Finite Group Databasebeta

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Cayley diagram

D185 #67305

order370
isoclinismisoclinic class of D185
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD185
unambiguous semidirect productC5⋊D37
unambiguous semidirect productC37⋊D5

Splittings

Fitting, socle, derived, HallC5×C37 ⋊ C2
SylowC37 ⋊ D5
SylowC5 ⋊ D37

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C37C2
LCSG ▷ C5×C37
UCSC1

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

Polycyclic presentation:

2,5,37|b^a=b^4,c^a=c^36

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10625191, #10625192, D5×D37, …
maximal normal subgroup of#10625191, #10625192, D5×D37, …
maximal quotient ofDic185, C2×D185, C3×D185, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(370, 3);