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G = D102order 204 = 22·3·17

Dihedral group

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

Aliases: D102, C2×D51, C34⋊S3, C6⋊D17, C172D6, C32D34, C1021C2, C512C22, sometimes denoted D204 or Dih102 or Dih204, SmallGroup(204,11)

Series: Derived Chief Lower central Upper central

C1C51 — D102
C1C17C51D51 — D102
C51 — D102
C1C2

Generators and relations for D102
 G = < a,b | a102=b2=1, bab=a-1 >

51C2
51C2
51C22
17S3
17S3
3D17
3D17
17D6
3D34

Smallest permutation representation of D102
On 102 points
Generators in S102
(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 99 100 101 102)
(1 102)(2 101)(3 100)(4 99)(5 98)(6 97)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 84)(20 83)(21 82)(22 81)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(41 62)(42 61)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)

G:=sub<Sym(102)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,102)(2,101)(3,100)(4,99)(5,98)(6,97)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102), (1,102)(2,101)(3,100)(4,99)(5,98)(6,97)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,90)(14,89)(15,88)(16,87)(17,86)(18,85)(19,84)(20,83)(21,82)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102)], [(1,102),(2,101),(3,100),(4,99),(5,98),(6,97),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,90),(14,89),(15,88),(16,87),(17,86),(18,85),(19,84),(20,83),(21,82),(22,81),(23,80),(24,79),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(41,62),(42,61),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52)])

D102 is a maximal subgroup of   D512C4  C3⋊D68  C17⋊D12  D204  C517D4  C2×S3×D17
D102 is a maximal quotient of   Dic102  D204  C517D4

54 conjugacy classes

class 1 2A2B2C 3  6 17A···17H34A···34H51A···51P102A···102P
order12223617···1734···3451···51102···102
size115151222···22···22···22···2

54 irreducible representations

dim111222222
type+++++++++
imageC1C2C2S3D6D17D34D51D102
kernelD102D51C102C34C17C6C3C2C1
# reps12111881616

Matrix representation of D102 in GL2(𝔽103) generated by

4185
984
,
104
193
G:=sub<GL(2,GF(103))| [41,9,85,84],[10,1,4,93] >;

D102 in GAP, Magma, Sage, TeX

D_{102}
% in TeX

G:=Group("D102");
// GroupNames label

G:=SmallGroup(204,11);
// by ID

G=gap.SmallGroup(204,11);
# by ID

G:=PCGroup([4,-2,-2,-3,-17,98,3075]);
// Polycyclic

G:=Group<a,b|a^102=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D102 in TeX

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