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G = D98order 196 = 22·72

Dihedral group

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

Aliases: D98, C2×D49, C98⋊C2, C49⋊C22, C7.D14, C14.2D7, sometimes denoted D196 or Dih98 or Dih196, SmallGroup(196,3)

Series: Derived Chief Lower central Upper central

C1C49 — D98
C1C7C49D49 — D98
C49 — D98
C1C2

Generators and relations for D98
 G = < a,b | a98=b2=1, bab=a-1 >

49C2
49C2
49C22
7D7
7D7
7D14

Smallest permutation representation of D98
On 98 points
Generators in S98
(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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98)
(1 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)

G:=sub<Sym(98)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98), (1,98)(2,97)(3,96)(4,95)(5,94)(6,93)(7,92)(8,91)(9,90)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98)], [(1,98),(2,97),(3,96),(4,95),(5,94),(6,93),(7,92),(8,91),(9,90),(10,89),(11,88),(12,87),(13,86),(14,85),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,62),(38,61),(39,60),(40,59),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50)])

D98 is a maximal subgroup of   D196  C49⋊D4
D98 is a maximal quotient of   Dic98  D196  C49⋊D4

52 conjugacy classes

class 1 2A2B2C7A7B7C14A14B14C49A···49U98A···98U
order122277714141449···4998···98
size1149492222222···22···2

52 irreducible representations

dim1112222
type+++++++
imageC1C2C2D7D14D49D98
kernelD98D49C98C14C7C2C1
# reps121332121

Matrix representation of D98 in GL2(𝔽197) generated by

145178
19113
,
145178
16352
G:=sub<GL(2,GF(197))| [145,19,178,113],[145,163,178,52] >;

D98 in GAP, Magma, Sage, TeX

D_{98}
% in TeX

G:=Group("D98");
// GroupNames label

G:=SmallGroup(196,3);
// by ID

G=gap.SmallGroup(196,3);
# by ID

G:=PCGroup([4,-2,-2,-7,-7,626,514,2691]);
// Polycyclic

G:=Group<a,b|a^98=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D98 in TeX

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