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G = D49  order 98 = 2·72

Dihedral group

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

Aliases: D49, C49⋊C2, C7.D7, sometimes denoted D98 or Dih49 or Dih98, SmallGroup(98,1)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C49 — D49
Chief series
C1 — C7 — C49 — D49
Lower central
C49 — D49
Upper central
C1

Generators and relations for D49
 G = < a,b | a49=b2=1, bab=a-1 >

C1
49C2
C7
7D7
C49
D49

Character table of D49

 class 127A7B7C49A49B49C49D49E49F49G49H49I49J49K49L49M49N49O49P49Q49R49S49T49U
 size 149222222222222222222222222
ρ111111111111111111111111111    trivial
ρ21-1111111111111111111111111    linear of order 2
ρ320222ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ420222ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ520222ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ620ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4927+ζ4922ζ4929+ζ4920ζ4936+ζ4913ζ4943+ζ496ζ4948+ζ49ζ4941+ζ498ζ4930+ζ4919ζ4937+ζ4912ζ4944+ζ495ζ4947+ζ492ζ4940+ζ499ζ4933+ζ4916ζ4926+ζ4923ζ4945+ζ494ζ4946+ζ493ζ4939+ζ4910ζ4932+ζ4917ζ4925+ζ4924ζ4931+ζ4918ζ4938+ζ4911ζ4934+ζ4915    orthogonal faithful
ρ720ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4946+ζ493ζ4925+ζ4924ζ4945+ζ494ζ4932+ζ4917ζ4938+ζ4911ζ4939+ζ4910ζ4936+ζ4913ζ4934+ζ4915ζ4943+ζ496ζ4927+ζ4922ζ4948+ζ49ζ4929+ζ4920ζ4941+ζ498ζ4944+ζ495ζ4933+ζ4916ζ4937+ζ4912ζ4940+ζ499ζ4930+ζ4919ζ4947+ζ492ζ4926+ζ4923ζ4931+ζ4918    orthogonal faithful
ρ820ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4945+ζ494ζ4932+ζ4917ζ4938+ζ4911ζ4939+ζ4910ζ4931+ζ4918ζ4946+ζ493ζ4948+ζ49ζ4929+ζ4920ζ4941+ζ498ζ4936+ζ4913ζ4934+ζ4915ζ4943+ζ496ζ4927+ζ4922ζ4926+ζ4923ζ4944+ζ495ζ4933+ζ4916ζ4937+ζ4912ζ4940+ζ499ζ4930+ζ4919ζ4947+ζ492ζ4925+ζ4924    orthogonal faithful
ρ920ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4940+ζ499ζ4926+ζ4923ζ4937+ζ4912ζ4947+ζ492ζ4933+ζ4916ζ4930+ζ4919ζ4939+ζ4910ζ4945+ζ494ζ4931+ζ4918ζ4932+ζ4917ζ4946+ζ493ζ4938+ζ4911ζ4925+ζ4924ζ4934+ζ4915ζ4948+ζ49ζ4936+ζ4913ζ4927+ζ4922ζ4941+ζ498ζ4943+ζ496ζ4929+ζ4920ζ4944+ζ495    orthogonal faithful
ρ1020ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4943+ζ496ζ4948+ζ49ζ4941+ζ498ζ4934+ζ4915ζ4927+ζ4922ζ4929+ζ4920ζ4926+ζ4923ζ4930+ζ4919ζ4937+ζ4912ζ4944+ζ495ζ4947+ζ492ζ4940+ζ499ζ4933+ζ4916ζ4939+ζ4910ζ4932+ζ4917ζ4925+ζ4924ζ4931+ζ4918ζ4938+ζ4911ζ4945+ζ494ζ4946+ζ493ζ4936+ζ4913    orthogonal faithful
ρ1120ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4925+ζ4924ζ4945+ζ494ζ4932+ζ4917ζ4938+ζ4911ζ4939+ζ4910ζ4931+ζ4918ζ4943+ζ496ζ4927+ζ4922ζ4948+ζ49ζ4929+ζ4920ζ4941+ζ498ζ4936+ζ4913ζ4934+ζ4915ζ4940+ζ499ζ4930+ζ4919ζ4947+ζ492ζ4926+ζ4923ζ4944+ζ495ζ4933+ζ4916ζ4937+ζ4912ζ4946+ζ493    orthogonal faithful
ρ1220ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4948+ζ49ζ4941+ζ498ζ4934+ζ4915ζ4927+ζ4922ζ4929+ζ4920ζ4936+ζ4913ζ4937+ζ4912ζ4944+ζ495ζ4947+ζ492ζ4940+ζ499ζ4933+ζ4916ζ4926+ζ4923ζ4930+ζ4919ζ4931+ζ4918ζ4938+ζ4911ζ4945+ζ494ζ4946+ζ493ζ4939+ζ4910ζ4932+ζ4917ζ4925+ζ4924ζ4943+ζ496    orthogonal faithful
ρ1320ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4933+ζ4916ζ4930+ζ4919ζ4944+ζ495ζ4940+ζ499ζ4926+ζ4923ζ4937+ζ4912ζ4945+ζ494ζ4931+ζ4918ζ4932+ζ4917ζ4946+ζ493ζ4938+ζ4911ζ4925+ζ4924ζ4939+ζ4910ζ4943+ζ496ζ4929+ζ4920ζ4934+ζ4915ζ4948+ζ49ζ4936+ζ4913ζ4927+ζ4922ζ4941+ζ498ζ4947+ζ492    orthogonal faithful
ρ1420ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4929+ζ4920ζ4936+ζ4913ζ4943+ζ496ζ4948+ζ49ζ4941+ζ498ζ4934+ζ4915ζ4944+ζ495ζ4947+ζ492ζ4940+ζ499ζ4933+ζ4916ζ4926+ζ4923ζ4930+ζ4919ζ4937+ζ4912ζ4932+ζ4917ζ4925+ζ4924ζ4931+ζ4918ζ4938+ζ4911ζ4945+ζ494ζ4946+ζ493ζ4939+ζ4910ζ4927+ζ4922    orthogonal faithful
