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G = D37order 74 = 2·37

Dihedral group

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

Aliases: D37, C37⋊C2, sometimes denoted D74 or Dih37 or Dih74, SmallGroup(74,1)

Series: Derived Chief Lower central Upper central

C1C37 — D37
C1C37 — D37
C37 — D37
C1

Generators and relations for D37
 G = < a,b | a37=b2=1, bab=a-1 >

37C2

Character table of D37

 class 1237A37B37C37D37E37F37G37H37I37J37K37L37M37N37O37P37Q37R
 size 137222222222222222222
ρ111111111111111111111    trivial
ρ21-1111111111111111111    linear of order 2
ρ320ζ37243713ζ373637ζ37263711ζ37233714ζ3735372ζ37273710ζ37223715ζ3734373ζ3728379ζ37213716ζ3733374ζ3729378ζ37203717ζ3732375ζ3730377ζ37193718ζ3731376ζ37253712    orthogonal faithful
ρ420ζ3730377ζ3729378ζ37233714ζ373637ζ37213716ζ3731376ζ3728379ζ37243713ζ3735372ζ37203717ζ3732375ζ37273710ζ37253712ζ3734373ζ37193718ζ3733374ζ37263711ζ37223715    orthogonal faithful
ρ520ζ37203717ζ3730377ζ3734373ζ37243713ζ37233714ζ3733374ζ3731376ζ37213716ζ37263711ζ373637ζ3728379ζ37193718ζ3729378ζ3735372ζ37253712ζ37223715ζ3732375ζ37273710    orthogonal faithful
ρ620ζ37273710ζ37223715ζ37203717ζ37253712ζ3730377ζ3735372ζ3734373ζ3729378ζ37243713ζ37193718ζ37233714ζ3728379ζ3733374ζ373637ζ3731376ζ37263711ζ37213716ζ3732375    orthogonal faithful
ρ720ζ3732375ζ37263711ζ37273710ζ3731376ζ37223715ζ373637ζ37203717ζ3733374ζ37253712ζ3728379ζ3730377ζ37233714ζ3735372ζ37193718ζ3734373ζ37243713ζ3729378ζ37213716    orthogonal faithful
ρ820ζ3728379ζ3732375ζ37193718ζ3733374ζ37273710ζ37243713ζ373637ζ37223715ζ3729378ζ3731376ζ37203717ζ3734373ζ37263711ζ37253712ζ3735372ζ37213716ζ3730377ζ37233714    orthogonal faithful
ρ920ζ3729378ζ37253712ζ37213716ζ37203717ζ37243713ζ3728379ζ3732375ζ373637ζ3734373ζ3730377ζ37263711ζ37223715ζ37193718ζ37233714ζ37273710ζ3731376ζ3735372ζ3733374    orthogonal faithful
ρ1020ζ37263711ζ3735372ζ37223715ζ3728379ζ3733374ζ37203717ζ3730377ζ3731376ζ37193718ζ3732375ζ3729378ζ37213716ζ3734373ζ37273710ζ37233714ζ373637ζ37253712ζ37243713    orthogonal faithful
ρ1120ζ37223715ζ3733374ζ3730377ζ37193718ζ3729378ζ3734373ζ37233714ζ37253712ζ373637ζ37273710ζ37213716ζ3732375ζ3731376ζ37203717ζ3728379ζ3735372ζ37243713ζ37263711    orthogonal faithful
ρ1220ζ37213716ζ37243713ζ3732375ζ3734373ζ37263711ζ37193718ζ37273710ζ3735372ζ3731376ζ37233714ζ37223715ζ3730377ζ373637ζ3728379ζ37203717ζ37253712ζ3733374ζ3729378    orthogonal faithful
ρ1320ζ37233714ζ37213716ζ3728379ζ3735372ζ3732375ζ37253712ζ37193718ζ37263711ζ3733374ζ3734373ζ37273710ζ37203717ζ37243713ζ3731376ζ373637ζ3729378ζ37223715ζ3730377    orthogonal faithful
ρ1420ζ37193718ζ37273710ζ373637ζ3729378ζ37203717ζ37263711ζ3735372ζ3730377ζ37213716ζ37253712ζ3734373ζ3731376ζ37223715ζ37243713ζ3733374ζ3732375ζ37233714ζ3728379    orthogonal faithful
ρ1520ζ3735372ζ3734373ζ3733374ζ3732375ζ3731376ζ3730377ζ3729378ζ3728379ζ37273710ζ37263711ζ37253712ζ37243713ζ37233714ζ37223715ζ37213716ζ37203717ζ37193718ζ373637    orthogonal faithful
ρ1620ζ3734373ζ37233714ζ3731376ζ37263711ζ3728379ζ3729378ζ37253712ζ3732375ζ37223715ζ3735372ζ37193718ζ373637ζ37213716ζ3733374ζ37243713ζ3730377ζ37273710ζ37203717    orthogonal faithful
ρ1720ζ37253712ζ37193718ζ37243713ζ3730377ζ373637ζ3732375ζ37263711ζ37203717ζ37233714ζ3729378ζ3735372ζ3733374ζ37273710ζ37213716ζ37223715ζ3728379ζ3734373ζ3731376    orthogonal faithful
ρ1820ζ373637ζ37203717ζ3735372ζ37213716ζ3734373ζ37223715ζ3733374ζ37233714ζ3732375ζ37243713ζ3731376ζ37253712ζ3730377ζ37263711ζ3729378ζ37273710ζ3728379ζ37193718    orthogonal faithful
ρ1920ζ3731376ζ3728379ζ37253712ζ37223715ζ37193718ζ37213716ζ37243713ζ37273710ζ3730377ζ3733374ζ373637ζ3735372ζ3732375ζ3729378ζ37263711ζ37233714ζ37203717ζ3734373    orthogonal faithful
ρ2020ζ3733374ζ3731376ζ3729378ζ37273710ζ37253712ζ37233714ζ37213716ζ37193718ζ37203717ζ37223715ζ37243713ζ37263711ζ3728379ζ3730377ζ3732375ζ3734373ζ373637ζ3735372    orthogonal faithful

Smallest permutation representation of D37
On 37 points: primitive
Generators in S37
(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)
(1 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)

G:=sub<Sym(37)| (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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)>;

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), (1,37)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20) );

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)], [(1,37),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,25),(14,24),(15,23),(16,22),(17,21),(18,20)])

D37 is a maximal subgroup of   C37⋊C4  C37⋊C6  D111  D185
D37 is a maximal quotient of   Dic37  D111  D185

Matrix representation of D37 in GL2(𝔽149) generated by

31148
10
,
31148
66118
G:=sub<GL(2,GF(149))| [31,1,148,0],[31,66,148,118] >;

D37 in GAP, Magma, Sage, TeX

D_{37}
% in TeX

G:=Group("D37");
// GroupNames label

G:=SmallGroup(74,1);
// by ID

G=gap.SmallGroup(74,1);
# by ID

G:=PCGroup([2,-2,-37,289]);
// Polycyclic

G:=Group<a,b|a^37=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D37 in TeX
Character table of D37 in TeX

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