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G = D110order 220 = 22·5·11

Dihedral group

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

Aliases: D110, C2×D55, C22⋊D5, C10⋊D11, C52D22, C112D10, C1101C2, C552C22, sometimes denoted D220 or Dih110 or Dih220, SmallGroup(220,14)

Series: Derived Chief Lower central Upper central

C1C55 — D110
C1C11C55D55 — D110
C55 — D110
C1C2

Generators and relations for D110
 G = < a,b | a110=b2=1, bab=a-1 >

55C2
55C2
55C22
11D5
11D5
5D11
5D11
11D10
5D22

Smallest permutation representation of D110
On 110 points
Generators in S110
(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 99 100 101 102 103 104 105 106 107 108 109 110)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 100)(12 99)(13 98)(14 97)(15 96)(16 95)(17 94)(18 93)(19 92)(20 91)(21 90)(22 89)(23 88)(24 87)(25 86)(26 85)(27 84)(28 83)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)

G:=sub<Sym(110)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,110)(2,109)(3,108)(4,107)(5,106)(6,105)(7,104)(8,103)(9,102)(10,101)(11,100)(12,99)(13,98)(14,97)(15,96)(16,95)(17,94)(18,93)(19,92)(20,91)(21,90)(22,89)(23,88)(24,87)(25,86)(26,85)(27,84)(28,83)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)], [(1,110),(2,109),(3,108),(4,107),(5,106),(6,105),(7,104),(8,103),(9,102),(10,101),(11,100),(12,99),(13,98),(14,97),(15,96),(16,95),(17,94),(18,93),(19,92),(20,91),(21,90),(22,89),(23,88),(24,87),(25,86),(26,85),(27,84),(28,83),(29,82),(30,81),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,70),(42,69),(43,68),(44,67),(45,66),(46,65),(47,64),(48,63),(49,62),(50,61),(51,60),(52,59),(53,58),(54,57),(55,56)])

D110 is a maximal subgroup of   D552C4  C5⋊D44  C11⋊D20  D220  C557D4  C2×D5×D11
D110 is a maximal quotient of   Dic110  D220  C557D4

58 conjugacy classes

class 1 2A2B2C5A5B10A10B11A···11E22A···22E55A···55T110A···110T
order122255101011···1122···2255···55110···110
size11555522222···22···22···22···2

58 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D5D10D11D22D55D110
kernelD110D55C110C22C11C10C5C2C1
# reps12122552020

Matrix representation of D110 in GL2(𝔽331) generated by

133250
81100
,
133250
59198
G:=sub<GL(2,GF(331))| [133,81,250,100],[133,59,250,198] >;

D110 in GAP, Magma, Sage, TeX

D_{110}
% in TeX

G:=Group("D110");
// GroupNames label

G:=SmallGroup(220,14);
// by ID

G=gap.SmallGroup(220,14);
# by ID

G:=PCGroup([4,-2,-2,-5,-11,194,3203]);
// Polycyclic

G:=Group<a,b|a^110=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D110 in TeX

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