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Cayley diagram

D111 #6366

order222
isoclinismisoclinic class of D111
automorphismAut(G) = D3×AGL(1,𝔽37) (order 7992) · Inn(G) = G · Out(G) = C4×C9
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD111
unambiguous semidirect productC3⋊D37
unambiguous semidirect productC37⋊D3

Splittings

Fitting, socle, derived, HallC3×C37 ⋊ C2
SylowC37 ⋊ D3
SylowC3 ⋊ D37

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C37C2
LCSG ▷ C3×C37
UCSC1

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 40
transitive degreeμt(G) = 111

A minimal faithful permutation representation — 2 generators on 40 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);(2,3)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)

Polycyclic presentation:

2,3,37|b^a=b^2,c^a=c^36

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D37, C2×D111, D333, …
maximal normal subgroup ofD3×D37, C2×D111, C3⋊(D37⋊C3), …
maximal quotient ofDic111, C2×D111, D333, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(222, 5);