ρ1520ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4930+ζ4919ζ4944+ζ495ζ4940+ζ499ζ4926+ζ4923ζ4937+ζ4912ζ4947+ζ492ζ4932+ζ4917ζ4946+ζ493ζ4938+ζ4911ζ4925+ζ4924ζ4939+ζ4910ζ4945+ζ494ζ4931+ζ4918ζ4948+ζ49ζ4936+ζ4913ζ4927+ζ4922ζ4941+ζ498ζ4943+ζ496ζ4929+ζ4920ζ4934+ζ4915ζ4933+ζ4916    orthogonal faithful
ρ1620ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4931+ζ4918ζ4946+ζ493ζ4925+ζ4924ζ4945+ζ494ζ4932+ζ4917ζ4938+ζ4911ζ4929+ζ4920ζ4941+ζ498ζ4936+ζ4913ζ4934+ζ4915ζ4943+ζ496ζ4927+ζ4922ζ4948+ζ49ζ4930+ζ4919ζ4947+ζ492ζ4926+ζ4923ζ4944+ζ495ζ4933+ζ4916ζ4937+ζ4912ζ4940+ζ499ζ4939+ζ4910    orthogonal faithful
ρ1720ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4932+ζ4917ζ4938+ζ4911ζ4939+ζ4910ζ4931+ζ4918ζ4946+ζ493ζ4925+ζ4924ζ4941+ζ498ζ4936+ζ4913ζ4934+ζ4915ζ4943+ζ496ζ4927+ζ4922ζ4948+ζ49ζ4929+ζ4920ζ4937+ζ4912ζ4940+ζ499ζ4930+ζ4919ζ4947+ζ492ζ4926+ζ4923ζ4944+ζ495ζ4933+ζ4916ζ4945+ζ494    orthogonal faithful
ρ1820ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4947+ζ492ζ4933+ζ4916ζ4930+ζ4919ζ4944+ζ495ζ4940+ζ499ζ4926+ζ4923ζ4925+ζ4924ζ4939+ζ4910ζ4945+ζ494ζ4931+ζ4918ζ4932+ζ4917ζ4946+ζ493ζ4938+ζ4911ζ4936+ζ4913ζ4927+ζ4922ζ4941+ζ498ζ4943+ζ496ζ4929+ζ4920ζ4934+ζ4915ζ4948+ζ49ζ4937+ζ4912    orthogonal faithful
ρ1920ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4937+ζ4912ζ4947+ζ492ζ4933+ζ4916ζ4930+ζ4919ζ4944+ζ495ζ4940+ζ499ζ4946+ζ493ζ4938+ζ4911ζ4925+ζ4924ζ4939+ζ4910ζ4945+ζ494ζ4931+ζ4918ζ4932+ζ4917ζ4929+ζ4920ζ4934+ζ4915ζ4948+ζ49ζ4936+ζ4913ζ4927+ζ4922ζ4941+ζ498ζ4943+ζ496ζ4926+ζ4923    orthogonal faithful
ρ2020ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4926+ζ4923ζ4937+ζ4912ζ4947+ζ492ζ4933+ζ4916ζ4930+ζ4919ζ4944+ζ495ζ4931+ζ4918ζ4932+ζ4917ζ4946+ζ493ζ4938+ζ4911ζ4925+ζ4924ζ4939+ζ4910ζ4945+ζ494ζ4927+ζ4922ζ4941+ζ498ζ4943+ζ496ζ4929+ζ4920ζ4934+ζ4915ζ4948+ζ49ζ4936+ζ4913ζ4940+ζ499    orthogonal faithful
ρ2120ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4934+ζ4915ζ4927+ζ4922ζ4929+ζ4920ζ4936+ζ4913ζ4943+ζ496ζ4948+ζ49ζ4933+ζ4916ζ4926+ζ4923ζ4930+ζ4919ζ4937+ζ4912ζ4944+ζ495ζ4947+ζ492ζ4940+ζ499ζ4925+ζ4924ζ4931+ζ4918ζ4938+ζ4911ζ4945+ζ494ζ4946+ζ493ζ4939+ζ4910ζ4932+ζ4917ζ4941+ζ498    orthogonal faithful
ρ2220ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4939+ζ4910ζ4931+ζ4918ζ4946+ζ493ζ4925+ζ4924ζ4945+ζ494ζ4932+ζ4917ζ4927+ζ4922ζ4948+ζ49ζ4929+ζ4920ζ4941+ζ498ζ4936+ζ4913ζ4934+ζ4915ζ4943+ζ496ζ4933+ζ4916ζ4937+ζ4912ζ4940+ζ499ζ4930+ζ4919ζ4947+ζ492ζ4926+ζ4923ζ4944+ζ495ζ4938+ζ4911    orthogonal faithful
ρ2320ζ4928+ζ4921ζ4942+ζ497ζ4935+ζ4914ζ4944+ζ495ζ4940+ζ499ζ4926+ζ4923ζ4937+ζ4912ζ4947+ζ492ζ4933+ζ4916ζ4938+ζ4911ζ4925+ζ4924ζ4939+ζ4910ζ4945+ζ494ζ4931+ζ4918ζ4932+ζ4917ζ4946+ζ493ζ4941+ζ498ζ4943+ζ496ζ4929+ζ4920ζ4934+ζ4915ζ4948+ζ49ζ4936+ζ4913ζ4927+ζ4922ζ4930+ζ4919    orthogonal faithful
ρ2420ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4941+ζ498ζ4934+ζ4915ζ4927+ζ4922ζ4929+ζ4920ζ4936+ζ4913ζ4943+ζ496ζ4947+ζ492ζ4940+ζ499ζ4933+ζ4916ζ4926+ζ4923ζ4930+ζ4919ζ4937+ζ4912ζ4944+ζ495ζ4946+ζ493ζ4939+ζ4910ζ4932+ζ4917ζ4925+ζ4924ζ4931+ζ4918ζ4938+ζ4911ζ4945+ζ494ζ4948+ζ49    orthogonal faithful
ρ2520ζ4935+ζ4914ζ4928+ζ4921ζ4942+ζ497ζ4936+ζ4913ζ4943+ζ496ζ4948+ζ49ζ4941+ζ498ζ4934+ζ4915ζ4927+ζ4922ζ4940+ζ499ζ4933+ζ4916ζ4926+ζ4923ζ4930+ζ4919ζ4937+ζ4912ζ4944+ζ495ζ4947+ζ492ζ4938+ζ4911ζ4945+ζ494ζ4946+ζ493ζ4939+ζ4910ζ4932+ζ4917ζ4925+ζ4924ζ4931+ζ4918ζ4929+ζ4920    orthogonal faithful
ρ2620ζ4942+ζ497ζ4935+ζ4914ζ4928+ζ4921ζ4938+ζ4911ζ4939+ζ4910ζ4931+ζ4918ζ4946+ζ493ζ4925+ζ4924ζ4945+ζ494ζ4934+ζ4915ζ4943+ζ496ζ4927+ζ4922ζ4948+ζ49ζ4929+ζ4920ζ4941+ζ498ζ4936+ζ4913ζ4947+ζ492ζ4926+ζ4923ζ4944+ζ495ζ4933+ζ4916ζ4937+ζ4912ζ4940+ζ499ζ4930+ζ4919ζ4932+ζ4917    orthogonal faithful

