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Cayley diagram

C3×D37 #6365

order222
isoclinismisoclinic class of D37
automorphismAut(G) = C2×AGL(1,𝔽37) (order 2664) · Inn(G) = D37 · Out(G) = C22×C9
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D37

Splittings

Fitting, socle, HallC3×C37 ⋊ C2
derived, SylowC37 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3D37
commutator, cosocle, hypocenterC37C2×C3
FrattiniC1G
Fitting, socleC3×C37C2
LCSG ▷ C37
UCSC1 ◁ C3

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)(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);(4,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8,6)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#3866, C3⋊#3866, D3×D37, …
maximal normal subgroup ofC3×#3866, C3⋊#3866, D3×D37, …
maximal quotient ofC3×Dic37, C2×C3×D37, C9×D37, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(222, 4);