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Cayley diagram

D37 #655

order74
isoclinismisoclinic class of D37
automorphismAut(G) = AGL(1,𝔽37) (order 1332) · Inn(G) = G · Out(G) = C2×C9
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD37
unambiguous semidirect productC37⋊C2

Splittings

Fitting, socle, derived, SylowC37 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC37C2
LCSG ▷ C37
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 37

A minimal faithful permutation representation — 2 generators on 37 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);(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)

Polycyclic presentation:

2,37|b^a=b^36

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#3866, C2×D37, D37⋊C3, …
maximal normal subgroup of#3866, C2×D37, D37⋊C3, …
maximal quotient ofDic37, C2×D37, C3×D37, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(74, 1);