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Cayley diagram

C5×C9×D5 #90649

order450
isoclinismisoclinic class of D5
automorphismAut(G) = C2×C3×C4×AGL(1,𝔽5) (order 480) · Inn(G) = D5 · Out(G) = C22×C3×C4
homologyAbelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1

Aliases

direct productC5×C9×D5
unambiguous semidirect productC5⋊(C2×C5×C9)

Splittings

Fitting, HallC52×C9 ⋊ C2
derivedC5 ⋊ C2×C5×C9
SylowC52 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC5×C9D5
commutator, hypocenterC5C2×C5×C9
FrattiniC3C3×C5×D5
FittingC52×C9C2
socleC3×C52C2×C3
prime radicalC3×C5×D5C3
cosocleC3×C5C2×C3×C5
LCSG ▷ C5
UCSC1 ◁ C5×C9

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,3,4,5)(7,8)(9,10)(11,12,14,13,15,17,16,18,19);(6,7,9,10,8)

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×C9×AGL(1,𝔽5), C9×#1042, C5×C9⋊AGL(1,𝔽5), …
maximal normal subgroup ofC5×C9×AGL(1,𝔽5), C9×#1042, C5×C9⋊AGL(1,𝔽5), …
maximal quotient ofC5×C9×Dic5, C2×C5×C9×D5, C5×D5×C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(450, 14);