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G = D104order 208 = 24·13

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D104, C131D8, C81D13, C1041C2, D521C2, C2.4D52, C4.9D26, C26.2D4, C52.9C22, sometimes denoted D208 or Dih104 or Dih208, SmallGroup(208,7)

Series: Derived Chief Lower central Upper central

C1C52 — D104
C1C13C26C52D52 — D104
C13C26C52 — D104
C1C2C4C8

Generators and relations for D104
 G = < a,b | a104=b2=1, bab=a-1 >

52C2
52C2
26C22
26C22
4D13
4D13
13D4
13D4
2D26
2D26
13D8

Smallest permutation representation of D104
On 104 points
Generators in S104
(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 99 100 101 102 103 104)
(1 104)(2 103)(3 102)(4 101)(5 100)(6 99)(7 98)(8 97)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 90)(16 89)(17 88)(18 87)(19 86)(20 85)(21 84)(22 83)(23 82)(24 81)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)

G:=sub<Sym(104)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,104)(2,103)(3,102)(4,101)(5,100)(6,99)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,90)(16,89)(17,88)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,104),(2,103),(3,102),(4,101),(5,100),(6,99),(7,98),(8,97),(9,96),(10,95),(11,94),(12,93),(13,92),(14,91),(15,90),(16,89),(17,88),(18,87),(19,86),(20,85),(21,84),(22,83),(23,82),(24,81),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53)])

D104 is a maximal subgroup of
D208  C16⋊D13  C13⋊D16  C8.6D26  D1047C2  C8⋊D26  D8×D13  Q8⋊D26  D104⋊C2
D104 is a maximal quotient of
D208  C16⋊D13  Dic104  C1045C4  D525C4

55 conjugacy classes

class 1 2A2B2C 4 8A8B13A···13F26A···26F52A···52L104A···104X
order122248813···1326···2652···52104···104
size1152522222···22···22···22···2

55 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D4D8D13D26D52D104
kernelD104C104D52C26C13C8C4C2C1
# reps11212661224

Matrix representation of D104 in GL2(𝔽313) generated by

21309
196231
,
2853
5228
G:=sub<GL(2,GF(313))| [21,196,309,231],[285,52,3,28] >;

D104 in GAP, Magma, Sage, TeX

D_{104}
% in TeX

G:=Group("D104");
// GroupNames label

G:=SmallGroup(208,7);
// by ID

G=gap.SmallGroup(208,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,61,66,182,42,4804]);
// Polycyclic

G:=Group<a,b|a^104=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D104 in TeX

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