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Cayley diagram

D109 #6342

order218
isoclinismisoclinic class of D109
automorphismAut(G) = AGL(1,𝔽109) (order 11772) · Inn(G) = G · Out(G) = C2×C27
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD109
unambiguous semidirect productC109⋊C2

Splittings

Fitting, socle, derived, SylowC109 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC109C2
LCSG ▷ C109
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 109
transitive degreeμt(G) = 109

A minimal faithful permutation representation — 2 generators on 109 points:

(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);(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)

Polycyclic presentation:

2,109|b^a=b^108

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#89137, C2×D109, D109⋊C3, …
maximal normal subgroup of#89137, C2×D109, D109⋊C3, …
maximal quotient ofDic109, C2×D109, C3×D109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(218, 1);