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Cayley diagram

C5×D37 #67304

order370
isoclinismisoclinic class of D37
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D37
unambiguous semidirect productC37⋊(C2×C5)

Splittings

Fitting, socle, HallC5×C37 ⋊ C2
derived, SylowC37 ⋊ C2×C5

Subgroups

subgroupG/H
center, hypercenterC5D37
commutator, cosocle, hypocenterC37C2×C5
FrattiniC1G
Fitting, socleC5×C37C2
LCSG ▷ C37
UCSC1 ◁ C5

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 42
transitive degreeμt(G) = 185

A minimal faithful permutation representation — 2 generators on 42 points:

(1,2,3,4,5)(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);(6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,42,40,38,36,34,32,30,28,26,24,22,20,18,16,14,12,10,8)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#3866, #10625194, D5×D37, …
maximal normal subgroup ofC5×#3866, #10625194, D5×D37, …
maximal quotient ofC5×Dic37, C2×C5×D37, C3×C5×D37, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(370, 2);