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G = D49order 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

C1C49 — D49
C1C7C49 — D49
C49 — D49
C1

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

49C2
7D7

Character table of D49

 class 127A7B7C49A49B49C49D49E49F49G49H49I49J49K49L49M49N49O49P49Q49R49S49T49U
 size 149222222222222222222222222
ρ111111111111111111111111111    trivial
ρ21-1111111111111111111111111    linear of order 2
ρ320222ζ7572ζ7572ζ7572ζ7572ζ7572ζ7572ζ7473ζ7473ζ7473ζ7473ζ7473ζ7473ζ7473ζ767ζ767ζ767ζ767ζ767ζ767ζ767ζ7572    orthogonal lifted from D7
ρ420222ζ7473ζ7473ζ7473ζ7473ζ7473ζ7473ζ767ζ767ζ767ζ767ζ767ζ767ζ767ζ7572ζ7572ζ7572ζ7572ζ7572ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ520222ζ767ζ767ζ767ζ767ζ767ζ767ζ7572ζ7572ζ7572ζ7572ζ7572ζ7572ζ7572ζ7473ζ7473ζ7473ζ7473ζ7473ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ620ζ49354914ζ49284921ζ4942497ζ49274922ζ49294920ζ49364913ζ4943496ζ494849ζ4941498ζ49304919ζ49374912ζ4944495ζ4947492ζ4940499ζ49334916ζ49264923ζ4945494ζ4946493ζ49394910ζ49324917ζ49254924ζ49314918ζ49384911ζ49344915    orthogonal faithful
ρ720ζ4942497ζ49354914ζ49284921ζ4946493ζ49254924ζ4945494ζ49324917ζ49384911ζ49394910ζ49364913ζ49344915ζ4943496ζ49274922ζ494849ζ49294920ζ4941498ζ4944495ζ49334916ζ49374912ζ4940499ζ49304919ζ4947492ζ49264923ζ49314918    orthogonal faithful
ρ820ζ4942497ζ49354914ζ49284921ζ4945494ζ49324917ζ49384911ζ49394910ζ49314918ζ4946493ζ494849ζ49294920ζ4941498ζ49364913ζ49344915ζ4943496ζ49274922ζ49264923ζ4944495ζ49334916ζ49374912ζ4940499ζ49304919ζ4947492ζ49254924    orthogonal faithful
ρ920ζ49284921ζ4942497ζ49354914ζ4940499ζ49264923ζ49374912ζ4947492ζ49334916ζ49304919ζ49394910ζ4945494ζ49314918ζ49324917ζ4946493ζ49384911ζ49254924ζ49344915ζ494849ζ49364913ζ49274922ζ4941498ζ4943496ζ49294920ζ4944495    orthogonal faithful
ρ1020ζ49354914ζ49284921ζ4942497ζ4943496ζ494849ζ4941498ζ49344915ζ49274922ζ49294920ζ49264923ζ49304919ζ49374912ζ4944495ζ4947492ζ4940499ζ49334916ζ49394910ζ49324917ζ49254924ζ49314918ζ49384911ζ4945494ζ4946493ζ49364913    orthogonal faithful
ρ1120ζ4942497ζ49354914ζ49284921ζ49254924ζ4945494ζ49324917ζ49384911ζ49394910ζ49314918ζ4943496ζ49274922ζ494849ζ49294920ζ4941498ζ49364913ζ49344915ζ4940499ζ49304919ζ4947492ζ49264923ζ4944495ζ49334916ζ49374912ζ4946493    orthogonal faithful
ρ1220ζ49354914ζ49284921ζ4942497ζ494849ζ4941498ζ49344915ζ49274922ζ49294920ζ49364913ζ49374912ζ4944495ζ4947492ζ4940499ζ49334916ζ49264923ζ49304919ζ49314918ζ49384911ζ4945494ζ4946493ζ49394910ζ49324917ζ49254924ζ4943496    orthogonal faithful
ρ1320ζ49284921ζ4942497ζ49354914ζ49334916ζ49304919ζ4944495ζ4940499ζ49264923ζ49374912ζ4945494ζ49314918ζ49324917ζ4946493ζ49384911ζ49254924ζ49394910ζ4943496ζ49294920ζ49344915ζ494849ζ49364913ζ49274922ζ4941498ζ4947492    orthogonal faithful
ρ1420ζ49354914ζ49284921ζ4942497ζ49294920ζ49364913ζ4943496ζ494849ζ4941498ζ49344915ζ4944495ζ4947492ζ4940499ζ49334916ζ49264923ζ49304919ζ49374912ζ49324917ζ49254924ζ49314918ζ49384911ζ4945494ζ4946493ζ49394910ζ49274922    orthogonal faithful
