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G = D70  order 140 = 22·5·7

Dihedral group

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D70, C2×D35, C14⋊D5, C10⋊D7, C5⋊2D14, C7⋊2D10, C70⋊1C2, C35⋊2C22, sometimes denoted D140 or Dih70 or Dih140, SmallGroup(140,10)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C35 — D70
Chief series
C1 — C7 — C35 — D35 — D70
Lower central
C35 — D70
Upper central
C1 — C2

Generators and relations for D70
 G = < a,b | a70=b2=1, bab=a-1 >

C1
C2
35C2
35C2
C5
C7
35C22
C10
7D5
7D5
C14
5D7
5D7
C35
7D10
5D14
D35
D35
C70
D70

Smallest permutation representation of D70
►On 70 points
Generators in S70
(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)

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

D70 is a maximal subgroup of   D70.C2  C5⋊D28  C7⋊D20  D140  C35⋊7D4  C2×D5×D7
D70 is a maximal quotient of   Dic70  D140  C35⋊7D4

38 conjugacy classes

class 1 2A2B2C5A5B7A7B7C10A10B14A14B14C35A···35L70A···70L
order122255777101014141435···3570···70
size11353522222222222···22···2

38 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D5D7D10D14D35D70
kernelD70D35C70C14C10C7C5C2C1
# reps12123231212

Matrix representation of D70 ►in GL2(𝔽71) generated by

4611
6019
,
4611
5325
G:=sub<GL(2,GF(71))| [46,60,11,19],[46,53,11,25] >;
 

D70 in GAP, Magma, Sage, TeX 

D_{70}
 
% in TeX
 
G:=Group("D70");
 
// GroupNames label
 
G:=SmallGroup(140,10);
 
// by ID
 
G=gap.SmallGroup(140,10);
 
# by ID
 
G:=PCGroup([4,-2,-2,-5,-7,194,1923]);
 
// Polycyclic
 
G:=Group<a,b|a^70=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D70 in TeX

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