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G = D106order 212 = 22·53

Dihedral group

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

Aliases: D106, C2×D53, C106⋊C2, C53⋊C22, sometimes denoted D212 or Dih106 or Dih212, SmallGroup(212,4)

Series: Derived Chief Lower central Upper central

C1C53 — D106
C1C53D53 — D106
C53 — D106
C1C2

Generators and relations for D106
 G = < a,b | a106=b2=1, bab=a-1 >

53C2
53C2
53C22

Smallest permutation representation of D106
On 106 points
Generators in S106
(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 99 100 101 102 103 104 105 106)
(1 106)(2 105)(3 104)(4 103)(5 102)(6 101)(7 100)(8 99)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)

G:=sub<Sym(106)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106), (1,106)(2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106), (1,106)(2,105)(3,104)(4,103)(5,102)(6,101)(7,100)(8,99)(9,98)(10,97)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,90)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106)], [(1,106),(2,105),(3,104),(4,103),(5,102),(6,101),(7,100),(8,99),(9,98),(10,97),(11,96),(12,95),(13,94),(14,93),(15,92),(16,91),(17,90),(18,89),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,81),(27,80),(28,79),(29,78),(30,77),(31,76),(32,75),(33,74),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54)])

D106 is a maximal subgroup of   D212  C53⋊D4
D106 is a maximal quotient of   Dic106  D212  C53⋊D4

56 conjugacy classes

class 1 2A2B2C53A···53Z106A···106Z
order122253···53106···106
size1153532···22···2

56 irreducible representations

dim11122
type+++++
imageC1C2C2D53D106
kernelD106D53C106C2C1
# reps1212626

Matrix representation of D106 in GL2(𝔽107) generated by

634
5016
,
5621
8051
G:=sub<GL(2,GF(107))| [6,50,34,16],[56,80,21,51] >;

D106 in GAP, Magma, Sage, TeX

D_{106}
% in TeX

G:=Group("D106");
// GroupNames label

G:=SmallGroup(212,4);
// by ID

G=gap.SmallGroup(212,4);
# by ID

G:=PCGroup([3,-2,-2,-53,1874]);
// Polycyclic

G:=Group<a,b|a^106=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D106 in TeX

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