ρ1520ζ49284921ζ4942497ζ49354914ζ49304919ζ4944495ζ4940499ζ49264923ζ49374912ζ4947492ζ49324917ζ4946493ζ49384911ζ49254924ζ49394910ζ4945494ζ49314918ζ494849ζ49364913ζ49274922ζ4941498ζ4943496ζ49294920ζ49344915ζ49334916    orthogonal faithful
ρ1620ζ4942497ζ49354914ζ49284921ζ49314918ζ4946493ζ49254924ζ4945494ζ49324917ζ49384911ζ49294920ζ4941498ζ49364913ζ49344915ζ4943496ζ49274922ζ494849ζ49304919ζ4947492ζ49264923ζ4944495ζ49334916ζ49374912ζ4940499ζ49394910    orthogonal faithful
ρ1720ζ4942497ζ49354914ζ49284921ζ49324917ζ49384911ζ49394910ζ49314918ζ4946493ζ49254924ζ4941498ζ49364913ζ49344915ζ4943496ζ49274922ζ494849ζ49294920ζ49374912ζ4940499ζ49304919ζ4947492ζ49264923ζ4944495ζ49334916ζ4945494    orthogonal faithful
ρ1820ζ49284921ζ4942497ζ49354914ζ4947492ζ49334916ζ49304919ζ4944495ζ4940499ζ49264923ζ49254924ζ49394910ζ4945494ζ49314918ζ49324917ζ4946493ζ49384911ζ49364913ζ49274922ζ4941498ζ4943496ζ49294920ζ49344915ζ494849ζ49374912    orthogonal faithful
ρ1920ζ49284921ζ4942497ζ49354914ζ49374912ζ4947492ζ49334916ζ49304919ζ4944495ζ4940499ζ4946493ζ49384911ζ49254924ζ49394910ζ4945494ζ49314918ζ49324917ζ49294920ζ49344915ζ494849ζ49364913ζ49274922ζ4941498ζ4943496ζ49264923    orthogonal faithful
ρ2020ζ49284921ζ4942497ζ49354914ζ49264923ζ49374912ζ4947492ζ49334916ζ49304919ζ4944495ζ49314918ζ49324917ζ4946493ζ49384911ζ49254924ζ49394910ζ4945494ζ49274922ζ4941498ζ4943496ζ49294920ζ49344915ζ494849ζ49364913ζ4940499    orthogonal faithful
ρ2120ζ49354914ζ49284921ζ4942497ζ49344915ζ49274922ζ49294920ζ49364913ζ4943496ζ494849ζ49334916ζ49264923ζ49304919ζ49374912ζ4944495ζ4947492ζ4940499ζ49254924ζ49314918ζ49384911ζ4945494ζ4946493ζ49394910ζ49324917ζ4941498    orthogonal faithful
ρ2220ζ4942497ζ49354914ζ49284921ζ49394910ζ49314918ζ4946493ζ49254924ζ4945494ζ49324917ζ49274922ζ494849ζ49294920ζ4941498ζ49364913ζ49344915ζ4943496ζ49334916ζ49374912ζ4940499ζ49304919ζ4947492ζ49264923ζ4944495ζ49384911    orthogonal faithful
ρ2320ζ49284921ζ4942497ζ49354914ζ4944495ζ4940499ζ49264923ζ49374912ζ4947492ζ49334916ζ49384911ζ49254924ζ49394910ζ4945494ζ49314918ζ49324917ζ4946493ζ4941498ζ4943496ζ49294920ζ49344915ζ494849ζ49364913ζ49274922ζ49304919    orthogonal faithful
ρ2420ζ49354914ζ49284921ζ4942497ζ4941498ζ49344915ζ49274922ζ49294920ζ49364913ζ4943496ζ4947492ζ4940499ζ49334916ζ49264923ζ49304919ζ49374912ζ4944495ζ4946493ζ49394910ζ49324917ζ49254924ζ49314918ζ49384911ζ4945494ζ494849    orthogonal faithful
ρ2520ζ49354914ζ49284921ζ4942497ζ49364913ζ4943496ζ494849ζ4941498ζ49344915ζ49274922ζ4940499ζ49334916ζ49264923ζ49304919ζ49374912ζ4944495ζ4947492ζ49384911ζ4945494ζ4946493ζ49394910ζ49324917ζ49254924ζ49314918ζ49294920    orthogonal faithful
ρ2620ζ4942497ζ49354914ζ49284921ζ49384911ζ49394910ζ49314918ζ4946493ζ49254924ζ4945494ζ49344915ζ4943496ζ49274922ζ494849ζ49294920ζ4941498ζ49364913ζ4947492ζ49264923ζ4944495ζ49334916ζ49374912ζ4940499ζ49304919ζ49324917    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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