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G = D72order 144 = 24·32

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D72, C91D8, C81D9, C3.D24, C721C2, D361C2, 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

C1C36 — D72
C1C3C9C18C36D36 — D72
C9C18C36 — D72
C1C2C4C8

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

36C2
36C2
18C22
18C22
12S3
12S3
9D4
9D4
6D6
6D6
4D9
4D9
9D8
3D12
3D12
2D18
2D18
3D24

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  D727C2  C8⋊D18  D8×D9  D72⋊C2  D725C2  D216  C3⋊D72  D72⋊C3  C721S3
D72 is a maximal quotient of
D144  C144⋊C2  Dic72  C721C4  C2.D72  D216  C3⋊D72  C721S3

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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