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G = D36order 72 = 23·32

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D36, C4⋊D9, C91D4, C3.D12, C361C2, C6.8D6, D181C2, C12.2S3, C2.4D18, C18.3C22, sometimes denoted D72 or Dih36 or Dih72, SmallGroup(72,6)

Series: Derived Chief Lower central Upper central

C1C18 — D36
C1C3C9C18D18 — D36
C9C18 — D36
C1C2C4

Generators and relations for D36
 G = < a,b | a36=b2=1, bab=a-1 >

18C2
18C2
9C22
9C22
6S3
6S3
9D4
3D6
3D6
2D9
2D9
3D12

Character table of D36

 class 12A2B2C3469A9B9C12A12B18A18B18C36A36B36C36D36E36F
 size 11181822222222222222222
ρ1111111111111111111111    trivial
ρ2111-11-11111-1-1111-1-1-1-1-1-1    linear of order 2
ρ311-1-111111111111111111    linear of order 2
ρ411-111-11111-1-1111-1-1-1-1-1-1    linear of order 2
ρ52200222-1-1-122-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-20020-222200-2-2-2000000    orthogonal lifted from D4
ρ722002-22-1-1-1-2-2-1-1-1111111    orthogonal lifted from D6
ρ82200-12-1ζ9594ζ989ζ9792-1-1ζ989ζ9792ζ9594ζ9594ζ9792ζ989ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ92200-12-1ζ9792ζ9594ζ989-1-1ζ9594ζ989ζ9792ζ9792ζ989ζ9594ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ102200-1-2-1ζ9792ζ9594ζ98911ζ9594ζ989ζ97929792989959497929594989    orthogonal lifted from D18
ρ112200-1-2-1ζ989ζ9792ζ959411ζ9792ζ9594ζ9899899594979298997929594    orthogonal lifted from D18
ρ122200-12-1ζ989ζ9792ζ9594-1-1ζ9792ζ9594ζ989ζ989ζ9594ζ9792ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ132200-1-2-1ζ9594ζ989ζ979211ζ989ζ9792ζ95949594979298995949899792    orthogonal lifted from D18
ρ142-20020-2-1-1-1001113-3-3-333    orthogonal lifted from D12
ρ152-20020-2-1-1-100111-3333-3-3    orthogonal lifted from D12
ρ162-200-101ζ9594ζ989ζ97923-398997929594ζ4ζ954ζ94ζ4ζ974ζ92ζ43ζ9843ζ94ζ954ζ9443ζ9843ζ94ζ974ζ92    orthogonal faithful
ρ172-200-101ζ9792ζ9594ζ989-3395949899792ζ4ζ974ζ9243ζ9843ζ9ζ4ζ954ζ944ζ974ζ924ζ954ζ94ζ43ζ9843ζ9    orthogonal faithful
ρ182-200-101ζ9792ζ9594ζ9893-3959498997924ζ974ζ92ζ43ζ9843ζ94ζ954ζ94ζ4ζ974ζ92ζ4ζ954ζ9443ζ9843ζ9    orthogonal faithful
ρ192-200-101ζ989ζ9792ζ95943-39792959498943ζ9843ζ94ζ954ζ94ζ4ζ974ζ92ζ43ζ9843ζ94ζ974ζ92ζ4ζ954ζ94    orthogonal faithful
ρ202-200-101ζ989ζ9792ζ9594-3397929594989ζ43ζ9843ζ9ζ4ζ954ζ944ζ974ζ9243ζ9843ζ9ζ4ζ974ζ924ζ954ζ94    orthogonal faithful
ρ212-200-101ζ9594ζ989ζ9792-33989979295944ζ954ζ944ζ974ζ9243ζ9843ζ9ζ4ζ954ζ94ζ43ζ9843ζ9ζ4ζ974ζ92    orthogonal faithful

Smallest permutation representation of D36
On 36 points
Generators in S36
(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)
(1 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)

G:=sub<Sym(36)| (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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;

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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );

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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,30),(8,29),(9,28),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)])

D36 is a maximal subgroup of
C72⋊C2  D72  D4⋊D9  Q82D9  D365C2  D4×D9  Q83D9  D108  C3⋊D36  D36⋊C3  C36⋊S3  C22⋊D36  C12.4S4  C5⋊D36  D180
D36 is a maximal quotient of
Dic36  C72⋊C2  D72  C4⋊Dic9  D18⋊C4  D108  C3⋊D36  C36⋊S3  C22⋊D36  C5⋊D36  D180

Matrix representation of D36 in GL2(𝔽37) generated by

124
338
,
124
2925
G:=sub<GL(2,GF(37))| [12,33,4,8],[12,29,4,25] >;

D36 in GAP, Magma, Sage, TeX

D_{36}
% in TeX

G:=Group("D36");
// GroupNames label

G:=SmallGroup(72,6);
// by ID

G=gap.SmallGroup(72,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,803,138,1204]);
// Polycyclic

G:=Group<a,b|a^36=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D36 in TeX
Character table of D36 in TeX

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