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G = D99order 198 = 2·32·11

Dihedral group

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

Aliases: D99, C9⋊D11, C11⋊D9, C991C2, C3.D33, C33.1S3, sometimes denoted D198 or Dih99 or Dih198, SmallGroup(198,3)

Series: Derived Chief Lower central Upper central

C1C99 — D99
C1C3C33C99 — D99
C99 — D99
C1

Generators and relations for D99
 G = < a,b | a99=b2=1, bab=a-1 >

99C2
33S3
9D11
11D9
3D33

Smallest permutation representation of D99
On 99 points
Generators in S99
(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 99)
(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 93)(9 92)(10 91)(11 90)(12 89)(13 88)(14 87)(15 86)(16 85)(17 84)(18 83)(19 82)(20 81)(21 80)(22 79)(23 78)(24 77)(25 76)(26 75)(27 74)(28 73)(29 72)(30 71)(31 70)(32 69)(33 68)(34 67)(35 66)(36 65)(37 64)(38 63)(39 62)(40 61)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)

G:=sub<Sym(99)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99), (2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99), (2,99)(3,98)(4,97)(5,96)(6,95)(7,94)(8,93)(9,92)(10,91)(11,90)(12,89)(13,88)(14,87)(15,86)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99)], [(2,99),(3,98),(4,97),(5,96),(6,95),(7,94),(8,93),(9,92),(10,91),(11,90),(12,89),(13,88),(14,87),(15,86),(16,85),(17,84),(18,83),(19,82),(20,81),(21,80),(22,79),(23,78),(24,77),(25,76),(26,75),(27,74),(28,73),(29,72),(30,71),(31,70),(32,69),(33,68),(34,67),(35,66),(36,65),(37,64),(38,63),(39,62),(40,61),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51)])

D99 is a maximal subgroup of   D9×D11
D99 is a maximal quotient of   Dic99

51 conjugacy classes

class 1  2  3 9A9B9C11A···11E33A···33J99A···99AD
order12399911···1133···3399···99
size19922222···22···22···2

51 irreducible representations

dim1122222
type+++++++
imageC1C2S3D9D11D33D99
kernelD99C99C33C11C9C3C1
# reps111351030

Matrix representation of D99 in GL2(𝔽199) generated by

62156
43105
,
10
198198
G:=sub<GL(2,GF(199))| [62,43,156,105],[1,198,0,198] >;

D99 in GAP, Magma, Sage, TeX

D_{99}
% in TeX

G:=Group("D99");
// GroupNames label

G:=SmallGroup(198,3);
// by ID

G=gap.SmallGroup(198,3);
# by ID

G:=PCGroup([4,-2,-3,-11,-3,561,533,722,2115]);
// Polycyclic

G:=Group<a,b|a^99=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D99 in TeX

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