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G = D72  order 144 = 24·32

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D72, C9⋊1D8, C8⋊1D9, C3.D24, C72⋊1C2, D36⋊1C2, C24.2S3, C18.3D4, C2.5D36, C6.3D12, C4.10D18, C12.42D6, C36.10C22, sometimes denoted D144 or Dih72 or Dih144, SmallGroup(144,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — D72
Chief series
C1 — C3 — C9 — C18 — C36 — D36 — D72
Lower central
C9 — C18 — C36 — D72
Upper central
C1 — C2 — C4 — C8

Generators and relations for D72
 G = < a,b | a72=b2=1, bab=a-1 >

C1
C2
36C2
36C2
C3
C4
18C22
18C22
C6
12S3
12S3
C9
C8
9D4
9D4
C12
6D6
6D6
C18
4D9
4D9
9D8
C24
3D12
3D12
C36
2D18
2D18
3D24
D36
D36
C72
D72

Smallest permutation representation of D72
►On 72 points
Generators in S72
(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)

(1 72)(2 71)(3 70)(4 69)(5 68)(6 67)(7 66)(8 65)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 52)(22 51)(23 50)(24 49)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)
 
G:=sub<Sym(72)| (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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;
 
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), (1,72)(2,71)(3,70)(4,69)(5,68)(6,67)(7,66)(8,65)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,52)(22,51)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );
 
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)], [(1,72),(2,71),(3,70),(4,69),(5,68),(6,67),(7,66),(8,65),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,52),(22,51),(23,50),(24,49),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])
 

D72 is a maximal subgroup of
 D144  C144⋊C2  C9⋊D16  C9⋊SD32  D72⋊7C2  C8⋊D18  D8×D9  D72⋊C2  D72⋊5C2  D216  C3⋊D72  D72⋊C3  C72⋊1S3
D72 is a maximal quotient of
 D144  C144⋊C2  Dic72  C72⋊1C4  C2.D72  D216  C3⋊D72  C72⋊1S3

39 conjugacy classes

class 1 2A2B2C 3  4  6 8A8B9A9B9C12A12B18A18B18C24A24B24C24D36A···36F72A···72L
order12223468899912121818182424242436···3672···72
size113636222222222222222222···22···2

39 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D6D8D9D12D18D24D36D72
kernelD72C72D36C24C18C12C9C8C6C4C3C2C1
# reps11211123234612

Matrix representation of D72 ►in GL2(𝔽73) generated by

112
7113
,
5523
518
G:=sub<GL(2,GF(73))| [11,71,2,13],[55,5,23,18] >;
 

D72 in GAP, Magma, Sage, TeX 

D_{72}
 
% in TeX
 
G:=Group("D72");
 
// GroupNames label
 
G:=SmallGroup(144,8);
 
// by ID
 
G=gap.SmallGroup(144,8);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,79,218,50,2404,208,3461]);
 
// Polycyclic
 
G:=Group<a,b|a^72=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D72 in TeX

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