ζ13313332ζ13732ζ136137ζ3ζ13113ζ1321323ζ13113ζ1321311ζ32ζ13732ζ136136ζ3ζ13103ζ1331333ζ13123ζ131312ζ32ζ13832ζ135135ζ3ζ1393ζ134134ζ32ζ13932ζ134134    orthogonal faithful
ρ1620-1ζ1311132ζ139134ζ138135ζ137136ζ1310133ζ131213ζ3ζ13113ζ132132ζ3ζ13103ζ133133ζ3ζ1393ζ13413432ζ13832ζ13513832ζ13732ζ136137ζ32ζ13732ζ136136ζ32ζ13832ζ135135ζ32ζ13932ζ134134ζ32ζ131032ζ1331333ζ13113ζ13213113ζ13123ζ131312ζ3ζ13123ζ1313    orthogonal faithful
ρ1720-1ζ131213ζ1311132ζ139134ζ1310133ζ138135ζ137136ζ3ζ13123ζ1313ζ32ζ13832ζ135135ζ3ζ13113ζ132132ζ32ζ13932ζ134134ζ3ζ13103ζ133133ζ32ζ131032ζ133133ζ3ζ1393ζ1341343ζ13113ζ132131132ζ13832ζ1351383ζ13123ζ13131232ζ13732ζ136137ζ32ζ13732ζ136136    orthogonal faithful
ρ1820-1ζ131213ζ1311132ζ139134ζ1310133ζ138135ζ1371363ζ13123ζ13131232ζ13832ζ1351383ζ13113ζ1321311ζ3ζ1393ζ134134ζ32ζ131032ζ133133ζ3ζ13103ζ133133ζ32ζ13932ζ134134ζ3ζ13113ζ132132ζ32ζ13832ζ135135ζ3ζ13123ζ1313ζ32ζ13732ζ13613632ζ13732ζ136137    orthogonal faithful
ρ1920-1ζ139134ζ138135ζ1310133ζ131213ζ137136ζ1311132ζ32ζ13932ζ134134ζ32ζ13732ζ13613632ζ13832ζ135138ζ3ζ13103ζ133133ζ3ζ13123ζ13133ζ13123ζ131312ζ32ζ131032ζ133133ζ32ζ13832ζ13513532ζ13732ζ136137ζ3ζ1393ζ134134ζ3ζ13113ζ1321323ζ13113ζ1321311    orthogonal faithful
ρ2020-1ζ137136ζ131213ζ1311132ζ138135ζ139134ζ1310133ζ32ζ13732ζ136136ζ3ζ1393ζ134134ζ3ζ13123ζ13133ζ13113ζ1321311ζ32ζ13832ζ13513532ζ13832ζ135138ζ3ζ13113ζ1321323ζ13123ζ131312ζ32ζ13932ζ13413432ζ13732ζ136137ζ3ζ13103ζ133133ζ32ζ131032ζ133133    orthogonal faithful
ρ2120-1ζ137136ζ131213ζ1311132ζ138135ζ139134ζ131013332ζ13732ζ136137ζ32ζ13932ζ1341343ζ13123ζ131312ζ3ζ13113ζ13213232ζ13832ζ135138ζ32ζ13832ζ1351353ζ13113ζ1321311ζ3ζ13123ζ1313ζ3ζ1393ζ134134ζ32ζ13732ζ136136ζ32ζ131032ζ133133ζ3ζ13103ζ133133    orthogonal faithful

Smallest permutation representation of D39
On 39 points
Generators in S39
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)

G:=sub<Sym(39)| (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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)>;

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), (1,39)(2,38)(3,37)(4,36)(5,35)(6,34)(7,33)(8,32)(9,31)(10,30)(11,29)(12,28)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21) );

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)], [(1,39),(2,38),(3,37),(4,36),(5,35),(6,34),(7,33),(8,32),(9,31),(10,30),(11,29),(12,28),(13,27),(14,26),(15,25),(16,24),(17,23),(18,22),(19,21)])

D39 is a maximal subgroup of   S3×D13  D117  D39⋊C3  C3⋊D39  C13⋊S4  D195
D39 is a maximal quotient of   Dic39  D117  C3⋊D39  C13⋊S4  D195

Matrix representation of D39 in GL2(𝔽79) generated by

3823
5636
,
3823
3041
G:=sub<GL(2,GF(79))| [38,56,23,36],[38,30,23,41] >;

D39 in GAP, Magma, Sage, TeX

D_{39}
% in TeX

G:=Group("D39");
// GroupNames label

G:=SmallGroup(78,5);
// by ID

G=gap.SmallGroup(78,5);
# by ID

G:=PCGroup([3,-2,-3,-13,25,650]);
// Polycyclic

G:=Group<a,b|a^39=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D39 in TeX
Character table of D39 in TeX

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