Smallest permutation representation of D49
►On 49 points
Generators in S49
(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)

(1 49)(2 48)(3 47)(4 46)(5 45)(6 44)(7 43)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 35)(16 34)(17 33)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)
 
G:=sub<Sym(49)| (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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26)>;
 
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), (1,49)(2,48)(3,47)(4,46)(5,45)(6,44)(7,43)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,35)(16,34)(17,33)(18,32)(19,31)(20,30)(21,29)(22,28)(23,27)(24,26) );
 
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)], [(1,49),(2,48),(3,47),(4,46),(5,45),(6,44),(7,43),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,35),(16,34),(17,33),(18,32),(19,31),(20,30),(21,29),(22,28),(23,27),(24,26)]])
 

D49 is a maximal subgroup of   C49⋊C6  D147  D245
D49 is a maximal quotient of   Dic49  D147  D245

Matrix representation of D49 ►in GL2(𝔽197) generated by

12617
18054
,
109106
18988
G:=sub<GL(2,GF(197))| [126,180,17,54],[109,189,106,88] >;
 

D49 in GAP, Magma, Sage, TeX 

D_{49}
 
% in TeX
 
G:=Group("D49");
 
// GroupNames label
 
G:=SmallGroup(98,1);
 
// by ID
 
G=gap.SmallGroup(98,1);
 
# by ID
 
G:=PCGroup([3,-2,-7,-7,577,46,758]);
 
// Polycyclic
 
G:=Group<a,b|a^49=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D49 in TeX
Character table of D49 in TeX

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